Volume, shape-factor

The two density functions can be related through a simple shape factor as follows. Suppose the mass of a single crystal is and the characteristic dimension of that crystal is E. If the crystal is from a population in which shape is not a function of size, then the mass of any crystal from that population is related to characteristic dimension by a volume shape factor ... 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

Taking the crystal density as 1770 kg/m and the volume shape factor,/v, as 0.47 calculate... 

Take the crystal volume shape factor to be 0.4 and the crystal density to be 2600 kg/m. Solution... 

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. 

Volume shape factor. Figure 3 shows SEM microphotograph of the typical crystal of Mg(0H)2 obtained for series I. As the crystal form is composed of disklike units and the crystal structure of Mg(0H)2 is Cdl2 type, the standard unit of Mg(0H)2 crystal is considered to be a disk. The ratio of the length L to thickness D of the disk of crystal unit was measured for each experimental condition, so that it was found that L/D was nearly constant at 6.4. The crystal volumes were calculated for... 

Figure 4 Relation between volume shape factor and unit number...

More commonly, a generalised volume shape factor K is used to relate particle volume V to the cube of particle size... 

The left-hand side is a compliance parameter or bulk strain normalized for the centrifugal, gravitational, and inertial stresses exerted on the material during spheronization. The volume shape factor of pellets became closer to that of a sphere as the compliance of the extrudate increased, when measured in a creep test (56). 

Transforming a mass distribution to a number distribution, or vice versa, requires a relationship between the measured and desired quantities. The mass of a single crystal, mciys, is related to crystal size by the volume shape factor, feyoi (see Eq. (19)) ... 

Table 8—Values of Volume Shape-Factor for Different Materials...

Number of Particles in a Packing—From the average diameter d of the particles, it is possible to compute the number of particles present in a unit-volume of the packing. The solid material in a unit-volume is clearly (1 — ), where represents the voids. Hence, if a9 represents the volume shape-factor and N the number of particles per unit-volume (see Chapter 3), then... 

Volume-shape factor of particulate matter. Probable errors in the computation. Ind. Eng. Chem., Anal. Ed., 11 545-546. 

Calculate the total volume of crystals per volume of slurry. The volume of solids per volume of slurry VT = kv nV dL = kvM u where kv is a so-called volume shape factor (see below) and... 

M3 is the third moment of the distribution, i.e., the value of the integral in step 1 when j = 3. Some volume shape factors are as follows ... 

Calculate the third moment of the crystal size distribution. The third moment M3 of the crystal size distribution equals MT/pskv, where M-, is the total weight of the crystals and kv is the volume shape factor see step 4 of Example 10.6. In this case, M3 = 29.87/[1.77(l)(100 cc)] = 0.169 cm3 solids per cubic centimeter of slurry. 

To use the previous equations for non-spherical particles, the diameter d, must be the diameter of the equivalent spherical particle. The volume of the equivalent spherical particle Vs = r( ) must be equal to the volume of the non-spherical particle Vp = jidl, where is a volume shape factor. Expressing the equality and solving for the equivalent spherical diameter d produces... 

Volume shape factor also ratio of dissolved air saturation value in waste-... 

Oy = volume shape factor = volume of crystal/(length) ... 

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