Particles nonspherical

In principle, some types of nonspherical particles could be packed more tightly than spheres, although they would start to interact at lower concentrations. In reality, higher viscosities are normally found with nonspherical particles. The concentration law is approximately exponential at low to moderate concentrations, but equations similar to eq. 10.5.1 can still be used as well. The empirical value of 4 m can be much smaller than that for spherical particles (e.g., 0.44 for rough crystals with aspect ratios close to unity Kitano et al., 1981). If fibers are used, this value drops even further, down to 0.18 for an aspect ratio of 27 (see also Metzner, 1985). The decrease with aspect ratio seems to be roughly linear. Homogeneous suspensions of fibers with large aspect ratios are difficult to prepare and handle. As in dilute systems, the type of flow will determine the extent of the shape effect. Extensional flows are discussed below. 

Doi and Edwards (1986) have used a tube model to describe flow of semidilute suspensions of rods. Predicted behavior is qualitatively similar to their theory for entangled, flexible polymer chains (see Chapter 11). This approach has also been extended to describe the rheology of nematic liquid crystalline polymers (Doi and Edwards, 1986 Larson, 1988). 

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Although most cakes consist of polydisperse, nonspherical particle systems theoretically capable of producing more closely packed deposits, the practical cakes usually have large voids and are more loosely packed due to the lack of sufficient particle relaxation time available at the time of cake deposition hence the above-derived value of 17.6 pm becomes nearer the 10 pm limit when air pressure dewatering becomes necessary. 

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. 

Other Factors Affecting the Viscosity of Dispersions. Factors other than concentration affect the viscosity of dispersions. A dispersion of nonspherical particles tends to be more viscous than predicted if the Brownian motion is great enough to maintain a random orientation of the particles. However, at low temperatures or high solvent viscosities, the Brownian motion is small and the particle alignment in flow (streamlining) results in unexpectedly lower viscosities. This is a form of shear thinning. 

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 ... 

This development has been generalized. Results for zero- and second-order irreversible reactions are shown in Figure 10. Results are given elsewhere (48) for more complex kinetics, nonisothermal reactions, and particle shapes other than spheres. For nonspherical particles, the equivalent spherical radius, three times the particle volume/surface area, can be used for R to a good approximation. 

Nonsplierical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. 

For particles with y < 0.67, the correlations of Becker Can. J. Chem. Ertg., 37, 85-91 [1959]) should be used. Reference to this paper is also recommended for intermediate region flow. Setthng charac teristics of nonspherical particles are discussed by Clift, Grace, and Weber, Chaps. 4 and 6. 

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. 

Similarly equation 36 for slurries with nonspherical particles is... 

These equations are valid for spherical particles. For nonspherical particles, a more detailed model must be used i.e., the effect of the irregular shape of the particles must be taken into account by means of shape factors. 

Even if the peculiarities of net-formation of nonspherical particles are not taken into account, at least two fundamentally new effects arise during the flow of dispersion. First, this is the possibility to be oriented in the flow, as a consequence of which the medium becomes anisotropic. And second, this is the possibility to rotate the spherical particles in the flow (spherical particles can, of course, rotato too, but their rotation does not affect the structure of the system as a whole). 

An alternative description of nonspherical particles is often represented by the sphericity factor (tfi), which is the number that, when multiplied by the diameter of a sphere with the same volume as the particle (ds), gives the particle effective diameter (dp) ... 

The minimum bed porosity at incipient fluidization for nonspherical particles can be estimated from... 

Various methods are employed to size particles by optical microscopy. For spherical particles, the diameter suffices, but for nonspherical particles, altema-... 

To create a useful CFD simulation the model geometry needs to be defined and the proper boundary conditions applied. Defining the geometry for a CFD simulation of a packed tube implies being able to specify the exact position and, for nonspherical particles, orientation of every particle in the bed. This is not an easy task. Our experience with different types of experimental approaches has convinced us that they are all too inaccurate for use with CFD models. This leads to the conclusion that the tube packing must either be computergenerated or be highly structured so that the particle positions can be calculated analytically. 

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

Since carbon black and amorphous silica tend to form clusters of spheres (grasping effect), an additional modification of the Einstein equation was made to account for the nonspherical shape or aspect ratio (L/D). This factor (/) is equal to the ratio of length (L) to diameter (D) of the nonspherical particles (/= L/D). 

In this treatment we develop the conversion equations for spherical particles in which steps 1, 2, and 3, in turn, are rate-controlling. We then extend the analysis to nonspherical particles and to situations where the combined effect of these three resistances must be considered. 

For a single-domain ferromagnet, any nonspherical particle shape gives rise to shape anisotropy due to the internal magnetostatic energy. The magnetostatic energy, for an ellipsoid of revolution, is equal to... 

The van der Waals forces can be much higher in the actual case of nonspherical particles, which usually present more than one contact point with the substrate. 

Most particles of practical interest are irregular in shape, and so do not fall into the above categories. A variety of empirical factors have been proposed to describe nonspherical particles and correlate their flow behavior. Empirical description of particle shape is provided by identifying two characteristic parameters from the following (B3) ... 

In this chapter, we extend the discussion of the previous chapter to nonspherical shapes. Only solid particles are considered and the discussion is limited to low Reynolds number flows. The flow pattern and heat and mass transfer for a nonspherical particle depend on its orientation. This introduces complications not present for spherical particles. For example, the net drag force is parallel to the direction of motion only if the particle has special shape properties or is aligned in specific orientations. 

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. 

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