12.12. CRYSTALLIZATION THEORY

12.12A. Introduction

When crystallization occurs in a homogeneous mixture, a new solid phase is created. An understanding of the mechanisms by which crystals form and then grow is helpful in designing and operating crystallizers. Much experimental and theoretical work has been done to help understand crystallization.

The overall process of crystallization from a supersaturated solution is considered to consist of the basic steps of nucleus formation or nucleation and of crystal growth. If the solution is free of all solid particles, whether foreign or of the crystallizing substance, then nucleus formation must first occur, before crystal growth starts. New nuclei may continue to form while the nuclei present are growing. The driving force for the nucleation step as well as the growth step is supersaturation. These two steps do not occur in a saturated or undersaturated solution.

12.12B. Nucleation Theories

1. Solubility and crystal size

In a solution at a given temperature, the thermodynamic difference between small and large particles or crystals is that the small particle has a significant amount of surface energy per unit mass, whereas the large particle does not. As a result, the solubility of a small crystal, of less than micrometer size, is greater than that of a larger-size crystal.

The ordinary solubility applies only to moderately large crystals. Hence, in a supersaturated solution a small crystal can be in equilibrium. If a larger crystal is also present, the larger crystal will grow and the smaller crystal will dissolve. This effect of particle size is an important factor in nucleation.

2. Homogeneous nucleation

As a result of rapid random fluctuations of molecules in a homogeneous solution, the molecules may come together and associate into a cluster. This loose aggregate may quickly disappear. However, if the supersaturation is large enough, then enough particles may associate to form a nucleus which can grow and become oriented into a fixed lattice to form a crystal. The number of particles needed to form a stable nucleus ranges up to a few hundred. In solutions with high supersaturation and no agitation, homogeneous nucleation may be important.

3. Contact nucleation

This is due to two types of contact nucleation. In the first type, the formation of new nuclei by contact nucleation is due to interference of the contacting agent (walls of a container or agitator blades) with clusters of solute molecules becoming organized into the existing crystals and by actual breakage of microscopic growths on the surface of the growing crystals. In the second type, the formation of new nuclei occurs in collisions between crystals. The intensity of agitation is an important factor in contact nucleation.

This phenomenon of contact nucleation has been isolated and studied experimentally (B9, C4) and a correlation has been developed. In practice, experimental data from an actual crystallizer are required for design.

4. Nucleation in commercial crystallizers

In commercial crystallizers, supersaturation is low and agitation is used to keep the crystals suspended. At low supersaturation the crystal growth rate is at the optimum for more-uniform crystals. The predominant mechanism is contact nucleation. Homogeneous nucleation is largely absent because of the agitation and low supersaturation.

12.12C. Rate of Crystal Growth and ΔL Law

1. Rate of crystal growth and growth coefficients

The rate of growth of a crystal face is the distance moved per unit time in a direction that is perpendicular to the face. Crystal growth is a layer-by-layer process, and since growth can occur only at the outer face of the crystal, the solute material must be transported to that face from the bulk of the solution. The solute molecules reach the face by diffusion through the liquid phase. The usual mass-transfer coefficient ky applies in this case. At the surface the resistance to integration of the molecules into the space lattice at the face must be considered. This reaction at the surface occurs at a finite rate, and the overall process consists of two resistances in series. The solution must be supersaturated for the diffusion and interfacial steps to proceed.

The equation for mass transfer of solute A from the bulk solution of supersaturation concentration yA, mole fraction of A, to the surface where the concentration is, , is

Equation 12.12-1


where ky is the mass-transfer coefficient in kg mol/sm2mol frac, is rate in kg mol A/s, and Ai is area in m2 of surface i. Assuming that the rate of reaction at the crystal surface is also dependent on the concentration difference,

Equation 12.12-2


where ks is a surface-reaction coefficient in kg mol/s · m2 · mol frac and yAe is the saturation concentration. Combining Eqs. (12.12-1) and (12.12-2),

Equation 12.12-3


where K is the overall transfer coefficient.

The mass-transfer coefficient ky can be predicted by methods given in Section 7.4 for convective mass-transfer coefficients. When the mass-transfer coefficient ky is very large, the surface reaction is controlling and 1/ky is negligible. Conversely, when the mass-transfer coefficient is very small, diffusional resistance is controlling. Surface-reaction coefficients and overall transfer coefficients have been measured and reported for a number of systems (B4, H2, P3, V1). Much of the information in the literature is not directly applicable, because the conditions of measurement differ greatly from those in a commercial crystallizer. Also, the velocities and the level of supersaturation in a system are difficult to determine, and vary with position of the circulating magma in the crystallizer.

2. The ΔL law of crystal growth.

McCabe (M1) has shown that all crystals that are geometrically similar and of the same material in the same solution grow at the same rate. Growth is measured as the increase in length ΔL, in mm, in linear dimension of one crystal. This increase in length is for geometrically corresponding distances on all crystals. This increase is independent of the initial size of the initial crystals, provided that all the crystals are subject to the same environmental conditions. This law follows from Eq. (12.12-3), where the overall transfer coefficient is the same for each face of all crystals.

Mathematically, this can be written

Equation 12.12-4


where Δt is time in h and growth rate G is a constant in mm/h. Hence, if D1 is the linear dimension of a given crystal at time t1 and D2 at time t2,

Equation 12.12-5


The total growth (D2D1) or ΔL is the same for all crystals.

The ΔL law fails in cases where the crystals are given any different treatment based on size. It has been found to hold for many materials, particularly when the crystals are under 50 mesh in size (0.3 mm). Even though this law is not applicable in all cases, it is reasonably accurate in many situations.

12.12D. Particle-Size Distribution of Crystals

An important factor in the design of crystallization equipment is the expected particle-size distribution of the crystals obtained. Usually, the dried crystals are screened to determine the particle sizes. The percent retained on different-sized screens is recorded. The screens or sieves used are the Tyler standard screens, whose sieve or clear openings in mm are given in Appendix A.5-3.

The data are plotted as particle diameter (sieve opening in screen) in mm versus the cumulative percent retained at that size on arithmetic probability paper. Data for urea particles from a typical crystallizer (B5) are shown in Fig. 12.12-1. Many types of such data will show an approximate straight line for a large portion of the plot.

Figure 12.12-1. Typical particle-size distribution from a crystal-lizer. [From R. C. Bennett and M. Van Buren, Chem. Eng. Progr. Symp., 65(95), 46 (1969).]


A common parameter used to characterize the size distribution is the coefficient of variation, CV, as a percent:

Equation 12.12-6


where PD16% is the particle diameter at 16 percent retained. By giving the coefficient of variation and mean particle diameter, a description of the particle-size distribution is obtained if the line is approximately straight between 90 and 10%. For a product removed from a mixed-suspension crystallizer, the CV value is about 50% (R1). In a mixed-suspension system, the crystallizer is at steady state and contains a well-mixed-suspension magma with no product classification and no solids entering with the feed.

12.12E. Model for Mixed Suspension–Mixed Product Removal Crystallizer

1. Introduction and model assumptions

This model will be derived for a mixed suspension–mixed product removal (MSMPR) crystallizer, which is by far the most important type of crystallizer in use in industry today. Conditions assumed are as follows: steady state, suspension completely mixed, no product classification, uniform concentration, no crystals in feed, and the ΔL law of crystal growth applies. All continuous crystallizers have mixing by an agitator or by pump around. In ideal mixing, the effluent composition is the same as in the vessel. This is similar to a CSTR continuous-stirred tank reactor.

2. Crystal population-density function n

To analyze data from a crystallizer, an overall theory must consider combining the effects of nucleation rate, growth rate, and material balance. Randolph and Larson (R1, R2, R3) derived such a model. They plotted the total cumulative number of crystals N per unit volume of suspension (usually 1 L) of size L and smaller versus this size L, as in Fig. 12.12-2. The slope dN/dL of this line is defined as the crystal population density n:

Equation 12.12-7


Figure 12.12-2. Determination of population density n of crystals.


where n is the number of crystals/(L · mm). This characterizes the nuclear growth rate of this crystallizer. For this model a relation between population density n and size L is desired.

This population density is obtained experimentally by screen analysis of the total crystal content of a given volume, such as 1.0 L of magma suspension. Each sieve fraction by weight is obtained by collection between two closely spaced and adjacent screens. Then, Lav = (L1 + L2)/2, where L1 and L2 are the openings in mm in the two adjacent screens. Also, ΔL = (L1L2), where L1 is the size opening of the upper screen. Then the volume of a particle νp is

Equation 12.12-8


where νp is mm3/particle and a is a constant shape factor. Knowing the total weight of the crystals in this fraction, the density ρ in g/mm3, and the weight of each crystal, which is ρνp, the total number of crystals ΔN is obtained for the size range ΔL.

Then, rewriting Eq. (12.12-7) for this ΔL size,

Equation 12.12-9


This method permits the calculation of n for each fraction collected in the screen analysis with an average size of Lav mm.

3. Population material balance

To make a population material balance, in Δt time, Δn ΔL crystals are withdrawn. Since the effluent composition in the outflow of Q L/h is the same as that in the crystallizer, then the ratio (Δn ΔL)/(n ΔL), or fraction of particles withdrawn during Δt time, is the same as the volume ratio Q Δt withdrawn divided by the total volume V of the crystallizer. Hence,

Equation 12.12-10


During this time period Δt, the growth ΔL of a crystal is

Equation 12.12-11


where G is growth rate in mm/h. Combining Eqs. (12.12-10) and (12.12-11),

Equation 12.12-12


Letting ΔL → 0, Δn → 0, and integrating,

Equation 12.12-13


Equation 12.12-14


where n0 is population of nuclei when L = 0, n is population when the size is L, and V/Q is τ the total retention or holdup time in h in the crystallizer. A plot of Eq. (12.12-14) of lnn versus L is a straight line with intercept n0 and slope −1/. If the line is not straight, this could be an indication of the violation of the ΔL law. These calculations are well suited for being done with a computer spreadsheet.

4. Average particle size and nucleation rate

Further derivation of the population approach results in the equation for the average size La in mm of the mass distribution:

Equation 12.12-15


Here, 50% of the mass of the product is smaller or larger in size than this value. Also, the predominant particle size is given as

Equation 12.12-16


This predominant particle size means more of the mass is in this differential size interval than in any other size interval.

Another relation can be obtained from this model, which relates the nucleation rate B0 to the value of the zero-size particle population density n0 and the growth rate G. For the condition L → 0, the limit of dN/dt (nucleation rate) can be written as

Equation 12.12-17


However, when L → 0, the slope dL/dt = G, the slope dN/dL = n0, and B0 = dN/dt. Hence,

Equation 12.12-18


where B0 is nucleation rate in number of nuclei/h · L.

5. Prediction of cumulative weight fraction obtained

The population equation can be used to perform a reverse calculation when only values of G and τ are known. The following equation has been derived from the population density function (L4):

Equation 12.12-19


where x = L/, and (1 − Wf) is the cumulative wt fraction at opening L mm, which is in the same form of results as from a screen analysis.

6. Use of population-model approach for process design

The data for experimental crystal growth rate G and nucleation rate B0 obtained by means of the population material balance are for one given set of conditions in the crystallizer. Then additional experiments can be conducted to determine the effect of residence time τ (production rate) and, possibly, pump-around rate (mixing) on B0 and G. This is done until the desired distribution Wf from Eq. (12.12-19) is achieved. Or, in some cases, the goal may be for a desired dominant size Ld to be obtained. In many cases, as the residence time τ is increased, the experimental growth rate G decreases. Additional useful references are (B6, C2, C4, L5, M2, P1, R4, S2).

EXAMPLE 12.12-1. Growth and Nucleation Rates in MSMPR Crystallizer

Calculate the population density and nucleation growth rates for crystal samples of urea from a screen analysis. The slurry density (g of crystals) is 450 g/liter, the crystal shape factor a is 1.00, the crystal density ρ is 1.335 g/cm3, and the residence time τ is 3.38 h. The screen analysis from reference (B5) is as follows:

MeshWt %MeshWt %
−14, +204.4−48, +6515.5
−20, +2814.4−65, +1007.4
−28, +3524.2−1002.5
−35, +4831.6  

Solution: The data above are tabulated in Table 12.12-1 using data from Appendix A.5-3. The value of L is the screen opening. For the 14–20 mesh portion, Lav = (1.168 + 0.833)/2 = 1.001 mm and ΔL = 1.168 − 0.833 = 0.335 mm. For Lav = 1.001 mm, using Eq. (12.12-8), νp = = 1.00(1.001)3 = 1.003 mm3/particle. The density ρ = 1.335 g/cm3 = 1.335 × 103 g/mm3 and the mass/particle = ρνp = 1.335 × 103 (1.003) = 1.339 × 103 g. The total mass of crystals = (450 g/L)(0.044 wt frac).

Table 12.12-1. Data and Calculations for Example 12.12-1
MeshL (mm)MeshL (mm)Lav (mm)ΔL (mm)ln nwt %
141.168200.8331.0010.33510.6954.4
200.833280.5890.7110.24413.22414.4
280.589350.4170.5030.17215.13124.2
350.417480.2950.3560.12216.77831.6
480.295650.2080.2520.08717.44115.5
650.2081000.1470.1780.06118.0997.4

Using Eq. (12.12-9) to calculate n for the 14–20 mesh size range,


Then ln n = ln 4.414 × 104 = 10.695. For the 20–28 mesh size range, Lav = 0.711/mm, and ΔL = 0.244. Then,


Then ln n = 13.224. Other values are calculated in a similar manner using a computer spreadsheet and are given in Table 12.12-1.

The ln n is plotted versus L in Fig. 12.12-3. The equation of this line is Eq. (12.12-14), where the slope is −9.12 and the intercept = 19.79:

Figure 12.12-3. Plot of population density n versus length for Example 12.12-1.



The slope −9.12 = −1/ = −1/(G)(3.38). Hence, G = 0.03244 mm/h. The intercept ln n0 = 19.79. Hence, n0 = 3.933 × 108. From Eq. (12.12-18), the nucleation rate B0 is

B0 = Gn0 = 0.03244(3.933 ×108) = 1.26 × 107 nuclei/h · L

The average size is, from Eq. (12.12-15), La = 3.67 = 3.67(0.03244)(3.38) = 0.402 mm. The predominant size, from Eq. (12.12-16), is Ld = 3.00 = 3.00(0.03244)(3.38) = 0.329 mm.


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