Page 221 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN004D-ID159 June 8, 2001 15:47

118 Crystallization Processes

a constant crystal growth rate G and no nucleation; in an appropriate dependence of system temperature on time.
other words, supersaturation is to be held constant and The result depends upon the relationship of solubility to
only those crystals added at the beginning of the run are in temperature. If that relationship is linear, the cooling rate
the crystallizer. Model development proceeds as follows: varies with time in a parabolic manner; i.e.,
combining the solute balance, Eq. (89), with Eq. (88),
dT 2
− = C 1 t + C 2 t + C 3 (100)
d(V T C) 3ρ A T k vol G dt
+ = 0 (90)
dt k area An approximation to the temperature–time relationship
Recognizing that the process specification requires C to that serves as a good starting point for establishing a fixed
be a constant and taking the derivative of Eq. (90): protocol is given by:
2 3
d V T k vol dA T t
C + 3ρ G = 0 (91) T = T 0 − (T 0 − T final ) (101)
dt 2 k area dt τ
where τ is the overall batch run time.
⇓ Eq. (87)
It is clear that stringent control of batch crystallizers
is critical to obtaining a desired crystal size distribution.
2
d V T 2
C + 6ρk vol G L T = 0 (92) It is also obvious that the development of a strategy for
dt 2 generating supersaturation can be aided by the types of
Taking the derivative of the last equation: modelingillustratedabove.However,theinitialconditions
in the models were based on properties of seed crystals
3
d V T 2 dL T
C + 6ρk vol G = 0 (93) added to the crystallizer. In operations without seeding,
dt 3 dt
initial conditions are determined from a model of primary
nucleation.
⇓ Eq. (86)
3
d V T 3 D. Effects of Anomalous Growth
C + 6ρk vol G N T = 0 (94)
dt 3
Throughout this section, crystals have been assumed to
Suppose that the batch crystallizer is seeded with a mass grow according to the McCabe L law. This has simpli-
¯
ofcrystalswithauniformsizeof L seed .Thenumberofseed fied the analyses of both continuous and batch crystal-
crystals is N seed , and, as the operation is to be free from lizers and, indeed, crystal growth often follows the L
nucleation, the number of crystals in the system remains law. However, as outlined in Section III, size-dependent
the same as the number of seed crystals. The initial val- growth and growth-rate dispersion contribute to devia-
ues of total crystal length, total crystal surface area, total tions from the models developed here. Both of these phe-
crystal mass, and system volume are nomena lead to similar results: In continuous, perfectly
¯ mixed crystallizers, the simple expression for population
L T (0) = N seed L seed (95)
density given by Eq. (54) is no longer valid. Both size-
A T (0) = k area N seed L ¯ 2 (96)
seed dependent growth and growth-rate dispersion due to the
¯ 3
M T (0) = ρk vol N seed L (97) existence of a random distribution of growth rates among
seed
crystals in a magma lead to curvature in plots of ln n vs.
(98)
V T (0) = V T0 L. Models for both causes of this behavior exist but are
considered beyond the scope of the present discussion. In
On integrating Eq. (94), the following dependence of sys-
batch crystallization, the effects of anomalous growth lead
tem volume on time can be obtained:
to a broadening of the distribution, as was illustrated in
3 2 ¯

C(V T0 − V T ) = k vol ρN seed (Gt) + 3(Gt) L seed Fig. 6.
¯ 2
+ 3(Gt)L
(99)
seed
E. Summary
Therefore, for the specified conditions, the evaporation
rate (−dV T /dt) is a parabolic (second-order) function of The discussion presented here has focused on the princi-
time, and the rate of heat input to the crystallizer must be ples associated with formulating a population balance and
controlled to match the conditions called for by Eq. (99). applying simplifying conditions associated with specific
If a cooling mode is used to generate supersaturation, an crystallizer configurations. The continuous and batch sys-
analysis similar to that given above can be used to derive tems used as examples were idealized so that the principles