Sphericity factor

The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, u, of spherical particles by a correction factor, iji, called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume ... 

An alternative description of nonspherical particles is often represented by the sphericity factor (tfi), which is the number that, when multiplied by the diameter of a sphere with the same volume as the particle (ds), gives the particle effective diameter (dp) ... 

In Fig. 4.7, the peak potential and the half-wave potential corresponding to a spherical electrode are plotted versus the sphericity factor Rq = rs j /DqT2 using three values of y (5, 1, and 0.2). As expected, the and ETfpfe values... 

Kolmogoroff microscale of turbulence (m) Particle sphericity factor, i.e., ratio of surface area of a sphere of same volume as the particle to surface area of particle (-)... 

We present some values of the sphericity factor r/> for some particles sphere, 1.000 octahedron, 0.846 cube, 0.806 tetrahedron, 0.670. 

Equation (9.4-13) may be integrated numerically and summed over all matrix-block sizes. Braun el a l, applind this model to a 5.8 X lO6 g sample of primary copper are with matrix-block sizes between 0.01 and 16 cm. The effect of weathering was accounted for by a systematic correction of the sphericity factor. 

The foregoing is all for beds of uniformly sized spherical particles. For other shapes of uniform particles, some efforts have been made to relate the results to those shown here through defining an empirical sphericity factor [2, p. 45 et seq]. However, results have not been successful enough to allow one to caleulate the behavior of such porous media accurately without experimental test, because even for a completely uniform set of nonspherical particles the porosity is a strong function of how the bed is assembled, and if the particles are loose, the porosity can be significantly altered by simply shaking the bed, etc. 

Little is known about the influence of particle shape on the minimum fluidising velocity and bed expansion of liquid fluidised beds even for Newtonian liquids [Couderc, 1985 Flemmer et al., 1993]. The available scant data suggests that, if the diameter of a sphere of equal volume is used together with its sphericity factor, satisfactory predictions of the minimum fluidising velocity are obtainable from the expressions for spherical particles. Only one... 

The relationship between adhesive force and sphericity factor, as determined by direct detachment of loess particles with a double-mean radius of 100-160 ixm is illustrated by the following data ... 

As the sphericity factor increases from 0.4 to 0.9, the adhesive force drops off because of the decrease in the actual contact area for irregularly shaped particles. We also find a greater scatter of values of adhesive force with irregular particles than with spherical particles. For example, in the case of particles with a double-mean radius of 180 juni, the force of adhesion varies over a range of 2.8- 10" to 1.4- 10" dyn. 

The concept of the sphericity factor still cannot be used fully in evaluating the specific features of adhesion for irregularly shaped particles since it does not really account for the relationship among the height, length, and width of the particle. This relationship is taken into account more fully in a concept that we have considered previously [162], that of equivalent size of particles, as determined by means of Eqs. (III. 14)-(III. 16). 

The resistance (drag) coefficient Cx is a function of the Reynolds number, i.e., Cx = /(Re). In order to compare values of Cx for spherical and irregular particles, we introduce the sphericity factor k to account for the particle shape (see p. 168). For particles of various shapes with diameters from 1 to 100 fxm in water, with Reynolds numbers below 0.2, the sphericity factor has the following values ... 

Ergun then used published data to confirm the validity of this equation. Note that Eq. (4.100) is for spheres. For structured solids other than spheres, we modify the equation by including a sphericity factor P, which is defined as... 

P Subramanian, Vr Arunachalam. A simple device for the determination of sphericity factor. Ind Eng Chem Fundam 19 436 37, 1980. 

The shape of the particle can be given by the traditional sphericity factor, fractal analysis, or by Fourier transform representations. The latter is a bit involved, requiring several coefficients for complex definition. Fractal analysis is receiving more attention of late in representing the shape of the particles handled. Chapter 1, Particle Characterization and Dynamics, presents a more complete description of shape analysis. 

Introducing a sphericity factor similar to that of Feldberg (1980), in this case being the ratio of the diffusion layer to drop radius at time... 

Figure III.15 shows the adhesive forces measured by the method of direct detachment for loess particles as a function of the sphericity factor. As the sphericity factor rises from 0,4 to 0,9, the adhesive force diminishes as a result of the reduction in the actual contact area of regularly shaped particles. For particles of irregular shape there is a greater spread of adhesive-force values than for spherical particles. Thus, for particles with a double mean radius of 180 M, the adhesive force varies between 2.8 10 and 1,4 10 dyn.

As often mentioned before, the shape of the particle is of importance in adhesion as well as other phenomena of particle behavior. Flat platelets would have more surface contact than spherical or jagged particles. The sphericity given previously is one way of classifying this shape term. Another method is the procedure of Heywood (1963). Figure 2-5 shows the adhesion force for a particular system with varying sphericity factors. 

Figure 2-5 Adhesion force of particles with a mean radius of approximately 130 iim as a function of the sphericity factor (Zimon, 1969). Reproduced with permission of Plenum Publishing Corp.

For particles of regular geometric shape, values of and ip can be calculated readily from the equivalent sphere definition and by Eq. (6.20), as listed in Table 6.2 for a number of geometric solids. However, the catalyst most commonly used in ammonia synthesis is a crushed and classified material which consists of a mixture of irregular particles having linear dimensions defined by the sieves used in the classification. The surface area of individual catalyst particles is very irregular and complex, and does not permit direct measurement or calculation. Consequently, a nominal sphericity factor of 0.65, as reported in the literature for... 

Hamilton model [16] based on Maxwell one, take into account the particle sphericity factor y which is related to n factor (n = 3/i/r). The model is given by Eq 5. 

In Sects. 5.3.2 and 5.3.3.1, it was shown how the value of D may be determined from the Ml plateau after semiintegration/convolution of the experimental I-E data when the parameters Tq, n, and c are known. As highlighted in Sect. 5.3.1, convolution can also be employed to determine the constants D and nc simultaneously from two I-E datasets obtained under conditions with differing sphericity factors (a) ... 

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