Crystal number, area and mass

Crystal number, area and mass Mass distributions... 

Suppose that the batch crystallizer is seeded with a mass of crystals with a uniform size of Lseed. The number of seed crystals is Nseed, and, as the operation is to be free from nucleation, the number of crystals in the system remains the same as the number of seed crystals. The initial values of total crystal length, total crystal surface area, total... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

The population balance approach to measurement of nucleation and growth rates was presented by Randolph and Larson (1971, 1988). This methodology creates a transform called population density [n(L)], where L is the characteristic size of each particle, by differentiating the cumulative size distribution N versus L. shown in Fig. 4-22, where N is the cumulative number of crystals smaller than L. Per unit volume, the total number of particles, total surface area, and total volume/mass are calculated as the first, second, and third moments of this distribution. 

These two topics are treated together because they may both be considered as dealing with solids that are incomplete in some way. Clusters may be as small as three atoms or may contain many thousands of atoms. The study of clusters, particularly metal clusters, is a very active field.The interest is two-fold one purpose is to learn how the properties of solids emerge as the number of units increases to infinity the other is because clusters are important in heterogeneous catalysis. They have very large ratios of surface area to mass and are much more reactive than large crystals. 

As discussed earlier, the analytical solutions for the CSD for a batch or semibatch crystallizer are difficult to obtain unless both the initial condition for the CSD and appropriate kinetic models for nucleation and growth are known. An example of such an analytical solution—simple yet not overly restrictive—was given by Nyvlt (1991). It is assumed that the process, in which both external seeding and nucleation take place, occurs at constant supersaturation (G = constant, Bq = constant) in an ideally mixed crystallizer. An additional assumption of size-independent growth allows one to rewrite the time-dependent moments, Eqs. (10.12)-(I0.15), in terms of the physical properties such as the total number (A), length (L), surface area A), and mass of... 

A theoretical simulation of the powder decomposition rate was attempted by L vov et al. [2, 3] in 1998 (Chapter 6). Later L vov and Ugolkov [4] continued these theoretical and experimental studies with dolomite crystals and powders of various particle sizes. Calculations performed for powders with varied numbers of layers n = 10 and 100), emittance coefficients (from 0.01 to 1), and residual air pressures in the reactor (10 and 10 bar) showed that the differences in the particle size and powder mass do not noticeably affect the temperature distribution and the effective number of powder layers Ue (see Sect. 6.3) decomposing at the same rate as does the surface layer. Furthermore, it was found that the decomposition rates of crystals and powders with the same external surface area and coefficients ranging from 0.01 to 0.3 should differ by a factor of no more than 2. 

It is an ordinary differential equation (time dependence only) where the zeroth moment (//q) is the total crystal number (per unit volume of suspension), the first (//i) the total crystal length (lined up end to end), the second (//i) is related to total surface area, the third p ) total volume (and hence mass) and so on. 

It was shown above that the total crystal number, surface area and the mass mean size are affected by the mean residence time and the rates of nucleation and crystal growth respectively. Since both these kinetic processes depend upon the working level of supersaturation which will itself depend on the amount of surface area available and crystal mass deposited, the question arises what will be the effect of a change in residence time on crystallizer performance Consider the idealized MSMPR crystallizer depicted in Figure 7.8. 

---