Volume equivalent sphere diameters

The volume equivalent sphere diameter or equivalent volume sphere diameter is a commonly used equivalent sphere diameter. We will see later in the chapter that it is used in the Coulter counter size measurements technique. By definition, the equivalent volume sphere diameter is the diameter of a sphere having the same volume as the particle. The surface-volume diameter is the one measured when we use permeametry (see Section 1.8.4) to measure size. The surface-volume (equivalent sphere) diameter is the diameter of a sphere having the same surface to volume ratio as the particle. In practice it is important to use the method of... 

Calculate the equivalent volume sphere diameter %v and the surface-volume equivalent sphere diameter Xsv of a cuboid particle of side length 1, 2, 4 mm. 

The surface-volume equivalent sphere diameter of the cuboid, Xsy = 1.714 mm... 

In the lower part the differences are represented between the apparent distribution and that of the volume-equivalent sphere diameter. It becomes evident that in general the distribution of x does not lead to a sufficiently accurate estimation of xhe distribution of x. ... 

For nonspherical particles, values for the slip correction factor are available in slip flow (MU) and free-molecule flow (Dl). To cover the whole range of Kn and arbitrary body shapes, it is common practice to apply Eq. (10-58) for nonspherical particles. The familiar problem then arises of selecting a dimension to characterize the particle. Some workers [e.g. (H2, P14)] have used the diameter of the volume-equivalent sphere this procedure may give reasonable estimates for particles only slightly removed from spherical, or in near-con-tinuum flow, but gives the wrong limit at high Kn. An alternative approach... 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

Ap is the difference in material density between the liquid and gas phases. This situation is typically handled by describing the bubbles with a single internal coordinate (i.e. the equivalent-sphere diameter) and by introducing an aspect ratio, defined as the ratio between the minor and the major axes of the bubble. This aspect ratio E can be calculated by using the empirical equation proposed by Moore (1965) as a function of the Morton number E = 1/(1 + 0.043RCp Mo ). An alternative to this is the use of the correlation proposed by Wellek et al. (1966) for liquid-liquid droplets E = 1/(1 + 0.1613Eo° ), which is valid for Eo < 40 and Mo < 10 , whereas for Eo > 40 and RCp > 1.2 fluid particles are typically of spherical shape. Once the characteristic E value is known, the ratio of the real area of the bubble Ap and the area Aeq of a sphere with an equivalent volume can be calculated as follows ... 

An irregular particle can be described by a number of sizes. There are three groups of definitions the equivalent sphere diameters, the equivalent circle diameters and the statistical diameters. In the first group are the diameters of a sphere which would have the same property as the particle itself (e.g. the same volume, the same settling velocity, etc.) in the second group are the diameters of a circle that would have the same property as the projected outline of the particles (e.g. projected area or perimeter). The third group of sizes are obtained when a linear dimension is measured (usually by microscopy) parallel to a fixed direction. 

The shape of particles can have important effects e.g. on the specific surface. The concept of the equivalent here may be extended by shape ctors to take account of the real sur ce or volume riien compared with that of the equivalent sphere. Sur ce area is proportional to yp, i.e. sur ce area =fiP (for a sphere ) and volume is proportional to x, i.e., volume = kP, (for a sphere tix /6). Hence/ is a sur ce ctor or coefficient and is a volume fector or coefficient. Tlie coefficients / and k are fimctions of the geometrical shape and the relative proportions of the particle their values depend on the equivalent sphere diameter used. Sphericity is defined as ... 

The value of the shape coefficients can be calculated for various equivalent sphere diameter bases. Let subscript a = projected area diameter v = volume diameter s -surface area diameter St = Stokes diameter m = mesh size. The volume of particles may be expressed as kjc/= k,xj = Aye/ = fexs/ = krfcj. Hence K = k/Xt/x f and so on. 

In both cases we need the volume equivalent particle diameter, dp, which is the diameter of a sphere having the same volume as the actual particle (see Equation 3.29). 

Define the following equivalent sphere diameters equivalent volume diameter, equivalent surface diameter, equivalent surface-volume diameter. Determine the values of each one for a cuboid of dimensions 2 mm x 3 mm x 6 mm. 

The most appropriate particle size to use in equations relating to fluid-particle interactions is a hydrodynamic diameter, i.e. an equivalent sphere diameter derived from a measurement technique involving hydrodynamic interaction between the particle and fluid. In practice, however, in most industrial applications sizing is done using sieving and correlations use either sieve diameter, Xp or volume diameter, Xy, For spherical or near spherical particles Xv is equal to Xp. For angular particles, Xy l.lSXp. 

Fig. 11 Comparison of the sphericity of agglomerates as a function of the normalised diameter of the volume equivalent sphere with a the particle Stokes number St and b the particle Reynolds number Rep as a parameter considering both mono-disperse (open symbols) and poly-disperse (half-filled symbols) size distributions of the primary particles...

Fig. 14 Porosity distribution of agglomerates resulting from simulations with different a particle Stokes numbers St and b particle Reynolds numbers Rep plotted against the normalised diameter of the volume equivalent sphere in dependence on the initial particle size distribution (open- and half-filled symbols)...

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

Before concluding this discussion of the excluded volume, it is desirable to introduce the concept of an equivalent impenetrable sphere having a size chosen to give an excluded volume equal to that of the actual polymer molecule. Two such hard spheres can be brought no closer together than the distance at which their centers are separated by the sphere diameter de. At all greater distances the interaction is considered to be zero. Hence / = for a dey and fa = 0 for a[Pg.529]

The size of a spherical particle is readily expressed in terms of its diameter. With asymmetrical particles, an equivalent spherical diameter is used to relate the size of the particle to the diameter of a perfect sphere having the same surface area (surface diameter, ds), the same volume (volume diameter, dv), or the same observed area in its most stable plane (projected diameter, dp) [46], The size may also be expressed using the Stokes diameter, dst, which describes an equivalent sphere undergoing sedimentation at the same rate as the sample particle. Obviously, the type of diameter reflects the method and equipment employed in determining the particle size. Since any collection of particles is usually polydisperse (as opposed to a monodisperse sample in which particles are fairly uniform in size), it is necessary to know not only the mean size of the particles, but also the particle size distribution. 

---