Shape sphericity factors

Table 3.1-1. Shape Factors (Sphericity) of Some Materials...

Sphericity. Sphericity, /, is a shape factor defined as the ratio of the surface area of a sphere the volume of which is equal to that of the particle, divided by the actual surface area of the particle. 

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, u, of spherical particles by a correction factor, iji, called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume ... 

These equations are valid for spherical particles. For nonspherical particles, a more detailed model must be used i.e., the effect of the irregular shape of the particles must be taken into account by means of shape factors. 

Where the particle of B contains m molecules, AGB is the bulk free energy change per molecule, a is the shape factor (4irr2 for a spherical interface) and y is the strain energy per unit area of interface. For a spherical nucleus, where vm is the volume of product per molecule ,... 

In a study of the viscosity of a solution of suspension of spherical particles (colloids), suggested that the specific viscosity rjsp is related to a shape factor Ua-b in the following way ... 

Like any dynamic strain instrument, the RPA readily measures a complex torque, S (see Figure 30.1) that gives the complex (shear) modulus G when multiplied by a shape factor B = iTrR / ia, where R is the radius of the cavity and a the angle between the two conical dies. The error imparted by the closure of the test cavity (i.e., the sample s periphery is neither free nor spherical) is negligible for Newtonian fluids and of the order of maximum 10% in the case of viscoelastic systems, as demonstrated through numerical simulation of the actual test cavity." ... 

We will assiune a spherical shape factor and use the latter of the two equations. Since the first part of Ap.2.4. is the volume factor while the last part is the surface factor, we can do this with some justification. For a spherical nucleus, i.e.- Ap.2.5., we get ... 

This is Stokes Law for sedimentation where we have added a, a shape factor, just in case we do not have spherical particles. For spheres, a = 1. It is fractioncd otherwise. 

Equation (1) points to a number of important particle properties. Clearly the particle diameter, by any definition, plays a role in the behavior of the particle. Two other particle properties, density and shape, are of significance. The shape becomes important if particles deviate significantly from sphericity. The majority of pharmaceutical aerosol particles exhibit a high level of rotational symmetry and consequently do not deviate substantially from spherical behavior. The notable exception is that of elongated particles, fibers, or needles, which exhibit shape factors, kp, substantially greater than 1. Density will frequently deviate from unity and must be considered in comparing aerodynamic and equivalent volume diameters. 

One of the earliest defined shape factors is the sphericity, i//w, which was defined by Wadell [109] as the surface area of a sphere having the same volume as the particle, related to the surface... 

Several parameters have been seen to influence the crystallization of ice crystals in subcooled aqueous solutions. The primary factor is the extent of subcooling of the solution. Other factors include the agitation rate, the types and levels of solutes in solution. Huige (6) has summarized past work on conditions under which dusk-shaped and spherical crystals can be found in suspension crystallizers. The effects of heat and mass transfer phenomena on the morphology of an ice crystal growing in a suspension have not been fully understood. 

Bowen and Masliyah examined the axial resistance of cylinders with flat, hemispherical and conical ends, and of double-headed cones and cones with hemispherical caps, together with the established results for spheroids. Widely used shape factors (including sphericity) did not give good correlations, while Eqs. (4-26) and (4-27) were found to be inapplicable to particles other than cylinders and spheroids. The best correlation was provided by the perimeter-equivalent factor Yj defined in Chapter 2. With this parameter, the equivalent sphere has the same perimeter as the particle viewed normal to the axis. Based on their numerical results, Bowen and Masliyah obtained the correlation... 

No fully satisfactory method is available for correlating the drag on irregular particles. Settling behavior has been correlated with most of the more widely used shape factors. Settling velocity may be entirely uncorrelated with the visual sphericity obtained from the particle outline alone (B8). General correlations for nonspherical particles are discussed in Chapter 6. 

Since most irregular particles of practical concern tend to be oblate, lenticular, or rod-like with moderate aspect ratio, these comparisons generally support Heywood s approach. Combining this observation with the fact that the volumetric shape factor is more readily determined than sphericity, we conclude that Heywood s approach is preferred for the intermediate range. For convenience in estimating Uj, Table 6.4 gives correlations, fitted to Heywood s values, for 0.1 < k < 0.4 at specific values of Since is relatively insensitive to interpolation for at other values of is straightforward. In common with Heywood s tabulated values, the correlations in Table 6.4 do not extrapolate to = 1 for a sphere k = 0.524). 

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,... 

Dg is the geometric diameter, pp is the density of the particle, neglecting the buoyancy effects of air, p is the reference density (1 g cm 3), and k is a shape factor, which is 1.0 in the case of a sphere. Because of the effect of particle density on the aerodynamic diameter, a spherical particle of high density will have a larger aerodynamic diameter than its geometric diameter. However, for most substances, pp 10 so that the difference is less than a factor of 3 (Lawrence Berkeley Laboratory, 1979). Particle densities are often lower than bulk densities of pure substances due to voids, pores, and cracks in the particles. 

The range of specific surface area can vary widely depending upon the particle s size and shape and also the porosity.t The influence of pores can often overwhelm the size and external shape factors. For example, a powder consisting of spherical particles exhibits a total surface area, S, as described by equation (1.6) ... 

The biggest difference between biological particles and ceramic particles in the application of Eq. (4.20) is that while most ceramic particles are spherical ( Ch = 2.5), most biological particles can be modeled as either prolate ellipsoids or oblate spheroids (or ellipsoids). Ellipsoids are characterized according to their shape factor, ajb, for which a and b are the dimensions of the semimajor and semiminor axes, respectively (see Eigure 4.17). In a prolate ellipsoid, a > b, whereas in an oblate ellipsoid, b > a.ln the extremes, b approximates a cylinder, and b a approximates a disk, or platelet. 

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

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... 

---