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Crystallization Processes 97

and x L and x C . In addition, an energy balance must be where µ i is the chemical potential of solutei at the existing
satisfied: conditionsofthesystem,µ isthechemicalpotentialofthe
∗
i
solute equilibrated at the system conditions, and a i and a ∗
ˆ ˆ ˆ ˆ i
m F H F + Q = m V H V + m L H L + m C H C (9)
are activities of the solute at the system conditions and at
The specific enthalpies in the above equation can be deter- equilibrium, respectively. Less abstract definitions involv-
mined as described earlier, provided the temperatures of ing measurable system quantities are often used to approx-
the product streams are known. Evaporative cooling crys- imate supersaturation; these involve either temperature or
tallizers (described more completely in Section V) operate concentration (mass or moles of solute per unit volume or
at reduced pressure and may be considered adiabatic. In mass of solution or solvent) or mass or mole fraction of
such circumstances, Eq. (9) is modified by setting Q = 0. solute. Recommendations have been made that it is best to
As with many problems involving equilibrium relation- express concentration in terms of moles of solute per unit
ships and mass and energy balances, trial-and-error com- mass of solvent. For systems that form hydrates, the so-
putationsareofteninvolvedinsolvingEqs.(7)through(9). lute should include the water of hydration, and that water
should be deducted from the mass of solvent.
Consider, for example, a system at temperature T with
III. NUCLEATION AND GROWTH KINETICS a solute concentration C, and define the equilibrium tem-
∗
perature of a solution having a concentration C as T and
∗
The kinetics of crystallization have constituent phenom- the equilibrium concentration of a solution at T as C .
ena in crystal nucleation and growth. The rates at which These quantities may be used to define the following ap-
these occur are dependent on driving forces (usually ex- proximate expressions of supersaturation:
pressed as supersaturation), physical properties, and pro-
cess variables, but relationships between these quantities 1. The difference between the solute concentration and
and crystallization kinetics often are difficult to express the concentration at equilibrium, C i = C i − C ∗
i
quantitatively. As a result, empirical or qualitative links 2. For a solute whose solubility in a solvent increases
between a process variable and crystallization kinetics are with temperature, the difference between the
useful in providing guidance in crystallizer design and op- temperature at equilibrium and the system
∗
eration and in developing strategies for altering the prop- temperature, T = T − T
erties of crystalline products. 3. the supersaturation ratio, which is the ratio of the
Nucleation and growth can occur simultaneously in a solute concentration and the equilibrium
supersaturated environment, and the relative rates at which concentration, S i = C i /C ∗
i
these occur are primary determinants of the characteris- 4. The ratio of the difference between the solute
tics of the crystal size distribution; one way of influencing concentration and the equilibrium concentration to
product size distributions is through the control of vari- the equilibrium concentration, σ i = (C i − C )/C =
∗
∗
i i
ables such as supersaturation, temperature, and mixing S i − 1, which is known as relative supersaturation.
characteristics. Obviously, those factors that increase nu-
cleation rates relative to growth rates lead to a crystal size Anyoftheabovedefinitionsofsupersaturationcanbeused
distribution consisting of smaller crystals. In the discus- overamoderaterangeofsystemconditions,butasoutlined
sion that follows, an emphasis will be given to the general in the following paragraph, the only rigorous expression
effects of process variables on nucleation and growth, but is given by Eq. (10).
the present understanding of these phenomena does not The definitions of supersaturation ratio and relative su-
allow quantitative a priori prediction of the rates at which persaturation can be extended to any of the other variables
they occur. used in the definition of supersaturation. For example,
∗
= a i /a gives:
defining S a i i
A. Supersaturation µ i γ i C i
= ln (11)
= ln S a i
∗
RT γ C ∗
Supersaturation is the thermodynamic driving force for i i
both crystal nucleation and growth; and therefore, it is Therefore, for ideal solutions or for γ i ≈ γ ,
∗
i
the key variable in setting the mechanisms and rates by
µ i C i
which these processes occur. It is defined rigorously as the ≈ ln = ln S i (12)
RT C ∗
deviation of the system from thermodynamic equilibrium i
and is quantified in terms of chemical potential, Furthermore, for low supersaturations (say, S i < 1.1),
a i µ i
∗
µ i = µ i − µ = RT ln (10) ≈ S i − 1 = σ i (13)
i
a ∗ RT
i