Packed uniform spheres

Diatomaceous Silica Filter aids of diatomaceous silica have a dry bulk density of 128 to 320 kg/m (8 to 20 Ib/fU), contain paiiicies mostly smaller than 50 [Lm, and produce a cake with porosity in the range of 0.9 (volume of voids/total filter-cake volume). The high porosity (compared with a porosity of 0.38 for randomly packed uniform spheres and 0.2 to 0.3 for a typical filter cake) is indicative of its filter-aid ability Different methods of processing the crude diatomite result in a series of filter aids having a wide range of permeability. 

The holes that exist among closest packed uniform spheres, (a) The trigonal hole formed by three spheres in a given plane, (b) The tetrahedral hole formed when a sphere occupies a dimple in an adjacent layer, (c) The octahedral hole formed by six spheres in two adjacent layers. 

Aristov et al. (17) developed a method for calculating sorption/ desorption isotherms for beds of regularly-packed uniform spheres. 

The value of S is commonly taken to be in the range 1.2 to 1.3, the theoretical value for close-packed uniform spheres being 1.35 (Nielsen, 1967a). ... 

The polydispersity of particle size increases 0m. for example, of randomly packed uniform spheres from 0.62 to about 0.9. Thus, while for small and large monodispersed spheres 0m = 0.62, for the mixtures, depending on composition and size ratio, 0m 1-0. For polydispersed spheres of diameter di and average diameter d, Pishvaei et al. [49] used the model of Ouchiyama and Tanaka [50-55] to calculate the maximum packing volume fraction, 0m, and the average number of spherical particles, n ... 

The characteristics of different modes of packing uniform spheres have also been calculated by Avery and Ramsay (129), and values are given in Table 5.1. Although in actual gels no such regularity is likely, nevertheless from the porosity of a gel one can approximate the coordination number, n. The density of the gel permits an estimate of the pore diameter if the average particle diameter is known. This presupposes that the ultimate particles in a gel are all of uniform size and this is generally true unless special means are taken to have two particle sizes present. 

Recall Kepler s conjecture on the density of packing uniform spheres outlined in the opening section of this book. The proof by Hales [58] was based on an exhaustive check by a computer of a very large, but finite, number of different arranganents. For a finite number of possibilities, there is little conceptual distinction between an exhaustive check of all possibilities and use of the trial-and-error approach to a check of all possibilities. Both could be very inefficient, in contrast to the exact solutions, which do not involve searches, which make them fairly efficient. 

As the upward velocity of flow of fluid through a packed bed of uniform spheres is increased, the point of incipient fluidisation is reached when the particles are just supported in the fluid. The corresponding value of the minimum fluidising velocity (umf) is then obtained by substituting emf into equation 6.3 to give ... 

A.l 1 Calculate the average density of a single carbon atom by assuming that it is a uniform sphere of radius 77 pm and that the mass of a carbon atom is 2.0 X 10 23 g. The volume of a sphere is f-nr , where r is its radius. Express the density of carbon in grams per cubic centimeter. The density of diamond, a crystalline form of carbon, is 3.5 g-cm 3. What does your answer suggest about the way the atoms are packed together in diamond ... 

In these Equations, G is the modulus of the syntactic foam, G0 is the modulus of the polymer matrix, v0 is Poisson s ratio of the polymer matrix, and 9 is the maximum packing fraction of the filler phase. For uniform spheres, 9 0.64 (see Sect. 3.6). The volume fraction of spheres in the syntactic foam is 9sph. The slope of the G/G0 vs. 9sph curve depends strongly upon whether or not G/G0 is greater or less than 1.0. The slope is negative if the apparent modulus of the hollow spheres is less than the modulus of the polymer matrix. 

The above requirements are to some extent contradictory, which has led to the proposition of a large number of different catalyst shapes and arrangements. However, only a few of these have proved really effective in practical operation. Suitable catalyst forms and arrangements include random packings of spheres, solid cylinders, and hollow cylinders, as well as uniformly structured catalyst packings in the form of monoliths with parallel channels, parallel stacked plates, and crossed, corrugated-plate packets (Fig. 3). 

A metallic crystal can be pictured as containing spherical atoms packed together and bonded to each other equally in all directions. We can model such a structure by packing uniform, hard spheres in a manner that most efficiently uses the available space. Such an arrangement is called closest packing (see Fig. 16.13). The spheres are packed in layers in which each sphere is surrounded by six others. In the second layer the spheres do not lie direotlv over those in the first layer. Instead, each one occupies an indentation (or dimple) formed by three spheres in the first layer. In the third layer the spheres can occupy the dimples of the second layer in two possible ways. They can occupy positions so that each sphere in the third layer lies directly over a sphere in the first layer (the aba arrangement), or they can occupy positions... 

The closest packing arrangement of uniform spheres. In each layer a given sphere is surrounded by six others, (a) aba packing The second layer is like the first, but it is displaced so that each sphere in the second layer occupies a dimple in the first layer. The spheres in the third layer occupy dimples in the second layer so that the spheres in the third layer lie directly over those in the first layer (aba), (b) abc packing The spheres in the third layer occupy dimples in the second layer so that no spheres in the third layer lie above any in the first layer (abc). The fourth layer is like the first. 

Closest packing describes the most efficient method for arranging uniform spheres. We can calculate the fraction of the space actually occupied by the spheres (/ ),... 

Fig. 6. Computed lateral voidage profile for a three-dimensional column randomly packed with uniform spheres. After Zimmerman and Ng 4).

The manner in which mercury penetrates a bed of uniform spherical particles was examined in detail by Mayer and Stowe [2] who postulated that the breakthrough pressure Pb required to force mercury to penetrate the void spaces between packed uniform non porous spheres of diameter D is given by equation (3) ... 

For particles of a defined shape and size distribution there exists a volume fraction (0) corresponding to a state of close packing. For example, for uniform spheres in a condition of hexagonal close packing, 0 = 0.74. If the packing fraction is i(i0, the voidage between the particles represents a volume fraction of 1- Q, and the bulk volume factor is 1/ 0. 

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.

The simplest static property is the mean coordination number, , which is loosely defined as the average number of pore throats per pore body (Dullien, 1991). For spatially periodic systems, such as cubic packing of uniform spheres, it is relatively easy to determine . Methods of estimating for more heterogeneous porous media are reviewed by Sahimi (1995). 

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