Page 203 - 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

100 Crystallization Processes

provide a useful means of correlating nucleation kinetics Multicrystal-magma studies usually involve examina-
and using the resulting correlations in process analysis and tion of the rate of change of a characteristic crystal di-
control. The correlations generally take the form: mension or the rate of increase in the mass of crystals in
j
i
◦
B = k N σ M N k (18) a magma. The characteristic dimension in such analyses
T depends upon the method used in the determination of
where k N , i, j, k are positive parameters obtained from crystal size; for example, the second largest dimension is
data correlation, M T is the magma density (mass of solids measured by sieve analyses, while an equivalent spherical
per unit volume of slurry or solvent in the magma), and N diameter is determined by both electronic zone sensing
is the rotational velocity of the impeller or pump rotor. For and laser light scattering instruments. A relationship be-
convenience, either crystal growth rate or mean residence tween these two measured dimensions and between the
time, both of which are directly related to supersaturation, measured quantities and the actual crystal dimensions can
may be substituted for σ in Eq. (18). be derived from appropriate shape factors. Volume and
If primary nucleation dominates the process, i tends area shape factors are defined by the equations:
to larger values (say greater than 3), j and k approach 3 2
v crys = k vol L and a crys = k area L (19)
zero, and Eq. (18) approaches Eq. (17). Should crystal–
impeller and/or crystal–crystallizer impacts dominate, j where v crys and a crys are volume and area of a crystal,
approaches 1; on the other hand, if crystal–crystal contacts k vol and k area are volume and area shape factors, and L
dominate, j approaches 2. is the characteristic dimension of the crystal. Suppose an
The ease with which nuclei can be produced by con- equivalent spherical diameter L sphere is obtained from an
tact nucleation is a clear indication that this mechanism electronic zone-sensing instrument, and the actual dimen-
is dominant in many industrial crystallization operations. sions of the crystal are to be calculated. Assume for the
Research on this nucleation mechanism is continuing with sake of this example that the crystals have a cubic shape.
sphere
the objective of building an understanding of the phe- Let L cube be the edge length of the crystal and k vol and
nomenon that will allow its successful inclusion in models k cube be the volume shape factors for a sphere and a cube,
vol
describing commercial systems. respectively. Since the volume of the crystal is the same,
regardless of the arbitrarily defined characteristic dimen-
sion,
D. Fundamentals of Crystal Growth
sphere 3 cube 3
v crys = k L = k L (20)
Crystal growth rates may be expressed as (1) the linear vol sphere vol cube
advance rate of an individual crystal face, (2) the change sphere cube
Since k vol is π/6 and k vol is 1.0, the numerical rela-
in a characteristic dimension of a crystal, or (3) the rate of
tionship between L cube and L sphere is given by:
change in mass of a crystal or population of crystals. These
different expressions are related through crystal geometry; k sphere 1/3 1/3
π
vol
it is often convenient to use the method of measurement L cube = cube L sphere = L sphere (21)
k 6
as the basis of the growth rate expression or, in certain vol
instances, the method used to analyze a crystallization The rate of change of a crystal mass dm crys /dt can be
process will require that growth rate be defined in a spe- related to the rate of change in the crystal characteristic
cific way. For example, the use of a population balance dimension (dL/dt = G) by the equation:
to describe crystal size distribution requires that growth d ρk vol L 3

rate be defined as the rate of change of a characteristic dm crys = = 3ρk vol L 2 dL (22)
dimension. dt dt dt
Single-crystal growth kinetics involve the advance rate where ρ is crystal density. Since k area = a crys /L ,
2
of an individual crystal face normal to itself or the rate of
change in crystal size associated with exposure to a super- dm crys = 3ρ(k vol /k area )a crys G (23)
saturated solution. The advance rate of a single crystal face dt
can be quantified by observation of the face through a cali- At least two resistances contribute to the kinetics of
bratedeyepieceofanopticalmicroscope,whichallowsex- crystal growth. These resistances apply to (1) integration
aminationofthestructureoftheadvancingcrystalfaceand of the crystalline unit (e.g., solute molecules) into the crys-
isolation of surface-reaction kinetics from mass-transfer talsurface(i.e.,lattice),and(2)moleculardiffusionorbulk
kinetics (these phenomena will be discussed later). An transport of the unit from the surrounding solution to the
additional advantage of single-crystal systems is that it is crystalsurface.Asaspectsofmoleculardiffusionandmass
possible to examine crystal growth kinetics without inter- transfer are covered elsewhere, the current discussion will
ference from competing processes such as nucleation. focus only on surface incorporation.