Isometric particles

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,... 

Transparent red iron oxides containing iron oxide hydrate can also be produced directly by precipitation. A hematite content of > 85 % can be obtained when iron(II) hydroxide or iron(II) carbonate is precipitated from iron(II) salt solutions at ca. 30 °C and when oxidation is carried out to completion with aeration and seeding additives (e.g., chlorides of magnesium, calcium, or aluminum) [5.271], Transparent iron oxides can also be synthesized by heating finely atomized liquid pentacarbonyl iron in the presence of excess air at 580-800 °C [5.272], [5.273]. The products have a primary particle size of ca. 10 nm, are X-ray amorphous, and have an isometric particle form. Hues ranging from red to orange can be obtained with this procedure, however, it is not suitable for yellow hues. 

The sedimentation of isometric particles, such as cubes and octa-hedra, deviates only slightly from the Stokes equation. Significantly anisometric particles that can be approximated by ellipsoids of revolution are amenable to rigorous sizing by centrifugation if the axial ratios q are known. If the major semi-axis is a and the minor semi-axis is c, the sedimentation velocity dx/dt can be written as... 

Fibers axe particles with great length in one dimension compared to much smaller lengths in the other two dimensions. Examples are prisms, needles, and threads or mineral fibers such as asbestos. Recent concern over the health hazard posed by inhalation of asbestos fibers has prompted study of fiber properties in air. There is still not as much known about fibers as isometric particles. 

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. 

Platelets are particles that have two long dimensions and a small third dimension. Leaves or leaf fragments, scales, and disks fall into this class. Very little is known about platelet behavior in air, and care must be exercised in applying knowledge derived from studying isometric particles to platelets. 

To study charging mechanisms theoretically for either diffusion charging or field charging, it is necessary to make several assumptions regarding the aerosol. First, the particles are assumed to be spherical. This assumption is reasonable for isometric particles. Second, it is also assumed that the particles are monodisperse. The effect of polydispersity complicates but does not invalidate theory. Third, there are no interactions between individual particles. Finally, the ion concentration and electric field near each particle are assumed to be uniform. These last two assumptions are essentially true for all natural and industrial aerosols. Thus except in the most extreme cases, theoiy should be adequate without other modification. 

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. 

Pettyjohn and Christiansen [24] made an extensive experimental study of isometric particles and proposed the following relationship ... 

Despite these complicating factors, it may be assumed that the conclusions concerning the relationship between particle size and the time required for their complete dissolution will be applicable for the case of single-sized isometric particles distributed uniformly throughout the melt, the composition of which changes con-... 

For single-size spherical particles, porosity can be readily shown to be independent of particle radius for a particular arrangement. For approximately isometric particles, a porosity of about 40% is obtained in real systems. When a fine fraction is mixed with a coarse one, then the fine particles will fill the voids between the coarse particles. In an optimum case, the minimum porosity amounts to 0.4 x 0.4 x 100 = = 16 vol.% However, such a packing can seldom be accomplished in practice, because the ratio of particle sizes is not suitably high and the mixing is imperfect. 

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. ... 

The particle-to-gas heat transfer coefficient in dense-phase fluidization systems can be determined from correlation Eq. 13.3.1 [2] given in Table 13.3. The correlation indicates that the values of particle-to-gas heat transfer coefficient in a dense-phase fluidized bed lie between those for fixed bed with large isometric particles (with a factor of 1.8 in the second term [49]) and those for the single-particle heat transfer coefficient (with a factor of 0.6 in the second term of the equation). 

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