Nonspherical Particle Settling

The terminal velocity in the case of fine particles is approached so quickly that in practical engineering calculations the settling is taken as a constant velocity motion and the acceleration period is neglected. Equation 7 can also be appHed to nonspherical particles if the particle size x is the equivalent Stokes diameter as deterrnined by sedimentation or elutriation methods of particle-size measurement. 

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 ... 

The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, u, of spherical particles by a correction factor, iji, called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume ... 

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. 

Venumadhav, G. and Chhabra, R.P. Powder Technol. 78 (1994) 77. Settling velocities of single nonspherical particles in non-Newtonian fluids. 

No fully satisfactory method is available for correlating the drag on irregular particles. Settling behavior has been correlated with most of the more widely used shape factors. Settling velocity may be entirely uncorrelated with the visual sphericity obtained from the particle outline alone (B8). General correlations for nonspherical particles are discussed in Chapter 6. 

For particles with f < 0.67, the correlations of Becker (Can. J. Chem. Eng., 37, 85—91 [1959]) should be used. Reference to this paper is also recommended for intermediate region flow. Settling characteristics of nonspherical particles are discussed by Clift, Grace, and Weber, Chaps. 4 and 6. 

If the particle-size distribution of a powder composed of hard, smooth spheres is measured by any of the techniques, the measured values should be identical. However, many different size distributions can be defined for any powder made up of nonspherical particles. For example, if a rod-shaped particle is placed on a sieve, then its diameter, not its length, determines the size of aperture through which it will pass. If however, the particle is allowed to settle in a viscous fluid, then the calculated diameter of a sphere of the same substance that would have the same falling speed in the same fluid (i.e., the Stokes diameter) is taken as the appropriate size parameter of the particle. Since the Stokes diameter for the rod-shaped particle will obviously differ from the rod diameter, this difference represents added information concerning particle shape. The ratio of the diameters measured by two different techniques is called the shape factor. 

Nonspherical particles are subjected to a larger drag force compared to their volume equivalent spheres because x > 1 and therefore settle more slowly. The terminal settling velocity of a nonspherical particle is then [following the same approach as in the derivation of (9.42)]... 

Calculate the Stokes diameter of the NaCl particle of the previous example. The two approaches (dynamic shape factor combined with the volume equivalent diameter and the Stokes diameter) are different ways to describe the drag force and terminal settling velocity of a nonspherical particle. The terminal velocity of a nonspherical particle with a volume equivalent diameter Dve is given by (9.104),... 

Sedimentation assumes a spherical particle assumption to relate settling speed with particle size that does result in a bias of the size reported for nonspherical particles. 

The Andreasen pipette introduced in the 1920s is perhaps the most popular manual apparatus for sampling from a sedimenting suspension. Determination of the change in density of the sampled particle suspension with time enables the calculation of size distribution of the particles. As Stokes law applies only to spherical particles, the nonspherical particles give a mean diameter referred to as Stokes equivalent diameter. The size range measurable by this method is from 2 to 60 pm (8). The upper limit depends on the viscosity of liquid used while the lower limit is due to the failure of very small particles to settle as these particles are kept suspended by Brownian motion. 

A spherical particle is unique in that it presents the same projected area to the oncoming fluid irrespective of its orientation. For nonspherical particles, on the other hand, the orientation must be known before their terminal velocity or the drag force can be calculated. Conversely, nonspherical particles tend to attain a preferred or most stable orientation, irrespective of their initial state, in the free-settling process. Vast literature, although not as extensive as that for spherical particles, is also available on the hydrodynamic behavior of nonspherical—regular as well as irregular type— particles in incompressible Newtonian fluids and these studies have been summarized in the aforementioned references, whereas the corresponding aerodynamic literature has been comprehensively reviewed by Hoerner (1965). 

The only drag measurements on nonspherical particles (cylinders, bars) in viscoelastic fluids are due to Rodrigue et al. (1984). They measured free-settling velocities in a polyisobutylene in kerosene/polybutene solution (Boger fluid), with a fluid relaxation time of 7.2 msec. They found it necessary to modify Eq. (16) by incorporating the following correction factor due to viscoelasticity ... 

Effect of Particle Shape and Orientation to Flow. As indicated by Figure 10-2, the shape of the particle, and particularly its orientation to flow, affects the settling velocity. Particle shape is often quantifled by the sphericity, l , which is the ratio of the surface area of a spherical particle of the same volume to that of the nonspherical particle. Chapman et al. (1983) reported that for particles with sphericity between 0.7 and 1, it is sufficient to use eqs (10-3) and (10-4) and replace the particle diameter, dp, with the diameter of a sphere of equal volume. For particles with sphericity less than 0.7, the estimation of the settling velocity is complicated by the fact that the orientation to flow is a function of the Reynolds number. The effect of shape on the settling of such particles must be evaluated experimentally. Correlations presented by Pettyjohn (1948) and Becker (1959) are recommended only for preliminary estimates. 

Free-falling diameter Also known as sedimentation or Stokes diameter, the diameter of a sphere with the same terminal settling velocity and density as a nonspherical or irregular particle. 

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