Equivalent spheres diameter

Figure 6.7 Equivalent sphere diameter as a function of needle diameter, equivalent sphere, and needle having equal conductance. Needle half axis length L=0.5.

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. 

Thus the Stokes diameter of any partiele is that of an equivalent sphere having same terminal settling veloeity and is a useful additional partiele eharaeteristie for partieulate systems involving fluid motion. 

NPei and NRtt are based on the equivalent sphere diameters and on the nominal velocities ug and which in turn are based on the holdup of gas and liquid. The Schmidt number is included in the correlation partly because the range of variables covers part of the laminar-flow region (NRei < 1) and the transition region (1 < NRtl < 100) where molecular diffusion may contribute to axial mixing, and partly because the kinematic viscosity (changes of which were found to have no effect on axial mixing) is thereby eliminated from the correlation. 

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. 

To determine the settling characteristics of a sediment, you drop a sample of the material into a column of water. You measure the time it takes for the solids to fall a distance of 2 ft and find that it ranges from 1 to 20 s. If the solid SG = 2.5, what is the range of particle sizes in the sediment, in terms of the diameters of equivalent spheres ... 

Daj=611A DlH)l is the surface-volumetric diameter, equal to the diameter of sphere or cube with equivalent surface-to-volume ratio. 

In order to allow for a variation of interfacial area from that of an equivalent sphere, the eccentricity of the ellipsoidal drop must be taken into account. The area ratio of Eq. (44) does not exceed unity by a serious amount until an eccentricity of 1.5 is attained. An experimental plot of eccentricity as ordinate vs. equivalent spherical drop diameter as abscissa may result in a straight line (G7, Kl, K3, S12). A parameter is yet to be developed by which the lines can be predicted without recourse to experiment. Eccentricity is not an accurate shape description of violently oscillating drops and should therefore be used only for drop size below the peak diameter (region B of Fig. 5). 

It is common practice to define a hydraulic equivalent sphere as the sphere with the same density and terminal settling velocity as the particle in question. For a spheroid in creeping flow, the hydraulic equivalent sphere diameter is 2a- E/A and thus depends on orientation. 

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 1 Left Laminar flow field around a 400- xm diameter spherical steam bubble rising at 0.06 m/s. Right Mean flow around a 4-mm equivalent sphere diameter spherical-cap steam bubble rising at 20 cm/s in molten CaBr2 at 1 013 K.

Lienau s Theorem—If d is taken as the random mean diameter of a fractured element, the mean volume of an equivalent sphere is... 

Fig. 1. Quantification of shear-induced platelet aggregation by flow cytometry. Panel A corresponds to an unsheared blood specimen. Panel B corresponds to a blood specimen that has been subjected to a pathologically high level of shear stress for 30 sec. As can be seen in the figure there are three distinct cell populations. The upper population consists of platelets and platelet aggregates. The rbcs-plts population corresponds to platelets associated with erythrocytes and leukocytes. The wbcs population consists of some leukocytes that have elevated levels of FITC autofluorescence. The left vertical line separates single platelets (<4.5 xm in diameter) from platelet aggregates, whereas the right vertical line separates small from large platelet aggregates. The latter were defined to be larger than 10 xm in equivalent sphere diameter.

Strategy. Think about the size of a football—perhaps as a size-equivalent sphere—and about the size of a pig— perhaps as a big box—then divide one by the other. Let us assume that a football can be compressed into a sphere, and our best guess is that this sphere will have a diameter of about 25 cm (10 in.). Let us also imagine that a pig is a rectilinear box that is about 1 m long, 0.5 m high, and 0.5 m wide. This ignores the head, the tail, and... 

Reynolds number based on equivalent sphere diameter, Dp us p /y, dimensionless. 

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