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Crystallization Processes 113

typical population density data obtained from a crystal- are fixed, the crystal size distribution is determined in its
lizer meeting the stated assumptions. The slope of the plot entirety. In addition, such distributions have the following
of such data may be used to obtain the growth rate while characteristics:
the intercept can be used to estimate nucleation rate.
Many industrial crystallizers operate in a well-mixed or Mass density function (from Eq. (39)):
near well-mixed manner, and the equations derived above L
3
can be used to describe their performance. Also, the sim- m = ρk vol n L exp − (59)
◦
plicity of the equations describing an MSMPR crystallizer Gτ
make experimental equipment configured so as to meet Dominant crystal size (from Eq. (40)):
the assumptions leading to Eq. (54) useful in determining
L D = 3Gτ (60)
nucleation and growth kinetics. From a series of runs at
different operating conditions, correlations of nucleation Moments of n (from Eq. (37)):
and growth kinetics with appropriate process variables can i+1
◦
m i = i!n (Gτ) (61)
be obtained (see, for example, the discussions of Eqs. (18)
and (25)). The resulting correlations can then be used to Total number of crystals per unit volume:
guide either crystallizer scale-up or the development of an
∞

operating strategy for an existing crystallizer. N T = ndL = m 0 = n Gτ (62)
◦
It is often very difficult to measure supersaturation, 0
especially in systems that have high growth rates. Even Total length of crystals per unit volume:
though the supersaturation in such systems is so small
∞

that it can be neglected in writing a solute mass balance, L T = nL dL = m 1 = n (Gτ) 2 (63)
◦
it is important in setting nucleation and growth rates. In 0
such instances it is convenient to substitute growth rate Total surface area of crystals per unit volume:
for supersaturation by combining Eqs. (18) and (25). This
∞
2
gives: A T = k area nL dL = k area m 2 = 2k area n (Gτ) 3
◦
0
j
i
B = k nuc G M N k (58) (64)
◦
T
Total solids volume per unit volume:
The constant k nuc depends on process variables other than
∞
supersaturation, magma density, and intensity of mixing; 3 ◦ 4
V TS = k vol nL dL = k vol m 3 = 6k vol n (Gτ)
these include temperature and presence of impurities. If 0
sufficient data are available, these variables may be sepa- (65)

rated from the constant by adding more terms in a power- The coefficient of variation of the mass density
law correlation. k nuc is specific to the operating equipment function (from Eq. (42)) is 50%.

and not transferable from one equipment scale to another. The magma density M T (mass of crystals per unit
The system-specific constants i and j are obtainable from volume of slurry or liquor) is the product of the crystal
experimental data and may be used in scale-up, although density, the volumetric shape factor, and the third
j may vary considerably with mixing conditions. moment of the population density function:
As shown by Eq. (54), growth rate G can be obtained ◦ 4
M T = 6ρk vol n (Gτ) (66)
from the slope of a plot of the log of population density
against crystal size; nucleation rate B can be obtained System conditions often allow for the measurement of
◦
from the same data by using the relationship given by magma density, and in such cases is should be used as
Eq. (57), with n being the intercept of the population a constraint in evaluating nucleation and growth
◦
density plot. Nucleation rates obtained by these proce- kinetics from measured population densities. This
dures should be checked by comparison with values ob- approach is especially useful in instances of
tained from a mass balance (see the later discussion of uncertainty in the determination of population
Eq. (66)). densities from sieving or other particle sizing
The perfectly mixed crystallizer of the type described techniques.
in the preceding discussion is highly constrained. Alter-
ation of the characteristics of crystal size distributions
B. Preferential Removal of Crystals
produced by such systems can be accomplished only
by modifications of the nucleation and growth kinetics As indicated above, crystal size distributions produced in
of the system being crystallized. Indeed, examination of a perfectly mixed crystallizer are constrained by the na-
Eq. (54) shows that once nucleation and growth kinetics ture of the system. This is because both liquor and solids