Pore structure, bidispersed

In a particle having a bidispersed pore structure comprising spherical adsorptive subparticles of radius forming a macroporous aggregate, separate flux equations can be written for the macroporous network in terms of Eq. (16-64) and for the subparticles themselves in terms of Eq. (16-70) if solid diffusion occurs. 

For particles with a bidispersed pore structure, the mass-transfer parameter in the LDF approximation (column 2 in Table 16-12) can be approximated by the series-combination of resistances... 

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. 

The method can be applied to investigate the bidisperse pore structures, which consist of small microporous particles formed into macroporous pellets with a clay binder. In such a structure there are three distinct resistances to mass transfer, associated with diffusion through the external fluid film, the pellet macropores, and the micropores. Haynes and Sarma [24] developed a suitable mathematical model for such a system. 

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. 

Two porous catalysts in the form of cylindrical pellets were used industrial hydrogenation catalyst Cherox 42-00 with monodisperse pore structure (Chemopetrol Litvinov, Czech Rep. height x diameter = 4.9 x 5.0 mm) and laboratory prepared a-alumina, A5 (based on boehmite from Rural SB, Condea Chemie, Germany) with bidisperse pore structure (height x diameter = 3.45 x 3.45 mm). 

Crystals of microporous materials must be formed into pellets of siutable dimensions, porosity and mechanical strength, or be formed into a membrane on the surface of support materials when used in practice. Such composite pellets or membranes offer a bidispersed porous structure, with macro-or mesopores between the crystals and micropores permeating the crystals. The overall rate of the transport in such systems depends on the interplay of various processes occurring within the pellets or membranes. Jordi and Do [24,46] have developed a general theoretical model and seven relevant degenerate models to analyse the frequency response spectra of a system containing bidispersed pore structure materials for slab, cylindrical and spherical macro- and micropore geometry. Sun et al. [47] also reported the theoretical models of the FR for non-isothermal adsorption in biporous sorbents. 

Park, I.S., Kwak C., and Hwang Y.G., Frequency response of adsorption of a gas onto bidisperse pore-structured solid with modulation of inlet molar flow rate, Kor. J. Chem. Eng., 18, 330-335, 2001. 

The porous structure of most catalysts is polydisperse. Therefore, capillary condensate fills only part of the pore-space — mostly small pores. The bidisperse globular structure (Figure 23.1) is convenient to consider as a model for rough estimations of the influence of external mass transfer and intraparticle diffusion on the total reaction rate. Such an analysis was made by Ostrovskii and Bukhavtsova [8]. According to this model, the only pores inside globules (micropores) will fill with liquid, and space between globules (macropores) fill with gas. Then the total porosity can be written as... 

Model assumptions include the following (i) The adsorbent has an uniform bidisperse pore structure, (ii) The pellets have spherical geometry, (iii) The reactor behaves like a CSTR, (iv) Ideal pulse input, (v) Macropore diffusion is Fickian, (vi) No external film resistance, (vii) Linear equilibrium, (viii) First order irreversible reaction, (ix) The crystals are small (<0.2 pm) agglomerates and diffusion resistance in these can be neglected. The following differential mass balances for species i result ... 

Adsorbents can have bidisperse pore structures when they are produced by combining primary particles which themselves are porous (Fig 6.5). The resulting particle has two pore systems micropores within the small particles and macropores corresponding to the space between primary particles Generally, inadequate diffusivities may result if diffusion data in bidisperse adsorbents are not analyzed by modek which account for both micro- and macropores. 

For a set of six porous materials with a range of mean pore radii fixim 50 to 3000 nm, and mono- or bidisperse pore structure, transport characteristics and textural properties were compared. 

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... 

An alternative model for diffusion in porous media has been proposed by Wakao and Smith [39], who noted that many solids of interest have a bidisperse pore structure. That is, the solids consist of compacts of solid particles that are themselves porous. The solid therefore contains micropores (pores within the particles) and macropores (interstices between particles) and diffusion occurs through macropores, through micropores, and through micropores and macropores in series. For diffusion at constant pressure, we have... 

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. 

In the above considerations the total pressure should also be taken into account. With increasing pressure, the efficiency of a bidisperse catalyst decreases because diffusion in micropores turns from Knudsen to ordinary and the difference between Df n and Dejt disappears. At a pressure of 1-10 MPa, a uniform porous structure with a pore size close to the mean free path is the most favorable [6]. 

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]. 

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. 

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