Volume shape coefficient

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)... 

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... 

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

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. 

Particle sizes combined with shape factors have been the subject of many of the recent studies regarding flow of solids. Sphericity, circularity, surface-shape coefficient, volume-shape coefficient, and surface-volume-shape coefficient are some of the most commonly used shape factors. It is generally accepted that the flowability of powders decreases as the shapes of particles become more irregular. Efforts to relate various shape factors to powder bulk behavior have become more successful recently, primarily because of the fact that shape characterization techniques and methods for physically sorting particles of different shapes are... 

VOLUME-SHAPE COEFFICIENT is the coefficient of proportionality relating the volume of the particle with the cube of its measured diameter the same qualifications apply here as in the case of the surface-shape coefficient but this time related to particle volume. 

SURFACE-VOLUME SHAPE COEFFICIENT is the ratio of surface to volume shape coefficients, combining together the features of the two. 

Particle shape can be quantified by different methods. One popular method is through the use of Hey wood coefficients (5). The Hey wood shape coefficient is defined as the ratio of the surface shape coefficient (n for a sphere) to the volume shape coefficient (n/6 for a sphere) hence, the shape coefficient for a sphere would be 6.0. Applying this to a cube and using its projected area in its most stable position, the shape coefficient is 6.8. Cutting the cube in half in one dimension increases the shape factor to 9.0, whereas it increases to 26.6 if that cube was sliced one-tenth in one dimension. Further details of these types of calculations are provided by Rupp (5). 

Alternatively, if the specific surface is known, the surface-volume shape coefficient by sieving Usv = SvA).008205. 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

This can be done via Eqs. (29) through (32). From the dissolution data, the coefficient B2 is obtained and through the results from microscopy the moments r2 and p3 can be evaluated. By knowing N, the initial number of particles, and the density of the solid, the average initial volume shape factor for a polydisperse powder can be estimated. 

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]. 

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... 

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. 

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... 

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. 

Shape coefficients are alternative parameters that have been used to characterize the shapes of particles. In shape coefficients, the measured particle size, x, is correlated to the measured particle surface area A or volume V, which are, respectively, given by ... 

The value of the shape coefficients can be calculated for various equivalent sphere diameter bases. Let subscript a = projected area diameter v = volume diameter s -surface area diameter St = Stokes diameter m = mesh size. The volume of particles may be expressed as kjc/= k,xj = Aye/ = fexs/ = krfcj. Hence K = k/Xt/x f and so on. 

X refers to the method of measurement being A for a sieve analysis St for a Stokes analysis and so on, (e.g. dj is the sieve diameter and dst the Stokes diameter, a refers to the shape coefficients, the suffixes denoting surface (s), volume fv) and surface-volume f v)]. 

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