Shape factors ratio coefficients

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... 

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 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. 

Calculation of the frictional coefficient, shape factor and axial ratio for the RAGl fragment (ZDD) from the sedimentation coefficient, s°q, has been previously described in detail [5]. 

This section will discuss some of the commonly used and cited shape factors in the pharmaceutical industry the shape factors discussed in this section are Wadelfs true sphericity and circularity, rugosity coefficient, correction factor, Dallavalle s shape factor, Heywood s shape factor, Schneiderhohn aspect ratio, one plane critical stability (OPCS), and Podczeck s two- and three-dimensional factor. There are also many other shape factors but they are beyond the scope of this chapter (11,14,26-31). 

Where R = resistance to fracture initiation cTp = fracture stress V = Poisson s ratio E = Young s modulus a = coefficient of thermal expansion ATmax = critical quench temperature difference S = shape factor... 

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). 

This tube has a ratio of outside to inside surface of about 3.5 and is useful in exchangers when the outside coefficient is poorer than the inside tube coefficient. The fm efficiency factor, which is determined by fm shape and size, is important to final exchanger sizing. Likewise, the effect of the inside tube fouling factor is important to evaluate carefully. Economically, the outside coefficient should be about V5 or less than the inside coefficient to make the finned unit look attractive however, this break-even point varies with the market and designed-in features of the exchanger. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

A measurement of physical parameters in solution for isolated macromolecules provides a manner by which the shape of a macromolecule can be determined. The approximate dimensions and axial ratio or radius can be calculated by applying Equations (4.3) through (4.17). As shown in Figure 4.10, the particle scattering factor for collagen molecules depicted in Figure 4.9 is more sensitive to bends than is the translational diffusion coefficient. 

When free-molecule flow is encountered, Oppenheim [8] has given convenient charts for calculating recovery factors and heat-transfer coefficients for flow over standard geometric shapes. Figures 12-16 and 12-17 give samples of these charts. The molecular speed ratio S used in these charts is defined by... 

The diffusion coefficients associated with translational motions when the radii of the diffusing radicals are not much larger than that of the solvent are expressed more accurately by D = kTI6nrr T (where r is the radius of the diffusing radical assuming a spherical shape and r (=yxr ) is the microviscosity. The value of /, the microfriction factor, can be calculated or taken equal to DsE/f exptb the ratio between the Stokes-Einstein diffusion coefficient (that considers van der Waals volumes, but not interstitial volumes) and the experimentally measured diffusion coefficient, Dexpti- As will be discussed later, these relationships appear to hold even in some polymer matrices. 

Raymond and DeVries (1977) have recently postulated that the functional saturation can be realized even with the simpler theory previously developed by Kuhn (1956). Reasoning that l differential surface-to-volume ratio, an irregularly shaped surface will lead to larger surface area for the same spherical volume, these investigators showed that l = [2pac] 112, where p is related to polymer size, a is an entrapment coefficient of the polymer in ice, and c is the concentration of polymers in solution. If this value of l is used for the evaluation ofK in Eq. (27), then even when the Flory-Huggins factor is not incorporated (x = i), ATioweringrxcin can be seen. Qualitative... 

In Fig. 5.1.2 the Perrin factor is plotted as a function of the axial ratio, defined as the ratio of the long semiaxis to short semiaxis, equal to p for the prolate spheroid and p for the oblate spheroid. As seen, the Perrin factor is always greater than unity, which may have been anticipated, since for equal volumes the surface area of the spheroid will be greater than that of a sphere of the same volume, so the friction coefficients will be greater. Because rhe volume of a molecule is proportional to its molar mass, then, for a constant mass, the more a molecule deviates from a spherical shape, the larger will be its mean friction coefficient. 

---