Equivalent spherical diameters

Make a theoretical plot of surface tension versus composition according to Eq. III-53, and compare with experiment. (Calculate the equivalent spherical diameter for water and methanol molecules and take o as the average of these.)... 

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. 

Eor randomly packed spherical particles, the constants M and B have been deterrnined experimentally to be 150 and 1.75, respectively. Eor nonspherical particles, equivalent spherical diameters are employed and additional corrections for shape are introduced. 

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

Every method, with the exception of imaging technologies, provides the measurement of an equivalent spherical diameter in one form or another. The spherical diameter information can be deduced indirectiy from the behavior of the particles passing through restricted volumes or channels under the influence of gravity or centrifugal force fields, and from interaction with many forms of radiation. 

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

Stokes diameter The equivalent spherical diameter of the particle being considered. 

Nominal Size, In. Approximate Number per Ft Approximate Weight per Ft , Lb Approximate Surface Area FtVFt Percent Free Gas Space Equivalent Spherical Diameter Dp, In. 

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. 

Various techniques and equipment are available for the measurement of particle size, shape, and volume. These include for microscopy, sieve analysis, sedimentation methods, photon correlation spectroscopy, and the Coulter counter or other electrical sensing devices. The specific surface area of original drug powders can also be assessed using gas adsorption or gas permeability techniques. It should be noted that most particle size measurements are not truly direct. Because the type of equipment used yields different equivalent spherical diameter, which are based on totally different principles, the particle size obtained from one method may or may not be compared with those obtained from other methods. 

A small sample of a coal slurry containing particles with equivalent spherical diameters from 1 to 500 pm is introduced into the top of a water column 30 cm high. The particles that fall to the bottom are continuously collected and weighed to determine the particle size distribution in the slurry. If the solid SG is 1.4 and the water viscosity is 1 cP, over what time range must the data be obtained in order to collect and weigh all the particles in the sample ... 

It is desired to determine the size of pulverized coal particles by measuring the time it takes them to fall a given distance in a known fluid. It is found that the coal particles (SG = 1.35) take a time ranging from 5 s to 1,000 min to fall 23 cm through a column of methanol (SG = 0.785, p. = 0.88 cP). What is the size range of the particles in terms of their equivalent spherical diameters Assume that the particles are falling at their terminal velocities at all times. 

Many inert pigments (often known as fillers) are incorporated into paper in addition to the cellulosic fibres. They may be added to improve certain optical properties—in particular opacity and brightness—or simply as a cheap replacement for costly fibre. The two most common pigments are kaolin (china clay) and chalk (limestone), but talc and speciality pigments such as titanium dioxide are also used. The particle size for general purpose fillers is normally expressed as an equivalent spherical diameter (esd) and this is determined from sedimentation data. Values for the common paper-... 

Particle size will always be expressed in terms of diameter rather than radius in this chapter. In all developments in this chapter, the particles will be assumed to be spherical. Consequently, when applying the results to other than spherical particles, that equivalent spherical diameter must be used which would correspond to the phenomenon involved. 

In a test on a centrifuge all particles of a mineral of density 2800 kg/m3 and of size 5 xm, equivalent spherical diameter, were separated from suspension in water fed at a volumetric throughput rate of 0.25 m3/s. Calculate the value of the capacity factor E. 

Relatively little appears to be known about the influence of shape on the behaviour of particulate solids and it is notoriously difficult to measure. Whilst a sphere may be characterised uniquely by its diameter and a cube by the length of a side, few natural or manufactured food particles are truly spherical or cubic. For irregular particles, or for regular but non-spherical particles, an equivalent spherical diameter de can be defined as the diameter of a sphere with the same volume V as the original particle. Thus... 

Equations which predict the volume or equivalent spherical diameter of a formed drop are not sufficient for extraction calculations, in the light of the very high rate of mass transfer during drop formation. It is desirable that the equation also lend itself to mathematical manipulation for the calculation of instantaneous interfacial area. To do this, the shape of the drop throughout the formation period must be defined. 

The utilization of such a length term requires a knowledge of the eccentricity of the droplet as a function of drop size. This item is reviewed below. The length term usually used is the very convenient equivalent spherical diameter (De), defined as the diameter of a sphere having the same volume as that possessed by the drop, regardless of the actual shape of the latter. 

This equation suggests that plotting carrier density vs. IGk would yield a straight line with an intercept equal to the particle density, pp, and with a slope related to the equivalent spherical diameter d. 

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