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Crystallization Processes 111

distribution of the product or intermediate material. Such
balances are not independent of those on mass and energy,
and their solution requires an independent expression for
nucleation kinetics.
In formulating a population balance, crystals are as-
sumed sufficiently numerous for the population distribu-
tion to be treated as a continuous function. One of the key
assumptions in the development of a simple population
balance is that all crystal properties, including mass (or
volume), surface area, and so forth are defined in terms of
a single crystal dimension referred to as the characteris-
tic length. For example, Eq. (19) relates the surface area
and volume of a single crystal to a characteristic length L.
In the simple treatment provided here, shape factors are
taken to be constants. These can be determined by simple
measurements or estimated if the crystal shape is simple
and known—for example, for a cube k area = 6 and k vol = 1.
The beginning point for any balance is the following
statement:

input + generation − output − consumption
= accumulation (44)
FIGURE 15 Oslo crystallizer.
where each of the terms may be expressed as a rate or an
of mother liquor and crystals of the desired product. The amount. In a population balance, the number of entities
crystals are forced to move in a specific direction by grav- (such as crystals) is the balanced quantity and each of
ity or rotating blades. As they flow towards the appropriate the terms has dimensions of number of crystals per unit
end of the crystallizer, the crystals encounter a heated re- time for a differential balance or number of crystals for an
gion and are melted. A portion of the melt is removed integral balance. The principles involved in formulating
as product, while the remainder flows countercurrently to a balance are outlined in the following sections, and they
the crystals, thereby providing some refining and remov- provide guidance in developing corresponding balances
ing impure adhering liquid. for systems whose configurations do not conform to those
In a second method of operation, the feed material is cir- described here.
culated through a bank of tubes, each of which has a diam-
eter of up to about 8 cm. The walls of the tubes are cooled,
A. Perfectly Mixed, Continuous Crystallizers
and material crystallizes on them throughout a fixed op-
erating period. At the end of that period, the remaining The balance equation must be constructed for a control
liquid is sent to a holding tank for further processing, and volume, which for a perfectly mixed crystallizer may be
then the tubes are heated slowly to cause partial melting assumed to be the total volume of the crystallizer V T .
of the adhering solids. This step is known as “sweating,” Then, a balance on the number of crystals in any size
and the impure “sweated” liquid produced is removed range (say, L 1 to L 2 = L 1 + L) must account for crys-
from the crystallizer and held for further processing. Fi- tals that enter and leave that size range by: (1) convective
nally, the product is obtained by adding additional heat to flow, (2) crystal growth, (3) crystal agglomeration, and
the tubes and melting the remaining adhering solids. The (4) crystal breakage. Agglomeration and breakage can
actual sequencing of these steps and the reprocessing of be detected through careful inspection of product parti-
residual and sweated liquids may be quite complicated. cles, and they can be quite significant in some processes.
For simplicity, however, they will be assumed negligible
in the present analysis. The rate of crystal growth G will
VI. POPULATION BALANCES AND be defined as in Eq. (26); i.e., the rate of change of the
CRYSTAL SIZE DISTRIBUTIONS characteristic crystal dimension L:

A balance on the population of crystals in a crystallizer dL
G =
can be used to relate process variables to the crystal size dt