On numerical computation of size-dependent crystal growth rates

Compurers them. Engng, Vol. 13, No. 8, pp. 959-965, Printed in Great Britain. All rights reserved

1989 Copyright

0

1989 Maxwell

ON NUMERICAL COMPUTATION SIZE-DEPENDENT CRYSTAL GROWTH J.

MYDLARZt

0098-I 354/89 53.00 + 0.00 Pergamon Macmillan plc

OF RATES

and A. G. JONES~:

Department of Chemical and Biochemical Engineering, University College London, Torrington Place,

London WCIE 7JE, U.K.

(Received 8 February

1989; receivedfir

pubiicafion

, 20 March

1989)

comparison of current methods for the estimation of crystal growth rates in continuous MSMPR crystallizers exhibiting both linear and non-linear log population density distributions is presented Direct fitting of population density data using an exponential two-parameter model is examined in detail. This method is shown to give an excellent estimation of growth rates during MSMPR crystallization both for systems which obey, and those which violate, McCabe’s A,L law. Potential errors in growth rate estimation arising from numerical integration of discrete sieve test type data are highlighted. Abstract-A

several empirical equations proposed (e.g. Bransom, 1960; Canning and Randolph, 1967; Abegg et al., 1968; Rojkowski, 1977, 1978; Mydlarz and Jones, 1989a,b). Three general methods have been proposed for estimation of size-dependent MSMPR crystal growth rates using the population balance: (1) Sikdar Method (Sikdar, 1977); (2) White-Bendig-Larson Method (White et al., 1976); and (3) the so-called Direct Method (Canning and Randolph, 1967; Abbeg er al., 1968; Jan&L: and Garside, 1975; Rojkowski, 1977, 1978; Rojkowski and Larson, 1979; Mydlarz and Jones, 1989a,b; Jones and Mydlarz, 1989b). There is a little published information, however, comparing the alternative methods of CSD analysis with typical data from continuous crystallizers. Such a comparison is presented here both for systems which hold and which obey McCabe’s AL law, i.e. with invariant and size-dependent crystal growth. It has been shown that the magnitude of crystallization rates derived from the CSD in MSMPR crystallizers is very sensitive to the smallest size down to which the size analysis is performed (Jam% and Garside, 1975; Jones et al., 1986; Jones and Mydlarz, 1988a,b). Much of the recently published work in this area has therefore used either particle counters or laser light scattering particle sizers facilitating analysis of the size distribution of small crystals (down to 1 lrm). In this present paper, however, only discrete sieve test type data typical of those from the routine size analysis may be used for the larger crystal sizes and in many industrial situations are evaluated. The relative usefulness of the alternative methods of MSMPR crystal growth estimation is then explored.

INTRODUCTION

The simulation, design and control of bulk suspension crystallizers is dependent on accurate prediction of the crystal size distribution (CSD). It is well known that the CSD is dependent on the nucleation and growth kinetics and residence time distribution within the crystallizer and can be predicted using population balance techniques (Randolph and Larson, 1971; Mullin, 1972). Crystallization kinetics are commonly measured using the continuous mixed-suspension, mixed-product-removal (MSMPR) crystallizer technique which permits simultaneous determination of both growth and nucleation rates by analysis of the CSD at a given mean residence time. These kinetic data can then be correlated with appropriate growth and nucleation rate models. In a number of systems, non-linearity of log population density vs size is observed over a wide range of crystal sizes. Even when curvature over the entire size range is not observed, a sharp upward curvature frequently occurs at small crystal sizes. The potential causes of such curvature have been discussed by several authors (e.g. Garside et al., 1976; Janse and 1976) with size-dependent growth and deJong, growth rate dispersion emerging as the most likely mechanisms. Additionally, in the case of crystallization in an agitated vessel, hydrodynamical conditions related to slip velocity differ between large and small crystals. If volume (bulk) diffusion plays even a small role in crystal growth, then size-dependent growth rate might be detected. The effect of size-dependent growth on the CSD obtained in an MSMPR crystallizer has been analyzed theoretically by a number of author’s and TPermanent address: Institute of Chemical Engineering & Heating Equipment, Technical University of Wroclaw, ul. Nonvida 4/6, SO-373Wroclaw, Poland.

$Author to whom all correspondence should be addressed. c*ce

13,11--o

THEORY

For an MSMPR crystallizer operating at steady state, the general population balance equation re-

959

and A. G.

J. MYDLARZ

960 duces

(Note varies points

to:

n(L)1

d[G(L). ClL

+n(L)=()

(1)

T

JONES

this relation is true whether the growth rate with crystal size or not). Thus, if the data are used directly, the growth rate is given by:

is the differential population density at where n(L) size L. For size independent growth, crystal d[G(L)]/dL = 0 and equation (1) may be integrated to give the well-known equation: n(L)

= no exp( - L/(rG)].

I (Sikdur

Sikdar

(1977)

average crystal size in the where Ei + , is the arithmetic range L,, , , Li and the subscripts i and i + I refer to the two consecutive sizes (L,, , -=zL,).

(2)

Method

the simple

expression:

L..!YiL

G(L)

T

.

(3)

n(L)’

distribution with size for an MSMPR crystallizer expiicity. Direct fitting of the CSD data to this derived equation permits estimation of size-dependent crystal growth rate.

in which N(L) is the cumulative number of oversize distribution of particles above size L and is defined as: N(L)=

*n(L)dL. I L

(4)

SIZE-DEPENDENT

Thus when a curved CSD plot is obtained, the size dependence of growth rate G(L) can be quantified if the data are plotted with the cumulative number oversize N(L), in addition to the differential population density data n(L). Method

d[ln N(L)] dL

balance

leads

=m’ 1. The In n(L) c (pm)

In n(L)

In n(k)

number =

--aL

oooulation + In

dens&v

n*

G(t,) x 10’ (InS~ ’

WE,) (m -‘)

= --aL

+ ln(n*)

3.3333

2 3

1540 1300

1690 1420

8.3333 12.3333

2.066 2.7572

0.0205 1.8444

Ii10 925 780 655 5.50 462.5 389 326.5 275 230 195 152.5 II5 84 54 34.25 22.5

1200 1012.5 852.5 717.5 602.5 506.25 425.75 357.75 300.75 252.5 212.5 173.75 133.75 99.5 69 44.125 28.375

IS.6667 18.5833 21.0000 23.0833 24.8333 26.2917 27.5167 28.5583 29.4167 30.1667 30.75 3 32.0833 32.6 33.1 33.4292 33.625

3.2644 3.5865 3.9628 4.3408 4.4147 4.5830 4.7019 4.7809 4.8497 4.8847 4.9298 4.8970 4.9195 4.9448 4.9483 4.9775 4.9920

72.156 1.6423 2.3635 2.3185 I.5245 7.5823 2.9006 9.0091 2.3295 5.2060 1.0140 .9343 3.7675 6.6674 1.1084 1.6779 2.1816

I

x x x x x x x x x x x x x x x x

= P,exp(Pz&

In n(L,)

-

1840

1.4583

CALCULATlONS

(7)

data

In n(L) .-___

I

4 5 6 7 8 9 IO II 12 13 14 I5 I6 I7 18 19 20

KATE

with a = 16,666.667 m-’ and ln(n’) = 34. The second set exhibits a significant curvature (see later). A mean residence time of 20 min will be assumed in each case.

(5)

Table

I

to the

- 1

GHOWTH

In order to make an absolute comparison between the alternative methods of CSD analysis, “synthetic” data of known characteristics are used. Table 1 presents two plausible sets of such log popuIation density vs crystal size data. The former set is derived from the linear relation:

II

Appiication of the population equation (White et ol., 1976):

111

An alternative method permits direct estimation of size-dependent growth models from the population density distribution coupled to a size-dependent growth model (see later). This procedure employs empirical size-dependent growth models used in conjunction with the population balance [equation (l)]. The equations derived represent the variation of population density or cumulative number oversize

Method) proposed

(6)

]n(!$+fE$)’

+

Equation (2) indicates that size analysis performed on a representative sample of the crystallizer contents yields the necessary data to determine both crystal growth rate G (=dL/dt), and nuclei population density no (L = 0), the latter being related to the nucleation rate B” (=n”G). Method

-G--i+,

G(C + ,) =

IO’ 10” 10’ lo6 IO’ IO’ IO’ IO’ IO” IO’ lo* IO9 10y IO’0 IO’O IO’”

+ P,L)

G(t,) x 10’ (m s-‘)

3.8305

-

8.Y IYS 13.0713

1.9018 2.5482 2.9261 3.1056 3.2760 3.3261 3.2790 3.1910 3.0238 2.9476 2.5163 2.2914 2.089 l.S91 I .2627 0.9095 0.5347 0.3645 0.2794

16.6139 19.8091 22.5616 25.0541 27.28 29.28 31.1154 32.8431 34.4401 36.0257 37.4376 39.4707 41.7296 44.1694 47.4733 50.6919 53.4526

11 IO

t P,

t- P,L WE,) (m-‘) -

I I

0.0331 3.4237 1.5698 4.3879 8.2831 1.0892 1.0902 8.5755 5.4571 2.5982 I. I865 5.91 18 2.3402 1.0940 7.3755 5.7525 7.0494 1.2752 1.9387

x x x x x x x x x x x x x x x x x

IO’ IO’ IO’ lo6 10’ IO’ 108 IO’ IO”’ IO” IO” IO’2 10’1 IO” 10’4 lOI 10’7

Numerical computation 01 crystal growth rates Method

961

I

1. Data obeying McCabe’s AL law. Assuming that the mean residence time within a crystallizer is equal to 1200 s, then the “McCabe” size-independent growth rate is: G = -I/[t(dlnn(L))/dL]=5-

10-8ms-L.

The mean value of the “Sikdar” growth rate in the < Li) is calculated from the data rangeLi+,,Li(Li+, in Table 1 using the equation:

CCL,+

NC&+,)

L)=G(L;+,)=

I,

(8)

.n(-c+,>’

7

where zi + , ( = (Li + Li + , )/2) is the arithmetic average crystal size in the range (Li + 1, Li). The cumulative number oversize distribution N(&+ , ) for crystals of size above i?,+ , can be expressed as: m n(L) dL N(Li+,) = s 4 +, L

J

=

s

n(L)

+

4

n(L)

dL

dL + N(&).

(9)

The mean value of population density in the crystal Li+ , , Li is defined as:

range

n(&+,)

=

Li s&+,

n(L) dLl(Li

- L,, ,)-

(10)

The mean growth rate in the crystal size range (Lit,. Li) given in equation (8) now becomes: iu n(L) dL G(&+,)= L’Li+l I :,+I . (11)

7

n(L) s

dL

L, + I

integrals presented in equation (11) were first evaluated analytically (not shown). The growth rates calculated using equation (11) together with the cumulative number oversize evaluated from equation (9) are presented in Table 1. These growth rates are also depicted in Fig. 1. As can be seen from Fig. 1 a rather surprising result is obtained for data which obey McCabe’s AL law. The calculated growth rates exhibit an apparent negative size-dependent growth rate. It is thought that this is due to approximations inherent in the use of sieve analyses. When sieves are used, the spacing between sieve size increases geometrically with size. For example, the spacings between sieves for the next to largest size class and the largest size class are 240 and 300 pm, respectively. In general, since the population density declines exponentially with size in the case considered, there will be a tendency to overestiThe

800 Crystal

1200 SlZC,

L

1600

i

10

(pm)

Fig. 1. Results of the Sikdar method growth rate calculation for a system which holds McCabe’s AL law.

‘x

n(L)dL. 1,+1

=

400

mate the mean value of a population density of a size class when the usual practice of taking an arithmetic average is followed. In consequence the growth rate will be underestimated. Thus using a standard sieve test, the larger the crystal size the larger is the difference between growth rates calculated from equation (11) and the true values of growth rate (equal here to 5 . lo-’ m s-l). For the small crystal sizes, however, values of growth rates calculated using Sikdar’s method [equation (1 l)] are close to true values as can be predicted by equation (7). The mean growth rate was also calculated using equation (11) but with the integral evaluated numerically using the trapezoidal rule. The results of this calculation are also presented in Fig. 1. The local growth rate at size Li was then calculated using the trapezoidal rule in the following expression: au n(L) dL c

(12) These data are also presented in Fig. 1. In both cases, the growth rates calculated using Sidkar’s method with trapezoidal integration for the data which exactly obey McCabe’s AL law results in an apparent significant positive size-dependent growth rate as shown in Fig. 1. It is also interesting to note that with the direct use of equations (4) and (7), equation (3) becomes: cc exp(-aL +lnn”)dL 1 s’ G(L)= (13) 5 .exp(--al + InnO) =aS’ Putting a = 16,666.667 m-l and 7 = 1200 s, one can again obtain G(L) = G = 5. lo-* m s-l. Results obtained in the example considered here clearly indicate that employing Sikdar’s method of

3. MYDLARZ

and A. G. JONES

r

0

. 0 .

&_yIIfEf > 406

1208

800 Crystal

size.

L

1600

21

0

<,bm)

Fig. 2. Results of the Sikdar method growth rate calculation for a system which violates McCabe’s AL law.

growth rate calculation for the data from sieve analysis can lead to substantial errors: apparent sizedependence is indicated when the data in fact obeys McCabe’s AL law. 2. Data violating McCabe’s A.L law. Table 1 also presents values of log population density data vs crystal size for a system which violates McCabe’s AL law. The log population density was caiculated from the following equation: In[n(L)]

= P, exp(P,L”.‘+

P,L)

+ P4L

+ P,,

(14)

P, = - 130.777 mmo,s, P3 = with P, = 38.441, 1456.8 m-‘, P4 = - 1.66795. 104 m-land P, = 32.467. The actual size-dependent growth rate for the CSD represented by equation (14) can be calculated from the following equation: X n(L)d L IG(L) = (15)

lations, both for the case when log population density plot results in a straight line [equation (7)] and when it exhibits a significant curvature [equation (14)] are presented in Fig. 3. The cumulative number oversize distribution was determined by trapezoidal integration, which in both cases leads to an overestimation of N(L). For any crystal size range, L,, , - Li, however, the overestimations in the cumulative number oversize N(L, + I) and N(L,), respectively are approximately comparable, therefore their log ratio will be insensitive to these errors. For this reason the calculated mean growth rates are very closely theoretical. Method

III

In this method the growth rate is calculated by the equations representing the variation of population density or cumulative number oversize with size for an MSMPR crystallizer based on empirical size-dependent growth models and the population balance as follows. Power

law size-dependent

growth

models

As mentioned above, several power law size-dependent growth models are available in the literature, amongst which the Bransom (1960), CanningRandoiph (C-R) (1967) and AbegggStevens-Larson (ASL) (1968) models are most notable. Mydlarz and Jones (19&9a,c) showed that these last two models can be simply reduced to Bransom’s model. Thus only the more general ASL size-dependent growth model will be considered here. The ASL model used in conjunction with the population balance concept yields the following analytical expression relating population density data to crystal size: n(L)

=nO(l

+ bL) x exp([l

u - ((1 + bL)“-“‘j/(1

-a)},

(16)

7 Method

II

cEq.61:

JLT

_n(L)

Graphical interpretation of this equation (assuming a mean residence time within the crystallizer equal to I200 s) is shown in Fig. 2. This figure also contains values of growth rate calculated using equations (11) and (12). The integrals in equation (11) were evaluated numerically in each size range. firstly by Simpson’s rule (step = 1 pm), and secondly by the trapezoidal rule over all size ranges. The integral from equation (12) was also evaluated by means of the trapezoidal rule. These results again demonstrate that employing Sikdar’s method for sieve data from the MSMPR crystallizer can lead to significant overestimation of crystal growth rate. Method

II

The mean crystal growth rates were also calculated directly using equation (6). The results of such calcu-

0

400

800 Crystal

1200 SIZE.,

t

1600

2000

cJ$m,

Fig. 3. Comparison of theoretical size-independent and size-dependent growth rate with that calculated from equation (6).

Numerical computation of crystal growth rates

-

x f

35-

c

30-

g 5 I 3 z D k J

ECp.7.16.28

40-

252E15LB5-

I

*

00

-

8

400

-

8

500

Crystal

SlZ.3,

-0

1200 L

8.

1600

al

2000



Fig. 4. Reproduction of population density data given by equations (7) and (14) from ASL [equation (16)] and authors’s [equation (20)] steady state population density distribution.

size and (I is an empirical coefficient. Both these equations contain four parameters and it is thus rather difficult to determine unambiguous values of these parameters by direct fitting of equations (17) and (18) to realistic data. Additionally, equation (18) requires the cumulative number oversize distribution which, although it is possible to determine by numerical integration, usually leads to an overestimation, as shown above. Rojkowski (1977) suggested two methods by means of which estimation of parameters of proposed exponential size-dependent models is possible. These methods require numerical solution of a system of three equations which are non-linear in respect to the parameter a (Method A) or numerical solution of system of four implicit non-linear equations (Method B). Again, this can lead to numerical difficulties. Recently, Mydlarz and Jones (1989a) proposed a simpler exponential two-parameter size-dependent growth model: = G,,[l

G(L)

where no is the population density at zero size, u and b are parameters. Direct fitting of this equation to population density data which obeys McCabe’s AL law pable 1, equation (7)] gives the following values: In@‘) = 34, a = - 1.248 . lo-’ and b = 1.6666667 m-l. Note that the value of a is close to zero, thus equation (16) can be approximately reduced to the form of equation (2). In the case when log population density exhibits a significant curvature [Table 1, equation (1411direct fitting of equation (16) gives values: ln(nO) = 53.1971, a = 18.630 and b = 5.9606.10” m-l. Both the value of ln(n”) and of a are unrealistic. Equation (14) gives for L = 0 ln(nO) = 70.9075, while the value of exponent of a should be less than unity. Additionally, the ASL size-dependent growth model predicts the growth rate for values of coefficients In(nO) = 53.1971, a = 18.630, b = 5.9606. lo3 m-’ at least one order of magnitude higher. For these coefficients, however, equation (16) reproduces the original population density quite well (Fig. 4). Exponential

size-dependent

growth

models

Two exponential size-dependent growth models appear in the literature. Firstly, Rojkowski (1977) proposed an exponential three-parameter size-dependent growth model which, together with population balance equation [equation (l)], yields the following relations for population density or cumulative number oversize distributions, respectively: n(L)=n’exp[-L/(G,r)] x {G;AG/[G, N(L)

-AG

exp(-~L)]}‘+~‘(“~~~),

(17).

= ~“(ev{Ll(G17)1 x {G,AG/[G,

-AC

exp(-czL)])L’“G1r,

(18)

where AG = G, - G,,, G, is the limiting growth rate for large crystals, G, is the growth rate of crystal at zero

963

- exp(-aaL)],

(19)

where G,,, is the limiting growth rate for large crystals, L is the crystal size and a is an empirical parameter. Using the proposed model [equation (1911 together with the population balance concept [equation (l)] they obtained a simple steady state population density distribution: n(L)

= n* exp[a(L

- L*)]

exp(aL)

(

exp(uL*)

- 1 (-‘I-Wh - 1>

(20)

where superscript * refers to chosen size crystal L* and corresponding population density n*, and coefficient b is related to coefficient a, mean retention time r and limiting growth rate for a large crystal (b = arCmax). Taking logarithms of both sides of this equation leads to the expression: lnn(L)=In(n*)+a(L-L*) -1

+---

-b b

. In

exp(al) exp(aL*)

- 1 -

I

’

(21)

which is very convenient for fitting of experimental log population density data, Note that equation (21) contains only two parameters (a and b, respectively), thus this equation permits direct estimation of the paramters of the exponential size-dependent growth rate model [equation (1911 from the MSMPR CSD in a relatively simple way. This procedure was successfully applied recently to potassium sulphate (Mydlarz and Jones, 1989a), potash alum (Jones and Mydlarz, 1989b) and also to other systems for which the CSD data are available in the literature (Mydlarz and Jones, 1989d). Direct fitting of the second set of population density data (see Table 1) to equation (21) results in the following parameters of the size-dependent growth model: a = 3.88009 r lO’m_‘, b = 23.28054 for the system which obeys McCabe’s AL law [equation (7)] and CI= 2.80309 . 10” mm.‘, b = 0.168467 in

J. MYDLARZ

964 6

1

0

and

A.G.

JONES

the growth rate from the relative number distribution [equation (6)] gives a remarkably good estimation of growth rate, both in the case when the log population density distribution is linear and when significant curvature is exhibited. The best estimation of growth rate is obtained here, however, by direct fitting of population density data to the simple steady-state MSMPR population density distribution function given by equation (21). Calculations clearly indicate that this relation permits accurate recovery of the parameters of the proposed size-dependent growth rate model [equation (19)]. Acknorvledgemrnt-This under its Specially Technology.

408

900

1200

Fig. 5. Comparison of theoretical rate with that predicted by author’s model [equation

work was supported by the SERC Promoted Programme in Particulate

1600

size-dependent size-dependent (I 9)].

growth growth

the case when log population density exhibits significant curvature [equation (14)]. Assuming that the mean retention time within the crystallizer is 1200 s from values of parameters a = 3.88009. lo5 m ’ and b = 23.28054 = asG,,,, we can obtain exactly G = G,,, = 5 IO-sm s -‘. A comparison of the actual [equation (15)], and predicted [equation (19>] growth rates for the CSD represented by equation (14) is presented in Fig. 5. It is apparent from Fig. 5 that the proposed twoparameter exponential size-dependent growth model predicts almost exactly the theoretical values of growth rate over the entire size range, including both the small and large crystals. It is also interesting to note that, in both cases, use of equation (20), with parameter values e = 3.88009 . 10’ m-l, b = 23.28054 and a = 2.80309. IO’m-‘, b = 0.168467, reproduces the population density data represented by equations (7) and (14), respectively with excellent agreement as illustrated in Fig. 4. A limitation of the model is its inability to predict zero-size growth rates, and this is to be the subject of a future communication (Mydlarz and Jones, 1989b). CONCLUSION The results of this work demonstrate some important points to note in the numerical calculation of crystal growth rates from MSMPR population density data. Firstly, the use of Skdar’s method for growth rate calculation [equation (3)J is most suitable for crystals in the small crystal size range, particuiarly those measured using particle counters, i.e. giving cumulative number distribution type data. It can lead to substantial overestimation of the growth rate of larger crystals, however, using sieve test type data. Even when the data actually obey McCabe’s Af. law, application of Sikdar’s method can then indicate a significant apparent size-dependence. On the other hand, using White er nl.‘s method for calculation of

NOMENCLATURE a, b. c G L n IY z

= = = = = =

Parameters of growth models Linear growth rate, m s-’ Crystal size, pm, m Population density, m ’ Cumulative number oversize distribution, Mean residence time. s

mm3

REFERENCES Abegg G. F., J. D. Stevens and M. A. Larson, Crystal size distribution in continuous crystallizer when growth rate is size-dependent. AIChE JI 14, I88 (I 968). Bransom S. H., Factors in the design of continuous crystallizers. Br. Chem. Engng 5, 838 (1960). Canning T. F. and A. D. Randolph, Some aspects of crvstallization theorv: svstems that violate McCabe’s AL. law. AIChE JI 13, 5- (1967). Garside J., V. R. Philips and M. B. Shah, On size-dependent crystal growth I&. .%gr?g C/rem. Fundunt. 15,236 (1976). Jancic S. J. and J. Garside. On the determination of crystallization kinetics from crystal size distribution data. Chem. Engn~ Sri. 30, 1299 (1975). Janse A. H. and E, J. deJong, The occurrence of growth dispersion and its consequences. !ndustrial CrystaNization (J. Mullin, Ed.), p. 145. Plenum Press, New York (1976). Jones A. G., J, Budz and J. W. Mullin, Crystallization kinetics of potassium sulphate in an MSMPR agitated vessel. AZCAE Ji 12. 2002 (1936). Jones A. G. and J. Mydlarz. Conttnuous crystallization and subsequent solid- liquid separation of potassium sulphate. Part I. MSMPR kinetics. Chem. Engng Res. Des. In press (1989a). Jones A. G. and J. Mydlarz. Continuous crystallization and subsequent solid-liquid separation of potash alum. Part I. MSMPR kinetics. In press (1989b). Mullin J. W.. Cvvstallization. 2nd Edn. Butterworths. London (1972). ’ Mydlarz J. and A. G. Jones, Direct estimation of size-dependent crystal growth rate models from MSMPR poPulation density data In preparation (1989a). Mydlarz J. and A. G. Jones, A new size-dependent crystal growth rate model. In preparation (f989b). Mydlarz J. and A. G. Jones. Growth and dissolution kinetics ot’ potassium sulfate crystals in aqueous 2propanol solutions. C%ernr. Engng Sci. 6, 1391 (1989~). Mydlarz J. and A. G. Jones. Assessment of MSMPR kinetics data. In preparation (1989d). Randolph A. D. and M. A. Larson, Theory (I/‘Parriculate Processes. Academic Press. New York (1971).

Numerical computation of crystal growth rates Rojkowski Z., New empirical kinetic equation of size dependent crystal growth and its use. Krist. und Techn. 12, 1121 {1977). Rojkowski Z., Two parameter kinetic equation of size dependent crystal growth. Krist. wzd Techn. 13, 1277 (1978). Rojkowski 2. and M. A. Larson, Crystallization kinetics of ammonium alum in an MSMPR crystallizer. In Indusiriul

965

Crystnlization ‘78 (E. J. de Jong and S. J. Jam%, Eds), p. 335. North-Holfand. Amsterdam (1979). Sikdar S. K., Size-dependent growth rate from curved tin(L) vs L steady-state data. f&3. Engng Chem. Fundam. 16, 390 (X977). White E. T., L. L. Bendig and M. A. Larson. The effect of size on the growth rate of potassium sulfate crystals. AIChE

Symp.

Ser. No.

153 72, 41 (1976).