Spheroids definition

Heywood [Heywood, Symposium on Paiticle Size Analysis, lust. Chem. Engrs. (1 7), Suppl. 25, 14] recognized that the word shape refers to two distinc t charac teiistics of a particle—form and proportion. The first defines the degree to which the particle approaches a definite form such as cube, tetr edron, or sphere, and the second by the relative proportions of the particle which distinguish one cuboid, tetrahedron, or spheroid from another in the same class. He replaced historical quahtative definitions of shape by numerical shape coefficients. 

If T is based on volume-equivalent radius, rather than equatorial radius as used here, E has almost no effect on the trajectory for prolate spheroids (LI). However, this definition of t obscures the effect of shape for oblate particles. 

When prepared by chemical methods, y-sulphur is frequently accompanied by an apparently amorphous powder which is readily soluble in carbon disulphide. This has been regarded, by some investigators, as a definite form of sulphur and given the name soluble amorphous sulphur 2 in reality, however, it consists of minute spheroidal crystals of rhombic sulphur possibly together with nacreous sulphur.3 Another so-called modification of amorphous sulphur, described 4 as soluble in carbon disulphide but becoming insoluble on evaporation of the solvent, is probably no distinct form, but only a mixture of y-sulphur with finely divided crystalline sulphur. 

Perfect spheres are rare, but spheroidal particles are present in some powders produced at high temperature (e.g. pyrogenic silicas) or by the sol-gel process. The term sphericity is useful for some purposes. Sphericity has been defined in various ways, the simplest definition being the ratio of the surface area of a sphere of the same volume as a given particle to the actual surface area of that particle (Allen, 1990). 

Probably the most precious occurrence of anion recognition is that related to size exclusion selectivity. This takes place when the receptor, providing for instance a spheroidal cavity, includes only spherical anions of radius less than or equal to a definite value. In this context, the smallest anion, fluoride, has offered vast opportunities. 

To estimate which occurs in the definition of Eod an empirical correlation of the aspect ratio E for spheroidal bubbles in a contaminated system was used ... 

For all other situations, particle shape has an overwhelming effect on how they behave during a test or in any process. Shape is characterized by form and proportions. Form refers to the degree to which a particle approaches a definite form, such as a sphere, cube, tetrahedron (Fig. 5.36), or higher order polyhedron. The relative proportions distinguish one spheroid, cuboid, tetrahedron, or polyhedron from another of the same class. 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

In one sense it is immediately obvious that it will be difficult to separate integrals involving these functions the explicit appearance of rx with its square root definition does not bode well for separation into a product (or sum) of separate functions of x, y and z. However, for pairs of such STOS it is possible to find a coordinate system into which the transforms of the STOs do separate the prolate spheroidal system, usually denoted by (f, q, (f)... 

Bubbles are usually not perfectly spherical in shape. Accordingly, the mean chord length is used to represent the bubble size unless the bubble shape is radically different from spheroidal. For definition, the equivalent spherical bubble has the diameter d-Q. The mean bubble chord length Lb is given by [12] ... 

One should be cautious in reviewing this work. The definition of the aspect ratio for the oblate spheroids (the thickness divided by the long dimension) is the inverse of what is described in the bulk of this chapter. There is a difference in assigning the Young s modulus in the first article (178 GPa) and the second article (167 GPa) for montmoriiionite. 178 GPa is generally accepted as the Young s modulus for montmoriiionite in the bulk of the work featured in this chapter. The bulk and the shear moduli are calculated from the model and utilized to calculate the Young s modulus. 

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