Diameter surface/volume equivalent

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... 

Fig. 1. (a) Particle and examples of equivalent spherical diameters (b) volume, (c) surface, and (d) settling. 

Another method of describing particle size is in terms of equivalent diameter or the equivalent sphere dpe, which is the diameter of a sphere possessing the same ratio of surface to volume as the actual particle. Thus, from the equation, Vp/Sp = dp/6/ the equivalent diameter (dp e) is... 

To explore this further, we present some additional data about the 400 spheres in Table 1.5, namely, that the sample possesses a total surface area of 5.85 102 m2 or 5.85 102/ 400 = 1.46 fj.m2 per particle. Likewise, the total volume of the 400 spheres is 76 m or 0.19 m3 per particle. We see in Chapter 9 how the surface areas of actual powdered samples are measured and the volume is readily available from mass when the density of the bulk material is known. Now let us calculate the average diameter of an equivalent sphere from these data. 

Figure 1.2. Schematic illustration of multidimensions of a particle and its equivalent volume diameter, surface diameter, and sieve diameter.

The diameter of microcarrier beads is usually about 200/im so the surface area of 1 g of beads is 0.6 m2 which is equivalent to 7 roller bottles, 27 Roux flasks or 315 5-cm Petri dishes (see Table 3.1). With 1 g of beads in 500 ml medium in a suspension flask the surface volume ratio is 12 compared with 4-6 for traditional vessels which means that medium replacement must take place two to three times as often. 

EQUIVALENT SPHERICAL DIAMETER PARTICLE VOLUME/EXTERNAL SURFACE 0.2857 cm... 

FIGURE 4 (A) An irregularly shaped particle with an infinite number of diameters. (B) The equivalent volume, surface, and projected area diameter of (A). 

Example 1. Calculate the equivalent projected area diameter surface area diameter and volume diameter of a particle with a projected area of 30.00 pm, perimeter of 24.58 pm and thickness of 0.40 pm as shown in Figure 4A. 

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... 

SPHERICITY is the ratio of the surface area of a sphere having the same volume as the particle, to the actual particle surface area the reciprocal is known as the coefficient of rugosity or angularity. It can be shown that sphericity is also equal to the ratio of the surface-volume diameter to the equivalent volume diameter this makes sphericity a useful conversion factor between... 

The specific surface determined by any of the above methods can be converted into an equivalent mean spherical diameter xsv using the following simple equation xsv = 6/Sv, where Sv is volume specific surface. The equivalent mean diameter xsv is the size of a spherical particle which, if the powder consisted of only such particles, would have specific surface area the same as the actual sample. 

To determine the equivalent channel diameter the surface area for n parallel channels of length L is set equal to the surface-volume ratio times the particle volume, S L (1 — e), where is the cross-sectional area of the bed ... 

The specific surface or surface per unit volume, aPy of a porous medium is defined as the ratio of the total open pore surface area to the volume of the solids. The equivalent spherical diameter, dsy is the diameter of an equivalent sphere that has the same surface area per unit volume of the solid material forming the porous medium. 

Particle suspensions must be fairly dilute to avoid problems of coincident passage of several particles through the aperture. The volume sampled is set by electronic probes in the mercury column, which start and stop switches as the mercury passes. The instrument used for this work had settings for sample volumes of 50 /xL, 500 /xL, and 2,000 /xL, and was operated with four apertures, 30 /xm, 70 /xm, 140 /xm, and 280 /xm. The Coulter Counter was calibrated with polystyrene latex (PSL) spheres. A computer program was used to convert channel counts and calibration information to particle diameters, surface areas, and volumes after editing spurious data from the paper tapes. Particles were assumed to be solid spheres, since output from the Coulter Counter for sludge is that for spheres of volume equivalent to the randomly shaped particles in the suspension. 

Specific surface may be e q>ressed on a mass basis using the material density to modify the volume. The diameter of the here having the same equivalent ecific sur ce as the particle is sometimes termed the surface volume mean or the Sauter mean diameter. 

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. 

The particle size enters the equations through the volume equivalent particle diameter, dp, and so-called surface shape factor, /, defined as... 

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. 

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