Equivalent sphere volume

To = temperature of the solvent at which tan A goes through a maximum. These values are presented in Table II. VSE (the Stokes-Eeinstein volume) is calculated for a spherical molecule if the molecule is aspherical this calculation (VSE) is called Vapparent The Vapparent can be smaller or larger than the Stokes-Einstein volume and varies from the equivalent sphere volume obtained by solution of equations 3,4 and 5. 

Figure 25 shows the projection areas of growing particles after 0, 20, 40, and 60 min of polymerization (Fig. 25a) and demonstrates the particle growth evaluation (Fig. 25b). These collected images are processed to determine the projection area of each catalyst particle. Although the projection area is the primary quantity measured, it is easier to comprehend the size of the particles in terms of their diameter and volume. Hence, the projection area is used to estimate the diameter of a circle of equivalent area (equivalent circle diameter, BCD) and from that the volume of a sphere having an equivalent projection area (equivalent sphere volume, ESV).

For the representation of the complex geometry in a quantitative way different sets of shape parameters have been introduced, for instance based on the void surface area, equivalent sphere volume, radius of gyration. Each of these methods introduces an oversimplification of the very complex nature of the voids. For the same reason the photo-chromic probe method described above shows a much narrower distribution than one might expect on the basis of the clearly elongated voids displayed in Figure 4.7. This is a common limit of all experimental probing methods, none of which can probe the complex void structure to its full extent. 

A particle of equivalent sphere volume diameter 0.2 mm, density 2500kg/m and sphericity 0.6 falls freely under gravity in a fluid of density l.Okg/m and viscosity 2 X 10 Pas. Estimate the terminal velocity reached by the particle. (Answer 0.6 m/s.)... 

MetaMorph then uses the area defended to calculate a radius and then calculate the equivalent sphere volume. ... 

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. 

Beeause of the uniqueness and simplieity of the sphere, the eharaeteristies of non-spherieal partieles (whieh most real ones are) are often related baek in some way to the size of an equivalent sphere whieh has some shared ehar-aeteristie, sueh as the same volume or surfaee area. 

P 0 is the apparent limiting anisotropy and P is the anisotropy (17). The slope is read as the straight line portion of the curves in Figures la-f and applied in equation 10 to obtain the equivalent-sphere molar volume. The difference between the extrapolated intercept of the linear portion of the line on they ordinate and the extrapolated intercept of the curved line is attributed to the internal rotation of the fluorophores in the molecule (5). 

By relating the endpoint of crushed DBF absorption to the void space within and between equivalent spheres of aggregates, and assuming the spheres to be packed at random, Wang et al. obtained the following equation for the effective volume fraction of carbon black ... 

According to Eq. (18), the high polymer molecule should exhibit the frictional coefficient of an equivalent sphere (compare Eq. 15) having a radius proportional to the root-mean-square end-to-end distance (r ) (or to (s ) / ). Similarly, according to Eq. (23) its contribution to the viscosity should be that of an equivalent sphere (compare Eq. 16) having a volume proportional to (r ) / In analogy with Eq. (17 ), we might write... 

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... 

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. 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

Here is the maximum value of X and A is the surface area of the volume equivalent sphere. For a fluid particle the average Sherwood number is... 

An equivalent sphere is defined as the sphere with the same value of one of the above measures. The commonest referent is the volume-equivalent sphere, which many authors describe as the equivalent sphere without further definition. The particle shape factor is defined as the ratio of another measure from the above list to the corresponding value for the equivalent sphere. Of the many possible shape factors, those which have proved most useful are described below. All shape factors are open to the criticism that a range of bodies with different forms may have the same shape factor, but this is inevitable if regular or complex shapes are to be described by a single parameter. 

When these factors are based on the volume-equivalent sphere,... 

Figure 4.5 shows the variation of A with E for flow parallel and normal to the axis, and averaged over random orientations. Except for disk-like particles, the dependence of A on aspect ratio is rather weak. In axial motion, a somewhat prolate spheroid experiences less drag than the volume-equivalent sphere A passes through a minimum of 0.9555 for E = 1.955. For motion normal to the axis of symmetry, A 2 takes a minimum of 0.9883 at = 0.702. However, the average resistance A is a minimum for a sphere. 

Fig. 4.5 Drag ratios for spheroids compared to volume-equivalent spheres.

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... 

Fig. 4.2.9 Histograms of the size distributions of the particles shown in Fig. 4.2.8. Original and final size distributions are shown by broken and solid lines, respectively. The diameter of an equivalent sphere having the same volume as a nonspherical particle was obtained with a Coulter counter. (From Ref. 9.)...

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 3. Overlapping volume of two radical polymer chains represented by equivalent spheres...

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

---