CRYSTALLIZER DESIGN CRYSTAL SIZE DISTRIBUTION CSD Crystal size

CRYSTALLIZER DESIGN

CRYSTAL SIZE DISTRIBUTION (CSD) § Crystal size distribution (CSD) is measured with a series of standard screens. § The size of a crystal is taken to be the average of the screen openings of successive sizes that just pass and just retain the crystal. § The cumulative wt % either greater or less than a specified screen opening is recorded. § Typical size distribution data on the following figure are plotted in two cumulative modes, greater than or less than, and as differential polygons or histograms.

Cumulative wt % retained or passed, against sieve aperture

THE PROCESS OF CRYSTALLIZATION CONDITIONS OF PRECIPITATION § evaporation of the solvent § changing to a temperature at which the solubility is lower § chemical reaction between separately soluble gases or liquids § induced by additives (salting out)

SUPERSATURATION

GROWTH RATES NUCLEATION Nucleation rates are measured by counting the numbers of crystals formed over periods of time. The nucleation rate depends on the extent of supersaturation (2) Values of the exponent b have been found to range from 2 to 9, but have not been correlated to be of quantitative value for prediction.

CRYSTAL GROWTH The growth rates of crystals depend on their instantaneous surface and the linear velocity of solution past the surface as well as the extent of supersaturation, and are thus represented by the equation (3) Values of the exponent (g) have been found of the order of 1. 5, but again no correlation of direct use to the design of crystallizers has been achieved.

In laboratory and commercial crystallizations, large crystals of more or less uniform size are desirable. This condition is favored by operating at relatively low extents of supersaturation. The optimum extent of supersaturation is strictly a matter for direct experimentation in each case. As a rough guide, the data for allowable subcooling and corresponding supersaturation of the Table 1 may serve. Since the recommended values are one-half the maxima shown, it appears that most crystallizations under commercial conditions should operate with less than about 2 C subcooling or the corresponding supersaturation.

Table 1. Maximum Allowable Supercooling T ( C) and Corresponding Supersaturation C (g/100 g water) at 25 C

Growth rates of crystals also must be measured in the laboratory or pilot plant, although the suitable condition may be expressed simply as a residence time. Table 2 gives some growth rate data at several temperatures and several extents of supersaturation for each substance. In most instances the recommended supersaturation measured as the ratio of operating to saturation concentrations is less than 1. 1. It may be noted that at a typical rate of increase of diameter of 10– 7 m/sec, the units used in this table, the time required for an increase of 1 mm is 2. 8 hr.

Table 2. Mean Overall Growth Rates of Crystals (m/sec) at Each Face

Batch crystallizers often are seeded with small crystals of a known range of sizes. The resulting CSD for a given overall weight gain can be estimated by an approximate relation known as the Mc. Cabe Delta-L Law, which states that each original crystal grows by the same amount L: 1. All crystals have the same shape. 2. They grow invariantly, i. e. the growth rate is independent of crystal size. 3. Supersaturation is constant throughout the crystallizer. 4. No nucleation occurs. 5. No size classification occurs in the crystallizer 6. The relative velocity between crystals and liquor remains constant.

The relation between the relative masses of the original and final size distributions is given in terms of the incremental L by (4) When R is specified, L is found by trial, and then the size distribution is evaluated

EXAMPLE Seed crystals with this size distribution are charged to a batch crystallizer L 0, length (mm) w (wt fraction) 0. 251 0. 178 0. 127 0. 089 0. 064 0. 09 0. 26 0. 45 0. 16 0. 04 On the basis of the Mc. Cabe L law, find: a. The length increment that will result in a 20 -fold increase in mass of the crystals. b. The mass growth corresponding to the maximum crystal length of 1. 0 mm.

SOLUTION a. When L is the increment in crystal length, the mass ratio is By trial, the value of L = 0. 2804 mm b. When Lmax = 1 L = 1 – 0. 251 = 0. 749 0. 003935

THE IDEAL STIRRED TANK All continuous crystallizers are operated with some degree of mixing, supplied by internal agitators or by pumparound MSMPR (mixed suspension mixed product removal). By analogy with the terminology of chemical reactors it could be called CSTC (continuous stirred tank crystallizer). Several such tanks in series would be called a CSTC battery. A large number of tanks in series would approach plug flow, but the crystal size distribution still would not be uniform if nucleation continued along the length of the crystallizer.

(a) The single stage CSTC. (b) Multistage battery with overall residence time

THE POPULATION BALANCE The crystal population density, n (number of crystals per unit size per unit volume of system) is defined as: (5) Where N is the number of crystals in the size range L per unit volume. The value of n depends on the value of L at which the interval d. L is taken, i. e. n is a function of L. The number of crystals in the size range L 1 to L 2 is thus given by: (6)

Application of the population balance is most easily demonstrated with reference to the case of the continuously operated MSMPR crystallizer assuming: 1. Steady-state operation. 2. No crystals in the feed stream. 3. all crystals of the same shape, characterized by a chosen linear dimansion L. 4. No break-down of crystals by attrition. 5. Crystal growth rate dependent of crystal size.

A continuous MSMPR crystallizer

A population balance (input = output) in a system of volume V for a time interval t and size range L = L 2 – L 1 is (7) where Q : volumetric feed and discharge rate G : crystal growth rate (d. L/dt) n : average population density As L 0 (8)

Defining the liquor and crystal mean residence time = V/Q and assuming the growth is independent of size ( L Law), i. e. d. G/d. L 0, then: (9) Upon integration: (10) where x is the ratio of crystal size of a crystal that has grown for a period to the residence time . is the concentration of crystals of zero length which are the nuclei; it also is called the zero size population density.

Equation (10) is the fundamental relationship between crystal size L and population density n characterizing the CSD. The quantity n 0 is the population density of nuclei (zero-sized crystals). A plot of log n vs. L should give a straight line with intercept at L = 0 equals to ln n 0 and a slope – 1/G. Therefore, if the residence time is known, the crystal growth rate, G, can be calculated.

The number nucleation rate, B, can be expressed as a function of the supersaturation, c: (2) The crystal growth rate G can be expressed in a similar manner: (3) as

The nucleation rate may be expressed in terms of the growth rate by (11) or (12) where (13) consequently (14) So a plot of log n 0 vs. log G should give a straight line of slope i -1 or a plot of log B vs. log G should give a line of slope i. Thus the kinetic order of nucleation, b, may be evaluated if the kinetic order of growth, g, is known.

Population plots characterizing the CSD and the nucleation and the growth kinetics for a continuous MSMPR crystallizer

The nucleation rate is (15) The number of crystals per unit volume is (16) The total mass of crystals per unit volume is (17) where is the volumetric shape factor c the crystal density

Accordingly, the number of crystals per unit mass is (18) The mass of crystals per unit volume with length less than L or with dimensionless residence time less than x is (19) The value of the integral is

This expression has a maximum value at x = 3 and the corresponding length LD is called the predominant length (modal size) The cumulative mass distribution is (20) and the differential mass distribution is (21) which has a maximum value of 0. 224 at x = 3.

The median size of the mass distribution is defined as the size of the crystal of which 50% by mass of the product from an MSMPR crystallizer is larger or smaller than the size. It is obtained by trial that x = 3. 672

1. 0 0. 9 0. 8 0. 7 0. 6 median 0. 5 0. 4 modal 0. 3 0. 2 0. 1 0. 0 0 1 2 3 4 5 6 7 8 9 10

CSD may be conveniently classified by the median size (LM) and the coefficient of variation (CV). The CV, which quantifies the size spread, is a statistical property related to the standard deviation of a Gaussian distribution and is normally expressed as a percentage by: (22) The higher the CV the broader the spread, CV = 0 denoting a mono-sized distribution.

The nucleation rate must generate one nucleus for every crystal present in the product. In terms of M’, the total mass rate of production of crystals (23)

EXAMPLE Analysis of Size Distribution Data Obtained in a CSTC Differential distribution data obtained from a continuous stirred tank crystallizer are tabulated w 0. 02 0. 05 0. 06 0. 08 0. 10 0. 13 0. 10 0. 09 0. 04 0. 03 L (mm) 0. 340 0. 430 0. 490 0. 580 0. 700 0. 820 1. 010 1. 160 1. 400 1. 650 1. 980 2. 370 The volumetric shape factor is av = 0. 866, the density is 1. 5 g/m. L, and the mean residence time was 2. 0 hr. Find growth rate G and the nucleation rate B 0.

SOLUTION The number of crystals per unit mass smaller than size L is (a) It is also related to the CSTC material balance by (b) Integration of eq. (b) is (c)

The number of crystals per unit mass smaller than size L is calculated using eq. (a): wi 0. 02 0. 05 0. 06 0. 08 0. 10 0. 13 0. 10 0. 09 0. 04 0. 03 Li 0. 34 0. 43 0. 49 0. 58 0. 70 0. 82 1. 01 1. 16 1. 40 1. 65 1. 98 2. 37 N 0. 3917 0. 8759 1. 2685 1. 5841 1. 8085 1. 9900 2. 0872 2. 1513 2. 1793 2. 1948 2. 1987 2. 2005

The relation of N and L is represented by eq. (c): According to eq. (c), there are two unknowns, i. e. G and n 0. We have a set of data of Ni and Li (see previous table). Thus both unknowns can be determined by regression: G = 0. 3515 mm/hr n 0 = 3. 4528 nuclei/mm 4 = 3. 4528 1012 nuclei/m 4 Accordingly: B 0 = G n 0 = 1. 2137 109 nuclei/m 4 hr

EXAMPLE Crystallization in an MSMPR with Specified Predominant Crystal Size Crystals of citric acid monohydrate are to made in an MSMPR at 30 C with predominant size LD = 0. 833 mm (20 mesh). The density is 1. 54 g/m. L, the shape factor av = 1 and the solubility is 39. 0 wt %. A supersaturation ratio C/C 0 = 1. 05 is to be used. Take the growth rate, G = , to be the value given in Table 2. For a mass production rate of 15 kg/hr of crystals, M’ = 15, find the nucleation rate and draw the differential mass distribution of the crystal.

SOLUTION The predominant size is related to other quantities by The cumulative and differential mass distributions are represented by eqs. (16) and (17), respectively.

0. 16 Differential distribution 0. 14 0. 12 0. 10 0. 08 0. 06 0. 04 0. 02 0. 00 0 0. 5 1 1. 5 2 Length, mm 2. 5 3 3. 5

BATCH CRYSTALLIZATION The population equation for an MSMPR crystallizer oparated at steady state with crystal growth rate independent of size, is written in eq. (8): (8) For batch crystallizer operated at unsteady state, the simple population balance relationship must be modified to: (26)

PROGRAMMED (CONTROLLED) COOLING

If natural cooling is employed, e. g. by passing coolant through the jacket or coils at a steady rate and constant inlet temperature, the temperature in the vessel will fall exponentially as shown in the following figure. Supersaturation increase very quickly in the early stages and peaks when nucleation occurs after exceeding the metastable limit. This sequence of events leads to an uncontrolled performance and results in small crystals with a wide CSD.

Natural, controlled (constant nucleation) and size-optimal cooling modes in a batch crystallizer. (a) temperature profile. (b) supersaturation profile