Particle shape, factors defining

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. 

An equivalent sphere is defined as the sphere with the same value of one of the above measures. The commonest referent is the volume-equivalent sphere, which many authors describe as the equivalent sphere without further definition. The particle shape factor is defined as the ratio of another measure from the above list to the corresponding value for the equivalent sphere. Of the many possible shape factors, those which have proved most useful are described below. All shape factors are open to the criticism that a range of bodies with different forms may have the same shape factor, but this is inevitable if regular or complex shapes are to be described by a single parameter. 

One of the earliest particle shape factors used in the pharmaceutical industry was WadeU s true spheritcity, The true sphericity defines the proximity of the irregular particle measured to a perfect sphere and the relationship between the iiregular particles to the perfect sphere is given by ... 

Once an image has been segmented, the microstructural characteristics can be measured, for example ice crystal and air bubbles size. Figure 6.1 shows that the ice crystals and air bubbles are not spherical. Therefore, several different measurements of size can be made, for example the maximum diameter, the minimum diameter, or the equivalent circular diameter i.e. the diameter of a circle with the same area). The shape factor (defined by equation 6.3) is a measure of how far a particle is from being circular it takes a value of one for a sphere and becomes smaller as the particle becomes less spherical. 

The particle size enters the equations through the volume equivalent particle diameter, dp, and so-called surface shape factor, /, defined as... 

Particle shape factor. The particle shape factor is a dimensionless parameter. For non-hollow particles, the shape factor is defined as the ratio of the external surface area 1S5 of a spherical particle with the same particle volume to the actual surface area Sp of the particle. This is equivalent to the ratio of the particle diameter dp to the diameter dp of a spherical particle with the same volume, that is, = Ss/Sp = ds/dp, where d = 6Vp/Sp. The shape factor indicates how much a particle differs from a spherical one. For spherical particles, 1 for non-spherical particles, is less than 1. The particle shape factor can be calculated from the volume and external surface area. The volume and external surface area for particles with... 

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 ... 

The shape of an individual particle is expressed in terms of a shape factor, / This is independent of particle size, and bears connection with the major defining dimension of a... 

Similarly, a shape factor may defined which converts the surface area, 5, of a particle to its surface mean diameter, as ... 

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... 

Heywood (H5) proposed a widely used empirical parameter based on the projected profile of a particle. The volumetric shape factor is defined as... 

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... 

Heywood s volumetric shape factor k, defined in Chapter 2, can be estimated rapidly, even for irregular particles, using Eq. (2-2). Table 6.3 gives values for regular shapes and some natural particles. Heywood (H2, H3) suggested that k be employed to correlate drag and terminal velocity, using dj and the projected... 

He defined a shape factor f as the ratio of the average volume of all particles having a maximum linear dimension equal to the mesh size of a screen to that of a cube which will just pass the same screen, f = 1.00 for cubes and 0.524 for spheres. For most materials f 0.5. The particle size distribution factor g is the ratio of the upper size... 

Specific Shape-Factor—The shape-factor may be defined in another way. Let there be N particles of weight w, and density p. Then the diameter of the particles is... 

The fineness of the powder is characterized by a number (e.g., a diameter d). Particles, of course, will have different shapes so that there are different ways of defining a diameter. The technique for obtaining d, given above has been used microscopically. More conventional is the so-called surface mean diameter, which, is the diameter of a sphere that has the same surface area as the particle. The so-called single-particle volume mean diameter is possible if there are instruments that can measure the volume of an odd-shaped particle. If the shape factor is indepen-... 

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. 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

The d3mamic shape factor, k, is defined as the resistance of a particle to motion divided by the resistance of a sphere to motion when the particle and the equivalent sphere have the same volume. When the particle is settling under laminar flow, k is given by [4]... 

To allow for the influence of various particle shapes and size distributions within a defined sieve fraction, in lay-out calculations it is customary to employ an effective particle diameter, deff, as nominal size [32]. The diameter de is defined as the ratio of equivalent diameter A and a form factor j/. A is equal to the diameter of a sphere with a volume equal to the (average) volume of the particles, and ijj is the average ratio of the particle surface to the surface of a sphere of equal volume. 

Qualitative terms [10] may be used to give some indication of particle shape but these are of limited use as a measure of particle properties ( Fable 2.4). Such general terms are inadequate for the determination of shape factors that can be incorporated as parameters into equations concerning particle properties where shape is involved as a factor. In order to do this, it is necessary to be able to measure and define shape quantitatively. 

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