On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers

On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers

Thx Ch.emical Engineering Jaurn.al, 53 (1993) 125-135 125 On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers...

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Thx Ch.emical Engineering

Jaurn.al, 53 (1993)

125-135

125

On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers J. Mydlarz* Depart-

and A.G. Jones** of Chemical

and Bioch.ewGcal Engineering,

University

College Lmdm,

Ton-in&on

Place, London WCIE

7JE

w0 (Received October 14, 1992; in final form January 24, 1993)

Abstract A three-parameter exponential size-dependent crystal growth rate function G(L) = G,,,{1 - exp [ - a(L + c) 1) is proposed. The model has been examined in detail for the direct determination of size-dependent crystal growth rates from the population density data of continuous mixed-suspension mixed-product-removal (MSMPR) crystallizers using two sets of realistic data, both for systems which hold and which violate McCabe’s AL law respectively. It is shown that direct fitting of differential population density data using the proposed model gives an improved estimation of effective crystal growth rates over the whole size range during MSMPR cry.stallization compared with previous models tested.

1. Introduction

and previous

work

The simulation, design and control of bulk suspension crystallizers is dependent on an 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 analysed using population balance techniques [ 1, 21. The population balance provides a convenient method for studying crystallization processes in a continuous mixed-suspension mixed-product-removal (MSMPR) crystallizer. Under the conditions of steady state, constant crystallizer volume, no crystal breakage, no crystals in the feed stream(s) and a well-mixed crystallizer, the population balance can be written as [l]

d[G(L)n(L)l

a

+ n(L) = o 7

(1)

For size-independent (i.e. invariant) growth dG(L)/ dL= 0 and eqn. (1) may be integrated to give the well-known equation n(L) =n” exp

L - GT

( )

*Permanent address: Eastman Kodak Company, Rochester, NY 14659-1729, USA. **Author to whom correspondence should be addressed.

0923-0467/93/$6.00

Equation (2) indicates that a size analysis performed on a representative sample of the crystallizer contents yields the necessary data to determine both the crystal growth rate G = dL/dt and the nucleation population density no (L = 0), the latter being related to the nucleation rate B”=noG. A non-linearity in log population density vs. size plots has been observed in many crystallization systems. Such curvature can arise for several reasons, notably anomalous crystal growth (size dependence, dispersion), hydrodynamic classification, etc. One way of modelling this behaviour is to adopt a size-dependent growth function such that an “effective” crystal growth rate can be determined for design purposes. When crystal growth rates exhibit such size dependence, the relationship between the crystal size distribution and the crystallization kinetics becomes more complicated. In this case use of the population balance (eqn. (1)) to estimate the crystallization kinetics requires knowledge of the size-dependent growth function G(L). Unfortunately, no generally agreed theoretical description has yet been proposed and resort is therefore currently made to empirical methods. The effect of various size-dependent growth functions on the CSD obtained in an MSMPR crystallizer has been analysed theoretically by a number of authors and several semiempirical equations have

0 1993 - Elsevier Sequoia. All rights resewed

J. Mydlarz,

126

A.G. Jones

/ Size-dependmt

been proposed, most notably by Bransom [3], Canning and Randolph [ 4 1,Abegg et al. [ 5 ] and recently by Mydlarz and Jones [6, 71. Rosen and Hulbert [8] first utilized an exponential growth model, with another form being proposed by Rojkowski [ 91. Difficulties encountered in using well-known empirical size-dependent growth models were discussed recently by Mydlarz and Jones [6, 71 and an alternative simple exponential two-parameter sizedependent growth model (MJ-2) was proposed [6, 71: G(L)=G,[l

-exp(-aL)]

(3)

where G, is the limiting growth rate for large crystals, L is the crystal size and a (a> 0) is an empirical parameter. Using the proposed model (eqn. (3)) with the population balance concept (eqn. (l)), analytic steady state population density and cumulative number oversize distributions are obtained as 16, 71 n(L) =n* exp[a(L

-L*)]

b=arG,

(

exp(&)

exp(aL*)

-

1

(- 1- bYb

- 11 (4) (5)

where superscript * refers to the chosen crystal size L* and corresponding population density n* (or cumulative number oversize distribution N*). These equations thus permit direct estimation of the exponential size-dependent growth rate model parameters from the CSD (eqns. (4) and (5)) in a simple way [ 7, lo]. One deficiency of the twoparameter model, however, remains its inability to predict crystal growth rates at zero size. The aim of this paper is to propose another form of three-parameter exponential size-dependent growth function which permits direct estimation of size-dependent growth functions from the CSD.

2. The size-dependent

growth

G(L) = G,{ 1 - exp[ - a(L + c)]}

czo

growth

rate jimctiuns

where G, is the limiting growth rate for large crystals, L is the crystal size and a and c are empirical parameters. G(L) approaches limited values of G, as L-+L, and has the value G(L= 0) = G,[ 1 - exp( - UC)] for L = 0. Since only positive sizedependent crystal growth rates are observed in practice, the value of the parameter a should also be positive. It has been shown [ 141 that the MJ-3 model (eqn. (6)) is similar to the Rojkowski exponential model [9] but of a simpler form. Analysis of eqn. (6) leads to the conclusion that this sizedependent growth is also capable of describing systems which obey the McCabe AL law [ 151. Indeed, as a -+ ~0, G(L) + G,, = constant. Note that eqn. (6) can also be rewritten as G(L) = 1 -exp[ G,

-a(L+c)]

A graphical illustration of this equation for different values of the exponent a (with c = 5 pm) is presented in Fig. 1. The curves show that the growth rate becomes more strongly size dependent as a decreases. As can be seen from this figure, for values of a less than approximately 8 X lo4 m-l the proposed empirical size-dependent growth model predicts almost invariant (independent of size) growth rates. Equation (7) is also plotted in Fig. 2 for various values of the parameter c. It is interesting to note that the parameter c affects the growth rate mainly in the size range close to zero. The larger the crystal size, the smaller is the influence of the parameter c on the growth rate. For a crystal size larger than about 800 Frn and a = 5 X lo3 m- ’ the growth rate calculated from eqn. (7) is virtually

rate model

The requirements of size-dependent growth ftmctions for use in conjunction with the MSMPR population balance concept to predict CSDs have been discussed fully by previous authors [ 5, g-131. A three-parameter growth rate model (MJ-3) which satisfies all essential size-dependent growth model requirements [5, g-131 is

a>O,

crystal

c c ._

.2

: E” ii

00

400 Crystal

(6)

800 size,

Fig. 1. Variation in dimensionless different values of parameter a.

1200 L

1600

(pm)

growth

rate with size for

J. Mydtarz,

i&O. Jones / Size-dependent cr@al

N(L)

growth

= In(L)

127

rate fimctions

dL

L

.6

n(L)=

-

cw(L)

7

(11)

.6

Use of eqns. (6) and (11) in the steady state MSMPR population balance yields .4

~~~~L)] ,2

0 0

size,

480

JGl0

200

100

Crystal

L (pm)

2. Variationin dimensionlessgrowth rate with size for differentvalues of parameterc.

Fig.

~dependent of the vahres of the parameter G considered here. Using the proposed model (eqn. (6)), eqn. (1) can be integrated analytically to give the steady state MSMPR pop~a~on density ~~bution n(L) = no exp(uL)

exp[a(L+c)]

(

- 1 1-1-a70”Ya7Gm

exp(m) - 1

)

(8) where no is the “zero-size” population density, L is the crystal size, r is the mean residence time within the crystallizer and a, c (a > 0 and c > 0) and G, are the parameters of the proposed size-dependent growth rate model (eqn. (6)). Setting arG, = b and exp(uc) =A, one can rewrite eqn. (8) as n(L) =K exp(uL)[A

-%L

- ’

1

(12)

where No is the total number of crystals in the system and is related to both the mean residence time Q-and the zero-size nucleation rate B” =N’/r. Thus eqn. (12) permits simultaneous estimation of the growth and nucleation kinetics from MSMPR cumulative number dist~bution data in a relatively simple way. Introducing the dimensionless population density

n(L) Y(L)= yp-

(13)

eqn. (9) may be rewritten as

(-1-b)‘b ~(3 = exp(al)

(14)

The dimensionless population density defined by eqn. (14) is ~lus~ated in Pig. 3 for b =0.3, c =5 111



I

Ax

-5

. -20

(9)

Analysis of eqn. (9) leads to the conclusion that it describes the population density distribution when the values of the parameters a, b and c differ from zero. Note also that as a --+~0,the popula~on density distribution (eqn. (9)) reduces to !

(

exp(al) - 1 ](-1-b)‘b

K=no(A - l)(l +*)I*

n(L) =n” exp

5 In A ex~\~)

= lnn(N”) -

1

i.e. McCabe’s AL law is obeyed (see eqn. (2)). The CSD of the product from an MSMPR crystallizer is also frequently expressed in terms of the cumulative number oversize distribution N(L) defined as

2 Ei.-25 M

j

-30

is ._ :

-35

E .v -0

-40

? -I

II 0

I\

6.104 3.104

\

1.5.184

400

Crystal

\\

104

3.103

1200

600

size,

\

L

6.1a3

1601

(pm>

3. Steadystatedimensionfess populationdensitydistribution definedby eqn. (14) as a functionof parametera.

Fig.

128

J. Mydlm-z,

A.G.

Jones

/ Size-~

pm and selected values of the parameter a. This figure shows the effect of the parameter a on the crystal size distribution in the continuous MSMPR crystallizer when the growth rate is size dependent. These analytical considerations show that the curvature in the population density plot can be considerable, particularly at low values of a, i.e. when the growth rate is strongly size dependent. As can be seen from Fig. 3, for b = 0.3, c = 5 pm and values of a larger than approximately lo5 m-l the plot of ln[n(L)] VS. L is virtually linear, consistent with the distribution predicted from McCabe’s AL, law. As the value of the parameter a decreases, the population density function begins to deviate increasingly from a straight line. Equation (14) is illustrated in Fig. 4 for different values of c (a=6x lo3 m-‘, b =0.3). It should be emphasized that the value of the parameter c strongly affects the population density distribution, particularly for values of c close to zero (c > 0). As the value of the parameter c increases, the curvature of log population density becomes less. Inspection of Fig. 4 also shows that there is an inverse relationship between the value of the parameter c and the “critical” crystal size. This “critical” size (L,,J corresponds to that crystal size above which the log population density plot is virtually linear. The higher the value of the parameter c, the smaller is the “critical” crystal size.

,

~R-waC; b-arG,;

a-&10m-’

c-q&&

growth

rate

functions

?----.

A.10

0)

b-1.00

2.20

0130

s

0

400 Crystal

800 size,

1200 L

1601

(pm)

Fig. 5. Variation in dimensionless population density distribution defined by eqn. (17) with size for different values of parameter b.

Figure 5 shows the steady state dimensionless size distribution for different values of the parameter b (a=6x103 m-‘, c = 5 ym). These theoretical predictions show that curvature in the log population density plot is strongly related to the value of the parameter b. Again, an inverse relationship between the value of the parameter b and the “critical” crystal size is observed. The closer the value of the parameter b to unity, the smaller is the “critical” crystal size.

; b-0.3

3. Comparison of the two- and threeparameter size-dependent crystal growth models m

-20

ii. :

-25

0) _:

-30

E ._ :

-35

E ._ -u

-40

Is)

1

Pm

5

P’m

4 400

800

Crystal

size,

1200 L

1601

(pm)

Fig. 4. Steady state dimensionless population density distribution defined by eqn. (14) as a function of parameter c.

As mentioned above, a comparison of current methods of growth rate calculations and size-dependent growth models with the two-parameter sizedependent growth model proposed previously (eqn. (3)) has been presented elsewhere [6, 71. Thus for brevity in the present work we compare only that two-parameter size-dependent growth model with the new form of exponential three-parameter model proposed here (MJ-3) for two sets of realistic “synthetic” log population density VS. crystal size data as used previously [6]. These data are described briefly below. The llrst set of population density data obeys McCabe’s AL law and is derived from the linearized relation

J. Mydlarz,

A.G.

Jones

/ Size-dependent

crystal

growth

129

rate _function.s

TABLE 1. Comparison of two- and three-parameter exponential size-dependent crystal growth models Model

ln[n(L)]= c&x10-*

-aL+ln(d) b

(m-l) G(L)=G,[l-exp(-aL)] G(L) = G,,,{I- exp [ -

38.8009 a(L + c)]}

1.43397

23.28054 0.860384

In[n(L)] = - uL + ln(n”)

ln[n(L)] G,x

(m s-‘)

CZXIO-~ (m-l)

b

(m) 623.211

5.000 5.00001

2.80309 2.79932

0.168467 0.166174

(15)

+P& +Ps

exp(P&‘~‘+P,L)

cx106

with a = 16666.667 m- ’ and ln(n”> = 34. The second set violates McCabe’s AL law. The log population density is calculated from In[n(L)] =P1 exp(P&O.’ +P&)

=P,

7

E :

10’

+P4L+P5

(m)

10’ (m s-‘)

2.105

5.00837 4.94685

CXlOB

G, X

6 t

G(L)=Gn[l-exp(-a(L+cNl

(16)

withP1=38.441,P2= -130.77’7m-‘/2,P,=1456.8 m-‘, P4= -1.66795X104 m-’ and Pg=32..467. A residence time of 20 min wiII be assumed in each case. Direct fitting of the 6rst set of population density data to eqns. (4) and (9) respectively results in the following parameter values: a = 3.88009 X lo6 m- ’ and b = 23.2805 for the two-parameter size-dependent growth model [lo]; a=1.43397x104 m-‘, b =0.860384 and c = 623.211 pm for the proposed three-parameter size-dependent growth rate model (eqn. (6)). It is interesting to note that in the case when the log population density data are linear, the values of the coefficients a and b in the two- and three-parameter size-dependent growth models are significantly different. For the three-parameter sizedependent growth model the values obtained for both a and b are approximately one order of magnitude smaller than those obtained from the twoparameter model. Additionally, assuming a typical retention time within the crystallizer of 1200 s, then from values of the parameters presented in Table 1 the growth rate G,=5.000~ 10V8 m s-’ for the two-parameter size-dependent growth model, while G,=5.00001 X 10m8 m s-’ when the three-parameter model is employed. Parameter values obtained are also listed in Table 1. Finally, using eqn. (2), we can get for both cases G= l/(7 d{ln[n(L)]}/ dL)=5X10-8 m s-l. It is also worth pointing out that the accuracy of the two-parameter model (eqn. (3)) in the smaller size range depends on the lower limit of the crystal size range for curve fitting. This is clearly illustrated in Pig. 6, which shows increasing deviation as the lower size limit increases; obviously care should be exercised in extrapolation below the range of size measurement. Growth rates predicted by the threeparameter model (eqn. (6)) for the small crystal

II&::: 2

0

4 Crystal

G(L)=GmC1-exp(-aL)l

2

6 size,

8 L

10

(brn)

6. Illustration of influence of lower crystal size range Iimit on accuracy of two-parameter growth rat.6 model.

Fig.

size range are also presented in Fig. 6, As is seen in this figure, the accuracy of the three-parameter growth model is excellent over the whole size range and is independent of any lower limit of the crystal size range used for curve fitting. Direct fitting of the second set of population density data (see eqn. (16)) to the steady state population density functions given by eqns. (4) and (9) results in the following parameter values of the size-dependent growth models: a = 2.80309X 1 O3 m-l and b = 0.168467 for the two-parameter model; b=0.166174 and m-l, a = 2.79932 X lo3 A = 1.00591 for the three-parameter model. Note that when the log population density data exhibit significant curvature, the values of the parameters a and b are almost the same for the MJ-2 and MJ3 size-dependent growth rate models. A comparison of the growth rates obtained from the CSD represented by eqn. (16) is presented in Fig, 7. The theoretical size-dependent growth rate can be calculated from [ 161 G(L)=

N(L)

-

m(L)

(17)

130

J. Mydlm-z,

A.G. Jones / Size-dependent

crystal

growth

rate functions

Eqa.4,9,16

400 Crystal

size,

L

(pm)

Comparison of theoretical size-dependent growth rate with that predicted by two- and three-parameter exponential size-dependent growth models (eqns. (3) and (6)). F’ig. 7.

where the cumulative number oversize distribution is as defined by eqn. (11). The integral in eqn. (11) was evaluated numerically using Simpson’s rule (step 1 pm). A graphical interpretation of eqn. (16) (again assuming a mean residence time within the crystallizer equal to 1200 s) is also shown in Pig. 7. It is apparent from Pig. 7 that both the two- and three-parameter exponential size-dependent growth models predict almost exactly the theoretical values of growth rate over the entire crystal size range, including both small and large crystal sizes; the three-parameter crystal growth model is the more accurate, however. Again it should be pointed out that the accuracy of the two-parameter model in the smaller size range is related to the lower limit of the crystal size range used for curve fitting (not shown here). It is also interesting to note that both use of eqn. (4) with parameter values a = 2.80309 x lo3 m- ’ and b = 0.168467 and use of eqn. (9) with parameter values a=2.79932X103 m-‘, b=0.166174 and A = 1.00591 reproduce population density data represented by eqn. (16) excellently, as illustrated in Pig. 8.

4. Application

to experimental

Crystal

data

The three-parameter exponential size-dependent growth model (MJ-3) proposed here is now briefly examined for use in the correlation of real data available in the literature.

1200

600 size,

L

1600

200

(pm)

Reproduction of population density data given by eqn. (16) from steady state population density distribution (eqns. (4) and (3)).

Fig. 8.

Crystal

size,

L

(cm)

Steady state size distribution data obtained by Canning and Randolph [4] for Glauber salt (original data redrawn). Fig. 9.

4.1. Gluuber salt data The data reported by Canning and Randolph [4] were obtained in a bench-scale, continuous, drafttube MSMPR crystallizer for Glauber salt (sodium sulphate decahydrate) and are shown in Pig. 9. It should be noted that the log population density data exhibit curvature, particularly in the small size range. Direct fitting of the population density data presented in Fig. 10 to the MJ-3 model (eqn. (9)) results in the following parameter values:

J. M@larz,

A.G. Jones

/ Size-dependent crystal

growth rate

131

_functh

ln[n(L)] = ln(n’) +

ln(1 + uL)

B=ado

(19)

(3) the Abegg-Stevens-Larson (G(L) = Go( I + ajb),

(ASL)

model

ln[n(L)] = ln(n”) - b ln( 1 + uL) 1 (1 +nL)‘-b + a&0(1 - b) - arG,( 1 - b)

q-WBL

0

600

300 Crystal

size,

Parameter values obtained are also presented in Table 2. Graphical interpretation of eqns. (18)-(20) is also included in Pig. 9. It should be noted that the reproduction of the CSD is excellent, as shown in this figure. It is interesting to compare the growth rates obtained from the Canning-Randolph CSD data by direct fitting these data to the steady state distribution represented by eqns. (9) and (18)-(20). This is presented in Fig. 10. It should be pointed out that in the size range between about 100 and 450 pm all models tested predict similar values of G(L). Using the Bransom, C-R and ASL models, however, G(L) tends monotonically to infinity in the larger size range, as shown in Pig. 10. This is of course physically unrealistic. The MJ-3 size-dependent growth model, on the other hand, approaches a maximum growth rate asymptotically. Analysis of size-dependent growth rate calculations [6, that use of the 171 showed White-Bendig-Larson (WBL) equation [ 121

Method

120

900 L

(ym)

Fig. 10. Comparison of Glauber salt growth rates obtained by

direct fitting of CSD data to eqns. (4), (9) and (18)-(20).

a- 1.685~ 10 cm-‘, b=0.7142, A= 1.9224 and ln(n’)=6.836. The parameter values obtained are listed in Table 2. Use of eqn. (9) with these parameter values reproduces the original population density data reasonably well, as is shown in Pig. 9. The Canning-Randolph population density data were also fitted to steady state population distribution functions (assuming a mean residence time equal to 1200 s) obtained for (1) the Bransom model (G(L) =aLb), ln[n(L)]

= lr(n*> +

a.,(blL’-b

@i+J =

1) b

L-L1 7 In[N(L, + 1)/!(LJ

1

(21)

provides an accurate estimation of G(L), particularly in the small crystal size range. The values of G(L, + 1) calculated using eqn. (21) are also presented in Pig. 10. It should be noted that G(L) calculated from eqn. (2 1) approaches a maximum and constant value for large crystal sizes which is consistent with the value of G(L) predicted by the MJ-3 size-de-

(13) where L* refers to the smallest measured crystal size and n* =n(L*), (2) the Canning-Randolph (C-R) model (G(L) =Go(l

(20)

+d)h

TABLE 2. Parameters of size-dependent crystal growth rate models for Glauber salt Model

Parameters b

a (m-l> G(L) = aLb G(L)-G&l +aL) G(L) = G,( 1 + CIA)” G(L) = G,,,{l - exp[ - a(L + c)]}

4.802x 1.044 x 1.192x 1.685 x

1O-7 103 lo4 lo3

cx10* cm-‘)

G,x 10’ (m s-l)

G,x lo7 (m s-‘)

3.880

0 1.842 1.589 1.695

3.533

0.1897 0.2824 0.7142

J. Mydlam,

132

A.G. Jones / Size-dependent

pendent growth model. Thus direct fitting of Glauber salt log population density data to the MJ-3 steady state population distribution (eqn. (9)) provides a better estimation of G(L). 4.2. Potash alum data Figure 11 shows the continuous population density and cumulative number oversize distribution data for potash alum (potassium ahrminium sulphate) reported recently [lo]. Correlation of the ASL theoretical population density curve with both sieve data (S data) and sieve and Malvern sizer data (S & M data) is presented in this figure. These data were also fitted to the Bransom [3], Canning-Randolph [4] and MJ-2 [ 71 theoretical population distribution curves and the MJ-2 and MJ-3 cumulative number oversize distribution curves. It should be noted that the reproduction of eqns. (5), (9) and (12) is excellent, as shown in this figure. Values of the model parameters calculated using a non-linear regression are presented in Table 3. Inspection of Table 3 confirms that a much better fit of the data was obtained using the MJ-2 and MJ3 steady state functions. Additionally, use of the cumulative number oversize distribution functions (eqns. (5) and (12)) leads to smaller variance of fit in comparison with those obtained using popon-Slaves I6

.m-Malvsrn

data data

(S-data) (M-data)

-1Po=o. 092

12

I8

14 S- b M-data

!0-

s-

2-

8Crystal

size,

L

(pm)

11. Steady state size distribution data for potash alum crystals precipitated with aqueous acetone by Jones and Mydlarz

Fig.

1101.

crystal

growth

rate .fimction.s

I

n-WBL Sieves

&

Method

Malvern

q -WBL

data

Method

/ Sieves 0

200

400 Crystal

600

800 size,

1000 L

data 1200

1400

(pm)

Fig. 12. Comparison of potash alum growth rates obtained by direct fitting of CSD data to eqns. (4), (9) and (18)--(20).

ulation number distribution functions (eqns. (5) and (12)). It is also interesting to note that log population density data (as well as log cumulative number oversize distribution data) for crystals larger than about 350 pm are approximately linear. Thus in this size range the growth rate can be easily estimated from the slope of the population density (or cumulative number oversize distribution) data plot (- l/rG,d and the mean residence time. Linear regression of the log population density data for crystal sizes larger than about 350 pm results in a slope equal to - 9.578 x low3 pm-‘, so the maximum growth rate G,,, should approach a value of 8.4X1O-8 m s-l. It is interesting to compare the growth rates obtained from the data presented in Fig. 11 by direct fitting these data to steady state distributions represented by eqns. (4), (9) and (Is)-(20). This is presented in Fig. 12. It should be pointed out that in the case of wider CSD data (S & M data, Fig. 12a) the Bransom, Canning-Randolph and ASL models only predict values of G(L) consistent with those calculated from the WBL equation (eqn. (21)) for crystals smaller than about 400 pm, while use of both the MJ-2 and MJ-3 models gives a much more accurate prediction of G(L) over the whole size range. Moreover, the value of G,, obtained from the slope of log population data for larger crystals is consistent with the value of G, obtained

G(L)=G,,,{l-exp[-a(L+c)]}

G(L)=G,,,[l-exp(-aL)I

Sievesdata G(L)=aLb G(L)=G,(l+aL) G(L)=G,,( ~+uL)~

3, 3, 6, 6,

18 19 20

3, 3, 6, 6,

G(L)=G,Jl-exp(-a.L)]

a(L+c)]}

18 19 20

G(L) = aLb G(L) = G,,(1 + aL) G(L)= GO(l + aL)*

G(L)=G,,,{ 1-exp[-

uystal

4 5 9 14

4 5 9 14

Equation(s)

of size-dependent

Sieves and Malvern data

Model

TABLE 3. Parameters parameters

Model

2.330x 2.516 41.149 8.904 6.232 4.237 2.840

lo-’

3.304 x 10-s 132.542 803.984 5.838 6.140 6.483 6.661

(m-l>

aX 10e3

rate models

growth

0.420 0.103 0.443 0.879 0.663 0.467 0.335

0.759 0.350 0.780 0.638 0.660 0.687 0.700

b

for potash

39.81 103.3

(m1

cx10’

alum

1.1838 1.3411

A

35.608

34.072 34.919

49.152

40.826 41.832

Wno>

23.960

29.232

WNO)

0 32.725 19.457 0 0 13.690 24.025

0 2.111 0.995 0 0 0.0373 0.2870

(m s-l)

G,x 10’

7.907 8.517 8.818 9.446

8.742 8.604 8.483 8.410

G,xlO*

0.0486 0.1350 0.0568 0.904 0.0333 0.0297 0.0083

1.3347 0.7887 0.2008 0.0669 0.1716 0.0636

Sk,

0.849 1.290 0.859 1.103 1.303 0.593 0.612

3.049 4.285 3.176 1.404 1.276 1.145 1.344

134

J. Mydlarz,

A.G. Jones / Size-dependeni

by direct fitting of the data to the MJ-2 and MJ-3 steady state distributions. In the case of the S data (Fig. 12b) estimation of G(L) using the Bransom, C-R and ASL steady state distributions is significantly improved in comparison with use of the S & M data. Use of the MJ-2 and MJ-3 models, however, still gives a substantially better estimation of G(L).

5. Conclusions This paper has been primarily concerned with the application of another form of exponential threeparameter size-dependent crystal growth rate model (eqn. (6)) for direct model identification from MSMPR population density data and its comparison with a two-parameter exponential size-dependent crystal growth model reported earlier [ 6, 71. This comparison was performed for two sets of realistic “synthetic” log population density size distribution data together with some CSD data available in the literature. Inherent in the three-parameter size-dependent growth model is a prediction of both the growth rate of “zero-size” crystals and its asymptotic increase to a maximum growth rate with increasing size. The proposed three-parameter size-dependent crystal growth rate model (eqn. (6)) results in a relatively simple form of both population density (eqn. (9)) and cumulative number oversize (eqn. (12)) distributions and also facilitates direct curve fitting of population density data to determine the coefficients of the model in a very simple and accurate way. In the case where the log population density distribution data are linear, the two- and threeparameter size-dependent crystal growth models predict almost the same values of the common parameters (see Table 1). Parameter values of both growth models are also similar in the case when the log population density distribution data exhibit significant curvature (see Table 1) and the reproduction of the population density distribution data is almost exact (see Fig. 7). For the population density data considered in this work, however, the three-parameter size-dependent growth model predicts the growth rate more accurately than does the two-parameter size-dependent growth model. Application of the two- and three-parameter sizedependent growth models to the experimental Canning-Randolph data for Glauber salt [4] and to the authors’ data for potash alum [lo] demonstrates

crystal growth rate functions

improved correlation compared with other models tested, particularly in the larger crystal size range.

Acknowledgment This work was supported by the SERC under its Specially Promoted Programme in Particulate Technology.

References

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6

7

8

9

10

11

12

13

14 15 16

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A.D. Randolph and MA. Larson, Theory of Particulate Processes, Academic, New York/London, 2nd edn., 1988. J.W. Mullin, Crystallization, Butterworths-Heinemann, Oxford, 3rd edn., 1993. S.H. Bransom, Factors in the design of continuous crystallizer, Br. Chem. Eng., 5 (1960) 838. T.F. Canning and A.D. Randolph, Some aspects of crystallization theory: systems that violate McCabe’s AL+ AIChE J., 13 (1967) 5. G.F. Abegg, J.D. Stevens and M.A. Larson, Crystal size distribution in continuous crystallizer when growth rate is size-dependent, AICti J., I4 (1968) 188. J. Mydlarz and A.G. Jones, On numerical computation of size-dependent crystal growth rates, Compur. Ch.em.. Eng., 13 (1989) 959. J. Mydlarz and A.G. Jones, On modelling the size-dependent growth rate of potassium sulphate in an MSMPR crystallizer, Chem. Eng. Common., 90 (1990) 47. H.N. Rosen and H.M. Hulbert, Growth rate of potassium sulphate in a fluidised bed crystallizer, Chem. Eng. Prog. Symp. Ser. No. 110, 67 (1971) 27. Z. Rojkowski, New empirical kinetics equation of size-dependent crystal growth and its use, Krist. Techn., 12 (1977) 1121. A.G. Jones and J. Mydlarz, Continuous crystallization of potash alum, MSMPR kinetics, Can. J. Chem. Eng., 68 (1990) 250. J. Garside and S.J. Jan&C, Prediction and measurement of crystal size distribution for size-dependent growth, Chem. Eng. Sci., 33 (1978) 1623. E.T. White, L.L. Bendig and M.A. Larson, The effect of size on the growth rate of potassium sulphate crystals, AlChE J. Symp. Ser., 153 (1976) 41. R.W. Rousseau and R. Woo, Effect of operating variables on potassium alum crystal size distribution, AICi& J. Symp. Ser., 193 (1980) 27. Z. Rojkowski, personal communication with J. Mydlarz, 1989. W.L. McCabe, Kinetics of crystallization in solution, Ind. Eng. Chem., 21 (1929) 112. S.K. Sikdar, Size-dependent growth rate from curved In n(&j vs. L steady-state data, Ind. Eng. Chem. find., 16 (1977) 390. J. Mydlarz, Analysis of growth rate estimation methods in an MSMPR crystallizer and new size-dependent growth models, Monograph-Habilitation Thesis, Technical University of Wroclaw, 1991.

J. M@hz,

A.G. Jones

/ Size-dependent

Appendix A: Nomenclature

N

a, b, c

NO

B0

G K L n

parameters of growth model zero-size nucleation rate (mT3 s-l) linear growth rate (m s- ‘) parameter of growth model, eqn. (9) crystal size (m, pm) population density (m-“)

crystal growth rate functions

135

cumulative number oversize distribution (me3) total number of crystals in system (mP3)

Greek letters

612,g 7

variance of fit in log[n(L)] plane mean residence time (s)

or log[N(L)]