Page 215 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GAE/LSK P2: FLV Final Pages
Encyclopedia of Physical Science and Technology EN004D-ID159 June 8, 2001 15:47

112 Crystallization Processes

Then, If the crystallizer has a clear feed, growth is invariant, but
if the magma volume V T is allowed to vary, the population
(45)
growth rate into the size range = V T (Gn) L 1
balance gives:
growth rate out of the size range = V T (Gn) L 2 (46) ∂n ∂(nG) ∂(ln V T ) V out n
+ + n + = 0 (55)
∂t ∂L ∂t V T

L 2
ndL The system model that led to the development of the
removal rate of crystals in the size range = V out
last two equations is referred to as the mixed-suspension,
L 1
(47) mixed-product removal (MSMPR) crystallizer.
Under steady-state conditions, the rate at which crystals
L 2
n in dL
feed rate of crystals in the size range = V in are produced by nucleation must be equal to the difference
L 1
in rates at which crystals leave and enter the crystallizer.
(48) Accordingly, for a clear feed,
∂ L 2 ∞ 1 ∞
◦
◦
accumulation rate in the crystallizer = nV T dL V T B = V out ndL ⇒ B = ndL (56)
∂t τ
L 1 0 0
(49) For crystallizers following the constraints given above,
◦
◦
Substituting the terms from Eqs. (46) through (49) into B = n G (57)
Eq. (44) gives:
For a given set of crystallizer operating conditions, nu-

L 2 cleation and growth rates can be determined by measuring
+ V in n in dL
V T (Gn) L 1 the population density of crystals in a sample taken from
L 1
either the well-mixed zone of a crystallizer or the prod-
L 2 L 2
∂ uct stream flowing from that zone. Sample analyses are

= V T (nG) L 2 + V out ndL + nV T dL (50)
∂t
L 1 L 1 correlated with Eqs. (54) and (57), and nucleation and
growth rates are determined from those correlations. The
Manipulation of this equation leads to
sample must be representative of the crystal population
∂(nG) V out n V in n in ∂n in the crystallizer (or leaving the well-mixed unit), and
+ − =− (51)
∂L V T V T ∂t experience with such measurements is invaluable in per-
forming this analysis properly. Figure 16 shows a plot of
Equation (51) may be used as a starting point for the
analysis of any crystallizer that has a well-mixed active
volume and for which crystal breakage and agglomera-
tion can be ignored. As an illustration of how the equation
can be simplified to fit specific system behavior, suppose
the feed to the crystallizer is free of crystals and that it is
operating at steady state. Then, n in = 0 and ∂n/∂t = 0.
Now suppose that the crystal growth is invariant with
size and time; in other words, assume the system follows
the McCabe L law and therefore exhibits neither size-
dependent growth nor growth-rate dispersion. Then,
∂(nG) ∂n
= G (52)
∂L ∂L
Defining a mean residence time τ = V T /V out and applying
the aforementioned restrictions leads to
dn n
G + = 0 (53)
dL τ
(τ is often referred to as the drawdown time to reflect the
fact that it is the time required to empty the contents from
the crystallizer.) Integrating Eq. (53) with the boundary
◦
condition n = n at L = 0:

L
◦ FIGURE 16 Typical population density plot from perfectly mixed,
n = n exp − (54)
Gτ continuous crystallizer.