Particle shape isometric

Pettyjohn and Christiansen Chem. Eng. Prog., 44, 157-172 [1948]) present correlations for the effect of particle shape on free-settling velocities of isometric particles. For Re < 0.05, the terminal or free-setthng velocity is given oy... 

If dp is the mean projected diameter, the mean projected area is nd2/4 and the volume is k d2, where k is a constant whose value depends on the shape of the particle. For a spherical particle, k is equal to tt/6. For rounded isometric particles, that is particles in which the dimension in three mutually perpendicular directions is approximately the same, k is about 0.5, and for angular particles k is about 0.4. For most minerals k lies between 0.2 and 0.5. 

HEISS, J. F. and Coull, J. Chem. Eng. Prog. 48 (1952) 133. The effect of orientation and shape on the settling velocity of non-isometric particles in a viscous medium. 

Even if an estimate for is available, the volumetric shape factor can only be evaluated if the particle volume is known, and this may not be readily available for naturally occurring particles, or if a distribution of particle sizes or shapes is present. Heywood (H4) suggested that k may be estimated from the corresponding value, k, of an isometric particle of similar form by the relationship... 

Pettyjohn and Christiansen (P4) reported extensive data for isometric particles. Heywood s volumetric shape factor was not a good basis for correlation in the Newton s law range, but sphericity was found suitable. Subsequently,... 

A final assumption made in the derivation of Stokes law was that the particles of interest were spheres. In many cases this is not true. Particles may have irregular shapes, depending on how they were formed and the amount of agglomeration which may have taken place. Liquid aerosols are always spherical, so that for liquid aerosols the assumption of sphericity holds. For isometric particles this assumption can also be used with little error. For long chains of particles or flocculated particles, large deviations from Stokes law are possible. 

Isometric particles are those for which all three dimensions are roughly the same. Spherical, regular polyhedral, or particles approximating these shapes belong in this class. Most knowledge regarding aerosol behavior pertains mainly to isometric particles. 

The size and shape of ceria NCs are proven fo appreciably change the chemical and physical properties hence, their control in synthesis is one chief objective for study, and various nanoparticles, nanocubes, nanooc-tahedra, nanowires, and nanotubes have been obtained for this purpose. Owing to the cubic fluorite structure, ceria tends to form isometric particles, which present sphere-like morphology and are usually intermediates between the shape of cubes and octahedra. The major exposed crystal surfaces for ceria NCs are low index ones, that is, 100, llOj, and 111, with considerable surface relaxation and reconstructions. Figure 1 shows some typical morphologies of ceria NCs. 

The drag coefficients for irregularly shaped particles such as coal or sand appear to be about the same as for spheres of the same nominal size at Reynolds numbers less than 50. However, the curve of Cc versus N c levels out at X 100, and the values of are two to three times those for spheres in the range Wr, = 500 to A Re = 3000. Similar results for isometric particles such as cubes and tetrahedrons have been reported. ... 

Figure 27 displays the lateral concentration variation for slurries of much 1.4-mm polystyrene particles at 3.4 m/s. Although not spherical, these particles were approximately isometric with shapes resembling... 

At the point of incipient fluidisation, the bed voidage, s f, depends on the shape and size range of the particles, but is approximately equal to 0.4 for isometric particles. The minimmn fluidising velocity V f for a power-law fluid in streamline flow is then obtained by substituting s = Smf in equation (5.67). Although this equation applies oifly at low values of the bed Reynolds numbers (<1), this is not usually a limitation at the high apparent viscosities of most non-Newtonian materials. 

Only four of the above particle size definitions are of general interest for applications in packed beds and fluidized beds. They are the sieve diameter, d, the volume diameter, d, the surface diameter, (f, and the surface-volume diameter, d. The most relevant diameter for application in a fluidized bed is the surface-volume diameter, 4v For applications in catalytic reactors with diflferent isometric catalyst shapes. Rase (1990) suggested that we use the equivalent diameters summarized in Table 1. 

Pettyjohn and Christiansen (1948) determined the free-settling rates of isometric particles of the following shapes and sphericities spheres (< ) = 1), cube octahedron (< ) = 0.906), octahedron (< ) = 0.846), cube (< ) = 0.806), and tetrahedron (< ) = 0.670). Their results suggest that the correction factor should be... 

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