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

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.

(2)bckB 1

The growth rates of crystals depend on their instantaneoussurface and the linear velocity of solution past the surface as well as the extent of supersaturation, and are thus represented by the equation

CRYSTAL GROWTH

(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.

gckG 2

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 2C subcooling or the corresponding supersaturation.

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

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 1mm 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 McCabe 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 crystallizer6. 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

3

0

30

ii

ii

LwLLw

R (4)

When R is specified, L is found by trial, and then the size distribution is evaluated

Seed crystals with this size distribution are charged to a batch crystallizer

L0, length (mm) 0.251 0.178 0.127 0.089 0.064w (wt fraction) 0.09 0.26 0.45 0.16 0.04

On the basis of the McCabe 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.

EXAMPLE

SOLUTION

a. When L is the increment in crystal length, the mass ratio is

20

003935.0

30

30

30

LLw

LwLLw

R ii

ii

ii

By trial, the value of L = 0.2804 mm

b. When Lmax = 1 L = 1 – 0.251 = 0.749

79.181

003935.0

30

30

30

LLw

Lw

LLwR ii

ii

ii

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

k

ciVQt

1

1

THE POPULATION BALANCE

The crystal population density, n (number of crystals per unit size per unit volume of system) is defined as:

ndLdN

LN

L

0lim

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 dL is taken, i.e. n is a function of L.

The number of crystals in the size range L1 to L2 is thus given by:

2

1

L

L

dLnN

(5)

(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 = L2 – L1 is

tLnQtVGntVGn 2211

where Q : volumetric feed and discharge rateG : crystal growth rate (dL/dt)n : average population density

As L 0

VQn

dLnGd

(7)

(8)

Defining the liquor and crystal mean residence time = V/Q and assuming the growth is independent of size (L Law), i.e. dG/dL 0, then:

Gn

dLdn

xnGL

nn

expexp 00

Upon integration:

where x is the ratio of crystal size of a crystal that has grown for a period to the residence time .

Ln

nL d

dlim

0

0

is the concentration of crystals of zero length which are the nuclei; it also is called the zero size population density.

(9)

(10)

Equation (10) is the fundamental relationship between crystal size L and population density n characterizing the CSD.

The quantity n0 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 n0 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:

b

L

ckdtdN

B

10

(2)

The crystal growth rate G can be expressed in a similar manner:

g

L

ckdtdL

G

20

asdtdL

dLdL

dtdL

LL

.00

(3)

The nucleation rate may be expressed in terms of the growth rate by

GnB 0

or iGkB 3

where gbi

consequently 140

iGkn

So a plot of log n0 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.

(11)

(12)

(13)

(14)

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

The nucleation rate is

0

00

0 limlim GndLdn

dtdL

dtdn

BLL

The number of crystals per unit volume is

GndLtGL

ndLnnc0

0

0

0

exp

40

0

03

0

6exp

GndLGL

nLdLmnm ccc

The total mass of crystals per unit volume is

where is the volumetric shape factorc the crystal density

(15)

(16)

(17)

Accordingly, the number of crystals per unit mass is

36

1

Gmn

cc

c

The mass of crystals per unit volume with length less than L or with dimensionless residence time less than x is

x

xc

L

L dxexnGdLmnm0

304

0

The value of the integral is

62

11632

0

3 xxxedxex x

xx

(18)

(19)

This expression has a maximum value at x = 3 and the corresponding length LD is called the predominant length (modal size)

GLD 3

The cumulative mass distribution is

6211

32 xxxe

mm x

c

Lm

and the differential mass distribution is

6

3 xm exdx

d

which has a maximum value of 0.224 at x = 3.

3tGL

x D

(20)

(21)

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.

%5062

1132

xx

xemm x

c

Lm

It is obtained by trial that x = 3.672

GLM 672.3

0 1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

dxd m

m

x

median

modal

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:

%50

%16%84100L

LLCV

The higher the CV the broader the spread, CV = 0 denoting a mono-sized distribution.

(22)

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

330 '5.1

6

''

prcvcvcc LaM

tGa

MmnM

B

(23)

EXAMPLEAnalysis of Size Distribution Data Obtained in a CSTC

Differential distribution data obtained from a continuous stirred tank crystallizer are tabulated

The volumetric shape factor is av = 0.866, the density is 1.5 g/mL, and the mean residence time was 2.0 hr.

Find growth rate G and the nucleation rate B0.

w L (mm)0.02 0.340 0.05 0.430 0.06 0.490 0.08 0.580 0.10 0.700 0.13 0.820 0.13 1.010 0.13 1.160 0.10 1.400 0.09 1.650 0.04 1.9800.03 2.370

SOLUTION

The number of crystals per unit mass smaller than size L is

L

i

i

v L

w

aN

03

1

It is also related to the CSTC material balance by

GL

nndLdN

exp0

Integration of eq. (b) is

G

LnGdL

GL

nNL

exp1exp 0

0

0

(a)

(b)

(c)

wi Li N0.02 0.34 0.39170.05 0.43 0.87590.06 0.49 1.26850.08 0.58 1.58410.10 0.70 1.80850.13 0.82 1.99000.13 1.01 2.08720.13 1.16 2.15130.10 1.40 2.17930.09 1.65 2.19480.04 1.98 2.19870.03 2.37 2.2005

The number of crystals per unit mass smaller than size L is calculated using eq. (a):

G

LnGN i

i exp10

The relation of N and L is represented by eq. (c):

According to eq. (c), there are two unknowns, i.e. G and n0. 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

n0 = 3.4528 nuclei/mm4 = 3.4528 1012 nuclei/m4

Accordingly:

B0 = G n0 = 1.2137 109 nuclei/m4 hr

EXAMPLECrystallization in an MSMPR with Specified Predominant Crystal Size

Crystals of citric acid monohydrate are to made in an MSMPR at 30C with predominant size LD = 0.833mm (20 mesh). The density is 1.54 g/mL, the shape factor av = 1 and the solubility is 39.0 wt %. A supersaturation ratio C/C0 = 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.

v

SOLUTION

hrmm144.0sm104 8 dtdLG

The predominant size is related to other quantities by

GLD 3833.0

hr93.1144.03833.0

The cumulative and differential mass distributions are represented by eqs. (16) and (17), respectively.

LL

GL

x 60.393.1144.0

0 0.5 1 1.5 2 2.5 3 3.50.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Length, mm

Diff

eren

tial d

istr

ibuti

on

BATCH CRYSTALLIZATION

Gn

dLnGd

(8)

tn

dLnGd

(26)

The population equation for an MSMPR crystallizer oparated at steady state with crystal growth rate independent of size, is written in eq. (8):

For batch crystallizer operated at unsteady state, the simple population balance relationship must be modified to:

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