Page 220 - 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

Crystallization Processes 117

possibilities in selecting cooling profiles, T (t), or vapor The identification of an initial condition associated with
generation profiles, V (t), or time dependencies of precip- the crystal size distribution is very difficult. If the system
itant or nonsolvent addition rates. is seeded, the initial condition becomes:
For illustrative purposes, consider that the protocol for
¯ n(L, 0) = ¯ n seed (L) (82)
a cooling crystallizer can involve either natural cooling—
cooling resulting from exposure of the crystallizer con-
where ¯ n seed is the population density function of the seed
tents to a heat sink without intervention of a control
crystals. If the system is unseeded, the nuclei often are
system—or manipulation of cooling to reduce the sys-
assumed to form at size zero.
tem temperature in a specific manner. In both cases, the
The rate of cooling, or evaporation, or addition of dilu-
instantaneous heat-transfer rate is given by:
ent required to maintain specified conditions in a batch
crystallizer often can be determined from a population-
Q = UA(T − T sink ) (79)
balance model. Moments of the population density func-
where U is a heat-transfer coefficient, A is the area avail- tion are used in the development of equations relating the
able for heat transfer, T is the temperature of the magma, control variable to time. As defined earlier, the moments
and T sink is the temperature of the cooling fluid. If T sink is are
a constant, the maximum heat-transfer rate and, therefore, ∞ i
the highest rate at which supersaturation is generated are m i = L ¯ ndL (83)
0
at the beginning of the process. This protocol can lead to
excessive primary nucleation and the formation of encrus- Recognizing that the zeroth moment is the total number
tations on the heat-transfer surfaces. of crystals in the system, it can be shown that:
The objective of programmed cooling is to control the
dm 0 dN T
◦
◦
rate at which the magma temperature is reduced so that su- = ¯ n G = B = (84)
dt dt
persaturation remains constant at some prescribed value,
usually below the metastable limit associated with pri- Moment transformation of Eq. (80) leads to the following
mary nucleation. Typically the batch is cooled slowly relationship:
at the beginning of the cycle and more rapidly at the
∂m j
end. An analysis that supports this approach is presented = jGm j−1 (85)
∂t
later. In size-optimal cooling, the objective is to vary
the cooling rate so that the supersaturation in the crys- Combining Eq. (85) with the relationships of moments
tallizer is adjusted to produce an optimal crystal size to distribution properties developed in Section VI.A for
distribution. j = 1, 2, 3 gives:
Protocols similar to those described above for cool-
dm 1 m 0 =N T dL T
ing crystallizers exist for crystallization modes involving = Gm 0 −→ = GN T (86)
dt dt
evaporation of solvent and the rate at which a non solvent
m 1 =L T dA T
or a reactant is added to a crystallizer. dm 2 = 2Gm 1 −→ = 2Gk area L T (87)
A population balance can be used to follow the devel- dt dt

opment of a crystal size distribution in batch crystallizer, dm 3 k area m 2 =A T dM T k vol
= 3Gm 2 −→ = 3Gρ A T (88)
but both the mathematics and physical phenomena being dt dt k area
modeled are more complex than for continuous systems
where N T is the total number of crystals, L T is total crystal
at steady state. The balance often utilizes the population
length, A T is total surface area of the crystals, and M T is
density defined in terms of the total crystallizer volume,
the total mass of crystals in the crystallizer. In addition to a
rather than on a specific basis: ¯ n = nV T . Accordingly, the
population balance, a solute balance must also be satisfied:
general population balance given by Eq. (51) can be mod-
ified for a batch crystallizer to give: d(V T C) dM T
+ = 0 (89)
∂(nV T ) ∂(GnV T ) ∂ ¯ n ∂(G ¯ n) dt dt
+ = + = 0 (80)
∂t ∂L ∂t ∂L where V T is the total volume of the system, and C is solute
concentration in the solution.
The solution to this equation requires both an initial con-
The above equations can be applied to any batch crys-
dition (¯ n at t = 0) and a boundary condition (usually ob-
tallization process, regardless of the mode by which su-
tained by assuming that crystals are formed at zero size):
persaturation is generated. For example, suppose a model
B (t) is needed to guide the operation of a seeded batch crys-
◦
◦
¯ n(0, t) = ¯ n (t) = (81)
G(0, t) tallizer so that solvent is evaporated at a rate that gives