Volumetric shape factor

An MSMPR crystalliser operates with a steady nucleation rate of n = 1013/m4, a growth rate, Gd = 10-8 m/s and a mixed-product removal rate, based on clear liquor of 0.00017 m3/s. The volume of the vessel, again based on clear liquor, is 4 m3, the crystal density is 2660 kg/m3 and the volumetric shape factor is 0.7. Determine ... 

Table IV shows whole data relating to CSD and j, p values, the latters of which were mainly estimated by the relation between Sw-Sn (Figure 2), because that almost the same values were obtained with different methods, as shown in Table III. The data in Table IV are arranged in order of volumetric shape factor of crystals, fy. fv is unity...

The weight basis probability density function of crystals in exit stream, fw(L) is derived from in a similar way. Multiplying both sides of Equation a-6 by PcfyL, where rc and fy are the density and volumetric shape factor of crystals, respectively,... 

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

Even if an estimate for is available, the volumetric shape factor can only be evaluated if the particle volume is known, and this may not be readily available for naturally occurring particles, or if a distribution of particle sizes or shapes is present. Heywood (H4) suggested that k may be estimated from the corresponding value, k, of an isometric particle of similar form by the relationship... 

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... 

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,... 

The last column is of the summation Wj/Lj at corresponding values of crystal length L. The volumetric shape factor is av =0.866, the density is 1.5g/mL, and the mean residence time was 2.0 hr. The linear growth rate G and the mideation rate B° will be found. 

The magma density AfT (mass of crystals per unit volume of slurry or liquor) is the product of the crystal density, the volumetric shape factor, and the third moment of the population density function ... 

The following values of sand volumetric shape factors have been reported angular = 0.64, sharp = 0.77, worn = 0.86, and spherical = 0.52. 

A 0.65-m deep filter bed has a uniformly sized sand with diameter of 0.45 mm, specific gravity of 2.65, and a volumetric shape factor of 0.87. If a hydrostatic head of 2.3 m is maintained over the bed, determine the backwash flow rate at 20°C. Assume the porosity is 0.4. 

Here, p is the density of the solid and k is the volumetric shape factor that relates the volume of a particle to its characteristic... 

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