Coefficient shape

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. 

Height is the vertical distance from ground or water surface to the center of area. The shape coefficient C, for a derrick is assumed as 1.25. and were obtained from ABS, Rules for Building and Classing Offshore Drilling Units, 1968. ... 

Characterization of the particle shape is generally described by the deviation from sphericity, as in the case of ellipsoids where the ratio of the two radii is the measure of deviation. The surface and volume are important properties aflected by the overall shape of a particle. A more complicated relationship for particle characterization was described by Heywood, who introduced shape coefficients such as surface and volume coefficients and elongation and flatness ratios [42]. 

Particle shape plays an important role in particle size determination. The simplest definition of particle size diameter is based on a sphere, which has a unique diameter. In reality, however, many particles are not well represented by this model. Figure 1 illustrates the variety of shapes that may be found in particle samples [1]. As the size of a particle increases, so does its tendency to have an irregular shape [2], complicating statistical analysis. Particle shape coefficients have been derived for different geometries [3], and various equivalent diame-... 

The shape of curve is modulated by varying the Hill coefficient to fit the observed behavior. Sensitivity of a drug is how concentration translates into effect as described by the shape coefficient or the Hill coefficient at a value of 1, it is a classical parabola. Eg is... 

Since the desired shape of a pellet is a sphere, shape factors have been used to describe the pellets. These are characterized variously as sphericity, roundness, shape coefficient, elongation index, and aspect ratio (63-67). Using the volume diameter, d, and projected diameter, d, a good measure... 

The parameter P is called the effective Biot number and k is a shape coefficient defined as ... 

A number of methods have been proposed for particle shape analysis, including shape coefficients, shape factors, verbal descriptions, curvature signatures, moment invariants, solid shape descriptors, and mathematical functions (Fourier series... 

In the most simplistic means of defining particle shape, measurements may be classified as either macroscopic or microscopic methods. Macroscopic methods typically determine particle shape using shape coefficients or shape factors, which are often calculated from characteristic properties of the particle such as volume, surface area, and mean particle diameter. Microscopic methods define particle texture using fractals or Fourier transforms. Additionally electron microscopy and X-ray diffraction analysis have proved useful for shape analysis of fine particles. 

Shape Coefficients and Shape Factors There are various types of shape factors, the majority based on statistical considerations. In essence this translates to the use of shape factors that do refer not to the shape of an individual particle but rather to the average shape of all the particles in a mass of powder. However, a method developed by Hausner [38] that uses three factors—elongation factor, bulkiness factor, and surface factor—may be used to characterize the shape of individual particles (Table 5). 

Surface shape coefficient CXs where V = average particle volume d = mean particle diameter S s = ... 

Volume-surface shape coefficient ocvs where S = average particle surface d = mean particle diameter a. ccvs — av... 

Shape factor cc0 where av = volume shape coefficient as = surface shape coefficient a0 = avnvJn... 

Sphericity shape factor Circularity shape factor 1 W where a0 = shape factor for equidimensional particle and thus represents part of av which is due to geometric shape only av = volume shape coefficient m = flakiness ratio, or breadth/thickness n = elongation ratio, or length/breadth Sphericity = (surface area of sphere having same volume as particle) / (surface area of actual particle) Circularity = (perimeter of particle outline)2 / 4tr(cross-sectional or projection area of particle outline)... 

The symbol x denotes size, as opposed to diameter, and includes a shape coefficient. This artifact is found to be useful for general treatment of data]. 

The numerical relationships between the various sizes of a particle depend on particle shape, and dimensionless ratios of these are called shape factors the relations between measured sizes and particle volume or surface area are called shape coefficients. 

Macroscopically, shape may be derived using shape coefficients or shape factors. 

Volume shape coefficients may be determined from knowledge of the number, volume mean size, weight and density of the particles comprising a fraction graded between close limits e.g. by sieving. Further, if surface areas are also determined by permeametry, surface shape coefficients may... 

As an illustration of the application of shape coefficients and shape factors consider a cuboid of side x, x, kx, where A is a variable. 

Assuming that the surface-volume shape coefficient by projected area, Usv.a, is size independent over the size range under consideration ... 

This transformation is derived with the assumption that particle shape does not change with size. More correctly, a shape coefficient aix) needs to be introduced ... 

Fig. 6.5 Drag coefficient versus Reynolds number for particles of different volume shape coefficients by microscopy compared with data for a sphere...

Experimental data were embodied in tables presenting C Re in terms of Re/Cf) and vice versa. Since the former expression is independent of velocity and the latter is independent of particle diameter, the velocity may be determined for a particle of known diameter and the diameter determined for a known settling velocity. Heywood also presented data for non-spherical particles in the form of correction tables for four values of volume-shape coefficient from microscopic measurement of particle-projected areas. 

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