Other particle shapes

The relationship between effectiveness factor r) and Thiele modulus may be calculated for several other regular shapes of particles, where again the chjuracteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that 

The results of investigations with particles of a variety of shapes, mainly irregular ones, have been reportM by RESlER and and the results are shown as data points in 

Estimate the Thiele modulus and the effectiveness factor for a reactor in which the catalyst particles are  

The first-order rate constant is 5 x 10 and the effective diffusivity of the reactants in the pores of the 

A first-cffder chenaical reaction takes place in a reactor in which the catalyst pellets are platelets of tMekness 5 mm. The effective diffusivity Dg for the reactants in the catalyst particle is 10 - m /s and the first-order rate constant k is 14.4 s .  

Thus t]X s 1, corresponding to the region where mass transfer effects dominate. The concentration profile is given by equation 10.198 as  

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Hardness. The resistance of a fabricated mbber article to indentation, ie, hardness, is influenced by the amount and shape of its fillers. High loadings increase hardness. Fillers in the form of platelets or flakes, such as clays or mica, impart greater hardness to elastomers than other particle shapes at equivalent loadings. 

Size. The precise determination of particle size, usually referred to as the particle diameter, can actually be made only for spherical particles. For any other particle shape, a precise determination is practically impossible and particle size represents an approximation only, based on an agreement between producer and consumer with respect to the testing methods (see Size measurement of particles). 

In order to take account of the effect of temperature and to determine the heat transfer coefficient at realistic freezing temperatures, rather than the room temperature of the reported experiments, and to extend the model to other particle shapes and sizes, Vazquez and Calvelo (1980) then plotted their data according to the model proposed by Chang and Wen (1966)... 

Most emulsion polymers are spheres, generally the lowest-energy and therefore most stable configuration. However, there are other particle shapes and morphologies which can be obtained during emulsion poly-... 

Interconnected cylinders (L2 ) are formed in two water content range (5.5 < vv < 11 and 30 < vv < 35). Syntheses in these two regions of the phase diagram show very strong correlation and similar data. Spherical and cylindrical particles are formed in both cases. No other particle shapes... 

Both the macro and micro population balances just derived conserve the number of particles. In some cases, it is appropriate to perform balances where the particles length, area, or volume (or mass) is conserved. For example, length conservation is critical in grinding fibers and volume conservation is critical in grinding other particle shapes. Such conservation equations can also be developed under the umbrella of a population balance, but this population balance must be different than those previously derived, where particle number is conserved. The way to make them different is to couple to the population balance an appropriate conservation equation. The population based on length, area, md volume (or mass) can be derived from the population based on number as shown in Table 3.1. Let us illustrate this idea of property conservation with an example showing conservation of length. 

Three conductive-mix detonators are shown in Figure 4. Since lead azide is an electrical insulator, a conductant is added, and flake conductants have been observed to be more effective than other particle shapes. A typical mixture contains 95% lead azide and 5% flake graphite. This type of detonator fires rapidly with low energy input for example, the E.I. duPont de Nemours Company s product, designated X811, fires in 4 msec when initiated by a 2.2 uF... 

Here, (qA)r is the concentration of A in the catalyst at the surface, and Vca, and r0 are the volume and radius of the catalyst particle, respectively. Other reaction orders or one-plus rate equations, other particle shapes, and reversible reactions give more complex equations [18-25], but the behavior is qualitatively the same. 

The scattering-absorption of incident beams by a long circular cylinder has also been studied by van de Hulst [50], He also considers other particle shapes. Wang and Tien [62], Tong and Tien [63], and Tong et al. [64] consider fibers used in insulations. They use the efficiencies derived by van de Hulst [50] and examine the effects of ks and d on the overall performance of the insulations. The effect of fiber orientation on the scattering-phase function of the medium is discussed by Lee [65]. The effective radiative properties of a fiber-sphere composite is predicted by Lee et al. [66]. 

Flake aluminum is made by milling other particle shapes. The impacts during milling will cause the flakes to stick together, unless a lubricant is added. 

The form of the response curves is shown in Fig. 1. The corresponding expressions for other particle shapes are easily derived but there is little numerical difference from the response for a spherical particle of the same external area to volume ratio (i.e., based on an equivalent radius). 

Exact, analytical expressions for the scattering of very small ellipsoids were already derived by Lord Rayleigh (1897). Other particle shapes may require numerical solutions for Rayleigh scattering. Note that the calculation of the orientation averaged scattering behaviour benefits from symmetry relations (Bohren and Huffman 1983, pp. 154-157). 

Below is shown the derivation using spherical geometry based on [113]. The spherical geometry is seldom used in steam reforming, but conversion to other particle shapes can be carried out using the equivalent particle diameter as described by Aris [24]. 

In a polydisperse sample, the gyration radius will be always larger than the number-averaged radius. Gyration radii have been tabulated for various other particle shapes. For example, for a cylinder one has Rg = y/R 12 -F L I12, where R is the cylinder radius, and L its length. 

The first experimental proof of virtual linearity of response, up to some 80% of the aperture diameter, came from Barfield and Knight and Barfield, Wharton and Lines.27 They used spherical particles of polymer latex and the COULTER COUNTER model ZM. The experiment involved the use of different sizes of "mono-sized" latex particles measured by a range of different apertures, and therefore required no assumed or measured "real" sizes for the particles. The experiment has not been repeated for other particle shapes as no other series of suitable model particles appear to exist, so the linearity of response for other particle shapes has not yet been verified experimentally. It is reasonable to assume however, from all existing theory that no significant extra alinearity will exist for non-spherical shapes. 

All particles are spheres (other particle shapes might also be considered). 

Expressions for the effectiveness factor, similar to Eqn. (9-9), can be derived for other particle shapes, other reaction orders, and for reversible as well as irreversible reactions. Fortunately, if the Thiele modulus is redefined somewhat, all of these solutions can be approximated by a single curve of rj versus 0. 

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