Packed spheres, pore volume

Vt total pore volume of regular packing spheres (as volume of N2 at STP, cm /g). 

The model can be made somewhat more precise by considering the silica skeleton as a loosely packed system of spheres of equal radii that is derived from a close packing by leaving out a fraction 1 — 0 of the possible sites. If the sphere radius is Rs (A.), it can easily be shown, assuming complete accessibility of all surfaces, that the pore volume F/ is... 

The overall pore volume of the bed of packed spheres is mainly determined by the. size of the inner cavities, which is in turn dependent on the particle radius, R, and the coordination number, N. The effective Kelvin radius corresponding to this major stage of capillary condensation is approximately given by the radius, r, of the inscribed sphere within the cavity (Karnaukhov 1971). 

The surface porosity is equal to the ratio of the pore area to membrane area multiplied by the number of pores. In most cases volume flux through ceramic membranes can be best described by the Kozeny-Carman relationship, which corresponds to a system of close packed spheres (see Figure 6.8a) ... 

The manufacture of molded articles is usually carried out with mixtures of aluminum oxides with different particle size distributions. This is particularly important when pore-free end-products are required, because this enables a higher volume concentration of aluminum oxide to be obtained than the 74% of ideally cubic close packed spheres by filling the gaps with smaller particles. The particle size distributions used in practice are usually determined using empirically determined approximate formulae (Andreasen or Fuller distribution curves) which take into account the morphology of the individual particles. 

Figure 5. Hysteresis regions of nitrogen adsorption and desorption isotherms calculated for regular packings of spheres with uniform radii of 500 X. Total pore volume, Vj, given for each packing.

Theoretical isotherms calculated for regularly packed spheres look encouraging. In spite of simplifying assumptions, we obtained a qualitative agreement with experimental isotherms, mainly narrow sorption hysteresis in the regions close to p/Po - 1- Total pore volumes depend strongly on the choice of n (coordination number). More detailed analysis, allowing for polydisperslty of both r and n, is required for quantitative interpretation of Isotherms. 

A typical pore in a dense silica gel is shown in Figure 5.3a. Here the silica spheres are in cubic packing and the pore volume is 48% of the total volume. Since the 52% by volume of silica has a density of 2.2 g cm, the specific porosity is 0.42 cm g . 

It is noted that it is easy to compress particles to a state of approximating cubic packing where the body has about 50% by volume of pores. However, at 100 tons in.- the porosity drops to 0.204, which is even less than that of perfect close, which is 0.255. Under this pressure some of the particles may have become flattened together in which case the formulas are not applicable. 

The model developed by Conner et al. (ref. 7) that is based upon packing of spheres is of interest. We have found that a model based upon a simple cubic packing of nonporous spheres returns the best agreement among surface area, pore volume and average pore size data for nitrogen adsorption/desorption measurements with numerous metal oxide samples (ref. 9). It is of special interest to compare the data obtained from the two techniques with common materials to learn whether a model based on packings of spheres applies equally well. 

For beds composed of spheres of mixed sizes the porosity of the packing can change very rapidly if the smaller spheres are able to fill the pores between the larger ones. Thus Coulson 15) found that, with a mixture of spheres of size ratio 2 1, a bed behaves much in accordance with equation 4.19 but, if the size ratio is 5 1 and the smaller particles form less than 30 per cent by volume of the larger ones, then K" falls very rapidly, emphasising that only for uniform sized particles can bed behaviour be predicted with confidence. 

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. 

The new mesoporous materials have extremely high surface-to-volume ratios. An example of these materials is MCM41, which was invented by DuPont. A simple structure that can be manufactured in the laboratory is illustrated in Eigure 15.14. The structure initially contained a periodic array of polymer spheres. When close packed, these spheres leave 26% of the volume empty. We can then infiltrate a liquid into these pores, burn out the spheres, and convert the liquid to a polycrystalline ceramic. Another synthesized porous ceramic is the cordierite honeycomb structure used to support the Pt catalyst in automobile catalytic converters. In this case the cylindrical pores are introduced mechanically in the extrusion process. 

The pores are thus defined by the coordinates of the centers of the spheres. The properties of the pores were evaluated by computer. According to Mason, the random packing of spheres gives a volume density of 0.63 which is about halfway between 0.52 for open packing (cubic) and 0.72 for close packing hexagonal). 

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