- Email: [email protected]
Modeling of crystal size distribution in a mixed suspension, mixed product removal crystallizer when growth is size dependent
Modeling of crystal size distribution in a mixed suspension, mixed product removal crystallizer when growth is size dependent Jerzy Mydlan and Daina Briedis Department
of Chemical Engineering,
Michigan State University, East Lansing,
MI, USA
A revised empirical two parameter size-dependent growth rate model proposed recently is discussed in detail and used to illustrate the analysis of both simulated and real population density and cumulative number oversize distribution data from a mixed suspension, mixed product removal crystallizer. The two-parameter model has superior curve-fitting properties compared with other models tested and shows improved consistency in the prediction of the growth rate in the case in which log population density plot exhibits curvature across a large crystal size range.
Keywords: mixed suspension, gr&vth rate
mixed product
removal crystallizer;
Introduction In recent years, the population balance technique developed by Randolph and Larson’ for the continuous mixed suspension, mixed product removal (MSMPR) crystallizer has been used in an effort to understand the mechanism of crystallization from solution. This model predicts that, for an ideal MSMPR crystallizer, the plot of log population density vs. size will result in a straight line if growth rate is constant for all crystals and nuclei are near zero in size. In several systems, nonlinearity in log population density vs. size curves is observed over a wide range of crystal sizes. Various causes for this deviation have been suggested in the literature including (1) secondary nucleation in a small, finite size range; (2) size-dependent growth rate; and (3) growth rate dispersion. Several MSMPR crystallizer studies support the idea of a size-dependent growth rate.‘-’ Further support for size-dependent growth rates was obtained using batch suspension crystallizers.9-‘3 Measurements of growth kinetics in fluidized bed crystallizers also supported the idea of increasing growth rate with crystal size.i4-” Finally, measurements of single crystal growth kinetics for potash alum and nickel ammonium
Address reprint requests to Dr. J. Mydlarz at his present address: Eastman Kodak Company, Rochester, NY 14650-1708, USA. Received 25 February 1992; accepted 26 May 1993
212
Sep. Technol.,
1993, vol. 3, October
modeling of growth rate; size-dependent
sulfatei indicated an increase in the growth rate with the size of crystals. Size-dependent growth rate was successfully modeled for several MSMPR crystallizer systems.2*6*8,12+19-21 The effect of size-dependent growth on the crystal size distribution (CSD) obtained in MSMPR crystallizers has been recently analyzed in the literature.20.22-23 The proposed models discussed in these references, the MJ-2, and MJ-3 as well as the ASL model are summarized in Table 1. Recent crystallization studies have confirmed that growth rate dispersion may also be a source of nonlinearity in the log population density plot.24-33 Such curvature can also arise by several mechanisms including variant crystal growth, crystal size classification in the withdrawal stream, and secondary processes such as crystal breakage and agglomeration. Researchers in crystallization still debate about sizedependent growth and growth rate dispersion. As cited previously, experimental evidence exists for both phenomena. These two models have recently been compared for real MSMPR crystallizer data by Mydlarz and Briedis.31 This comparison has shown that in the case of curvature in log population density vs. size data plots, use of the current growth rate dispersion models leads to significant underestimation of growth rate whereas using recently proposed size-dependent growth models gives good estimation of the growth rate especially for large crystal size. In this article, we will discuss these size-dependent growth models. 0
1993 Buttenvorth-Heinemann
Modeling Table 1
The ASL, M-2, MJ-2, and MJ-3 size-dependent
Model
Abbrevi&on
G(L) = G, (1 + aLjb
ASL
+ aL)F%Jxp
Cumulative
function
1 - (1 +
G(L)
=
a+(1
G(L) = G,,,Il
oversize
_ (1 +
sL)‘-b
~(1 - b)rG,
expt-al)1
n(L) = n+exp[a(L
MJ-2
G(L) = G,,,(l - expl-a(L
is a simpler
+ ~11)
exponential
-
L’)l
/v(L) =/v
1 (-,-b),b - 1)
n(L) = Kexp(eLl(A
MJ-3t
form of the Rojkowski
-
model
(,$+b
exp(aLl
-
N(L) = K,&A c+ - l)-“b
I)‘-‘-@‘*
(private communication,
Mydlan
Jones
&
(19931
(2)
density. In thi.iase a semilog plot of population density versus size results in a straight line of slope - l/(e) and intercept In (no), and growth and nucleation kinetics are easily calculated.
= 0
Proposed hyperbolic growth rate model M-2 In this work, use of recently proposed empirical two-parameter hyperbolic growth rate model (M-2)*l is discussed in detail. The model satisfies most of the essential model requirements3:
(1)
where G(L) is the crystal growth rate, n(L) is the population density, L is the crystal size, and 7 is the mean
r--
Jones
where no is the lim n(L), i.e., the zero-size population-
balance
+ n(L)/7
Mydlan & (1999.1990)
n(L) = no exp( - LIGr)
For an MSMPR crystallizer operating at steady state, Randolph and Larson’ have shown that the general population balance equation reduces to d[G(L)n(L)]ldL
This work
retention time of suspension within the crystallizer. For size-independent growth, G(L) = G = constant, and Equation 1 can be integrated to give
Theory population
t-l* = n(L =.L”l W = N(L = L9 G* = G(L = L”) d = rG* oaas1 n+ = n(L = L*l N* = NIL = L”1 b = erG,,, a>O,b>O A=& b=ar(i, a>O,b>O.c>O K = #(A - l)‘+b”
1999).
The aim of this article is to examine recently proposed empirical size-dependent growth rate models for growth rate estimation from MSMPR crystallizer data. An improved form is proposed that permits direct estimation of model parameters from either the population density or cumulative number oversize distributions.
MSMPR
Abegg, Stevens and Larson (19681
)
-a)LIL*
-
a>0 O
- 1
M-2
exp(aL) exp(aL*L
tMJ-3
number
N(L) = NOewp
sLpb
1 -b (
G* L/L+
and D. Briedis
growth models
Population-density
n(L) = nw
of CSD: J. Mydlatz
Notation
a, c parameters
of size-dependent growth models constant of the MJ-3 model (= exp(ac)), constant of the MJ-2 and MJ-3 models (= aTG,J, B” nucleation rate, #/(s L) constant of the M-2 model (= 7G*), pm d ET dimensionless growth rate (= G(L)IG*), G linear growth rate, pm/s, m/s GWI limiting growth rate for large crystals, pm/s GO growth rate of crystals at zero site, pm/s G* growth rate of crystals at size L*, pm/s K constant of the MJ-3 model (= n”(A - l)“‘),#/(pm L) constant of the MJ-3 model KN (= N”(A - I)l’b),#/L L crystal size, pm nuclei size, pm LN
A b
n(L) no nN
N(L) N” X
Y
population density, #/(pm L) zero-size population density, #/km L) nuclei population density, #/(pm L) cumulative number oversize distribution, #IL zero-size cumulative number oversize distribution, #IL dimensionless crystal size (= L/L*), dimensionless population density (= n(L)ln*), -
Greek letters 7
mean residence time, s variance of fit, pm-* variance of fit in log n(L) (or log N(L)) plane, pm-*L-*, L-*
Sep. Technol.,
1993, vol. 3, October
213
Modeling G(L)
of CSD: J. Mydlam and 0. Briedis = G*(LIL*)I[a
+ (1 - a)LIL*];
0 I a 5 1 (3)
where L is the crystal size, L* is the chosen crystal size, G* is the growth rate at crystal size L*, and a is a coefficient. G(L) approaches values of G* as L j L* and zero for L = 0. The latter condition can present conceptual difficulties, however, in estimation of the zero-size nucleation rate. Because only positive sizedependent crystal growth rates are observed in practice, the value of parameter a should also be positive. For a = 0, G(L) = G* = constant demonstrating that this size-dependent growth model is also capable of describing systems for which McCabe’s AL law holds. To examine the influence of the parameter a on the predicted G(L), Equation 3 can be written in dimensionless form as g(x) = G(L)IG*
= xl[u + (1 - a)~]
(4)
where x = L/L*. Figure I shows the dimensionless growth rate curves with dimensionless size x for different values of parameter a. For any value of parameter u (except a = 0 or a = l), the growth rate increases with increasing size to an asymptotic maximum. The “critical” size for which the growth rate becomes size independent (G(L) reaches its maximum value) is strongly related to the value of the parameter a. The smaller the value of parameter a, the smaller the value of the “critical” size. For a = 0, however, the g(x) results in horizontal straight line, indicating size-independent growth rate, as discussed earlier. There is a large body of evidence in the literature that, in continuous crystallization, the secondary nuclei are a substantial source of new particles. Conversely, these secondary nuclei can only survive in a supersaturated solution when they are larger than the so-called critical size, Lcrit, which, of course, has a positive value. It is well known that the Lcrit is related to the actual supersaturation; the lower is the supersat-
a=0.90
uration, the larger the Lctit.34Thus in continuous crystallizer, the Lcritis fairly large, because continuous crystallizers operate with a small supersaturation. Logically, no = n(L = 0) is equal exactly to zero, because there are no particles that have a size equal to zero, or there are no crystals in a real MSMPR crystallizer that have size smaller than the Lcrit. The proposed M-2 models suffers from an inability to estimate zero-size crystal growth rate (G,) directly; however, there are no zero-size crystals in any real crystallizer, and the zero-size growth rate, G,, only has hypothetical meaning. In view of the preceding, the zero-size growth rate, G,, is only a mathematical concept that is frequently used in deducing a “hypothetical” nucleation rate at zero size, B”. Nevertheless, Jancic and Garside35 showed that the nucleation rate can also be calculated from the mean residence time and population density data only. MSMPR crystallizer population density distribution. Using the proposed hyperbolic size-dependent model (Equation 3), one can integrate the MSMPR population balance (Equation 1) between sizes L* and L (L > L* > 0) to give the simple steady-state population density distribution21 n(L) = n* a + (1 - (I)$
I(L1
ew
(-oL*-d)ld
L*
[
Y(L
- L*)
1
(5)
where the superscript * has the same meaning as in Equation 3, and TG* = d. Although this equation does not predict the population density at zero size, it is possible to choose a value of L* arbitrarily close to zero. In the practical application of Equation 5, L* is the smallest measured crystal size that is dependent on the size measurement technique used. Taking logarithms of both sides of this equation leads to an expression that is a very convenient form for fitting of experimental log population density data. Equation 5 contains only two parameters (a and d); thus this equation permits direct estimation of the parameters of the hyperbolic size-dependent growth rate model (Equation 3) from MSMPR crystallizer CSD in a simple way. This procedure was recently applied to different crystallization systems for which CSD data are available in the literature.21 Introducing the dimensionless variables y = n(L)ln*;
x = LIL*
(6)
into Equation 5 leads to a=0.60 Y a=030 La=0 0
(size-indepandant 50
100
Dimensionless
200
x (-)
Figure 1 Variation of dimensionless growth rate with dimensionless crystal size for different values of parameter a.
214
Sep. Technol.,
(7)
growth)
150 size.
y = [a + (1 - u)x](x)‘-“L’-d”d exp[(u - l)L* (x - 1)/d]
1993, vol. 3, October
Figure 2 presents the variation of the dimensionless population density distribution with dimensionless size for different values of parameter a and a constant value of parameter d (= 4 pm). When growth rate increases with size, a increases, and semilog plots that are con-
Modeling
of CSD: J. Mydlatz
and D. Briedis
Application of the proposed hyperbolic model for growth rate estimation Simulated
data
It has been shown that use of simulated data of known characteristics is very useful in making an absolute comparison between the alternative methods of CSD analysis. u-23,36 Two sets of such simulated log population density vs. crystal size data are again considered in the present work. The first set is derived from the linear relation In n(L) = -0.0125L
+ 13.28
(10)
The second set exhibits a significant curvature calculated from the following equation: p-10
L 0
In n(L) = P, exp(P,L”* 100 Dimensionless
200 size, x (-)
300
Figure 2 Dependence of dimensionless population density distribution defined by Equation 7 on dimensionless size for different values of parameter a.
cave upward are obtained. As growth rate becomes less size dependent, a decreases to a value of zero for which the semilog population density plot results in a straight line. For parameter u equal to zero, Equation 7 reduces to a form similar to Equation 2, thus McCabe’s AL law is obeyed. Figure 3 shows the dependence of the log-dimensionless population density distribution versus dimensionless size for different values of parameter d and a constant value of parameter u (= 0.90). As can be seen from this figure, the curvature of the log-population density plot can be considerable, particularly for higher values of parameter d and in the small crystal size range. For a larger size range, however, the steadystate population density obtained for the M-2 model (Equation 7) produces an approximately straight line when plotted on a semilogarithmic scale.
n(L) = -dN(L)ldL
+ P,L) + P,L + Ps (11)
with P, = 29.31, P, = -0.135 pm-“*, P, = 1.82 x 10m3 pm-‘, P4 = -0.0125 pm-’ and Ps = 3.73. The simulated log population data are presented in Figure 4. Direct regression fitting of the first set of log population density data (Equation 10) to Equation 5 results in the following parameters of the size-dependent growth model: a = 1.705 x 10-9, d = 80 pm. Because the value of a is close to zero, Equation 5 can be reduced to a similar form of Equation 2. Assuming that the mean retention time within the crystallizer is 1,080 s, from the value of parameter d = 80 pm = TG*, we can obtain G = G* = 0.08 pm/s. In the case in which the log population density exhibits a significant curvature (Equation ll), direct fitting of Equation 5 gives values of a = 0.179 and d = 88.362 pm. The actual growth rate for the CSD represented by Equation 11 can be calculated from the Sikdar equation3’:
L’=
1 /M-l7
0
0.90
=
7=
MSMPR crystallizercumulative number oversize distribution. The cumulative number oversize distribution, N(L), is defined as N(L) = 1x n(L)dL:
and is
1000
s
(8)
0
Substituting Equations 3 and 8 into the MSMPR population balance Equation 1 leads to N(L) = N*(LIL*)-“L”d exp[(u - l)(L* - L)ld]
(9)
where N* = N(L = L*). For a = 0, the cumulative number oversize distribution (Equation 9) reduces to a form that is linear when plotted in semilogarithmic scale; again, McCabe’s AL law is obeyed. As did Equation 5, this equation contains only two parameters (a and d) and thus also permits direct estimation of the parameters of the hyperbolic growth rate model (Equation 3) from MSMPR crystallizer CSD data.
0
100 DImensionless
200 size, x (-)
300
Figure 3
Variation of dimensionless population-density distribution defined by Equation 7 with dimensionless size for different values of parameter c/.
Sep. Technol.,
1993, vol. 3, October
215
Modeling of CSD: J. Mydlarz and D. Briedis mn(L)dL G(L) = IL 7 n(L)
l
(12)
The integral in Equation 12 was calculated numerically by Simpson’s rule. A comparison of the actual (Equation 12) and predicted (Equation 3) growth rates for the CSD represented by Equation 11 is shown in Figure 5. It is apparent from this figure that the proposed twoparameter hyperbolic size-dependent growth model predicts very closely the exact values of growth rate over the entire size range. It is also interesting to note that use of Equation 5 with parameters a = 0.179 and d = 88.362 pm reproduces the population density data represented by Equation 11 with excellent agreement (see Figure 4). The next section will describe the application of the hyperbolic model to several sets of experimental data.
l
5 0
I 10 Crystal
‘3‘
IdN(L)l
inIn
\
-
205 =:
I size,
20 L (pm)
3o”
Figure 6 CSD of potassium sulfate precipitated with acetone in MSMPR crystallizer.’ o
Equation
(10)
q
Equation
(11)
Experimental MSMPR crystallizer CSD data Eauotion
(51
-6
-12
_
--1Fl ,_ 0
Figure 4
3
The simulated
i a00
1200
600
Crystal
size,
L
(pm)
number population-density
data.
8-
\ 3 “o r 5. x
6-
4d
- n
5 f
-
-
-
MJ-2,MJ-3
G(L)=N(L)/[rn(L)I
2-
e 0 0
400 Crystal
1200
800 size,
L (pm)
Figure 5 Comparison of real growth rate with growth rate estimated by the hyperbolic size-dependent model for population density represented by Equation 11.
216
Sep. Technol.,
1993, vol. 3, October
Potassium sulfate data. The crystallization kinetics of potassium sulfate from aqueous solutions in a continuous MSMPR crystallizer were recently reported by Jones and Mydlarz.’ Figure 6 shows a curved log-population density and log-cumulative number distributions typical of those obtained in the case of drowning-out precipitation of potassium sulfate with pure and slightly diluted acetone. The size distribution presented in this figure was determined with the Malvem Size Analyzer. In the small crystal size range, an increase in population density is exhibited that is greater than what would be expected on the basis of Equation 2. The log population density and log cumulative number data were fitted to the proposed M-2 model (Equations 5 and 9). The values of parameters obtained are included in Table 2. Also included are the values of the MJ-2, MJ-3, and ASL parameters.21 The use of the M-2 population density and cumulative number oversize functions (Equations 5 and 9) gives a better fit to the experimental data (gives the smallest value of variance) than other model tested. Reproduction of logpopulation and log-cumulative number density data by the M-2 model is excellent as shown in Figure 6. Also shown is the best fit In n(L) curve of the continuous ASL population density equation. This curve also smooths the experimental data points very well. The estimated size-dependent growth kinetics of potassium sulfate precipitated from aqueous solution with acetone is presented in Figure 7. The predictions of potassium sulfate growth by fitting the experimental log-population density data to the steady-state population density functions of the MJ-2, MJ-3, M-2, and ASL models (see Table I) are shown in this figure. The predictions of the MJ-2, MJ-3, and M-2 models are quite similar; however, for the larger crystal sizes, the M-2 model predicts higher values of G(L) than both
of CSD: J. Mydlatz
Modeling Table 2
Comparison
of fit of different
model steady-state
a
Model
System
K2S04
M-2 MgSD,j
X 7H,D
MgS04
X 7ti,O
K2=34
MJ-3 MgSO.,
X 7ti,O
K2s04
ASL
MgS04
*Dimensionless
b
j.km-’
-
0.841* 0.882* 0.846’ 0.2211
-
0.154 0.144 5 x 10-s 4.1 x 10-S 0.148 0.078 4.4 x 10-S 3.6 x 1O-3 28.386 0.037
K2s04
MJ-2
X 7H2D
CSD functions to potassium
d
A
In(nO)
sulfate MSMR data In(NO)
pm-lL-l
-
pm
&g
-
35.181 -
24.837
-
1.0071 1.0954 1.0296 1.0775 -
9.592 -
16.768
-
7.435 1.854 1.220 0.203 7.525 3.314 1.301 0.579 9.251 1.756 1.487 0.354 13.600 -
-
-
31.657 12.495
x 102
L-2 prnm2Le2
L-1
-
0.219 0.421 33.196 163.38 -
0.378 0.388 0.712 0.649 0.364 0.248 0.854 0.566 0.711 0.51
and magnesium
and D. Briedis
value.
the MJ-2 and MJ-3 models. Moreover, the values of G(L) estimated using the M-2 model are consistent with the value of G(L) calculated using the WhiteBending-Larson (WBL) equation3: (13)
It has been shown that the WBL equation gives a good estimate of growth rate, particularly for the small crystal size range.i2 Figure 7 also shows the estimated size-dependent growth rate of potassium sulfate predicted by the ASL model. The ASL prediction is different from the MJ2, MJ-3, and M-2 models and from the prediction of the WBL equation (Equation 13). Use of the ASL model gives rise to overestimation of G(L) for larger
crystals and underestimation of G(L) for small crystals. The ASL equation predicts values of G(L) that continue to increase to infinity with increasing crystal size, because lim G(L) = Fi
L-b=
[GO(I + uL)~] + CO
(14)
which is physically unrealistic. This is a main disadvantage of the ASL model. Note that in the case of M-2, MJ-2, and MJ-3 models, G(L) approaches limited values as L + w. This behavior has been often observed.2-8.20”’ Magnesium sulfate beptehydrate data. Figure 8 shows both the population density and cumulative number oversize data that are typical of those reported by
. In[N(L)l . In[n@)l M-2 MJ-2
_---__________ ---
0
MJ_3
ASL 8 WBL method
5 Crystal
10 size,
15
20
L (pm)
Figure 7 Predicted overall growth rate of potassium precipitated with acetone in MSMPR crystallizer.
0
300 Crystal
sulfate
size.
600 L
900 (pm)
Figure 8 CSD of magnesium sulfate produced cooling crystallization in MSMPR crystallizer.”
Sep. Technol.,
by continuous
1993, vol. 3, October
217
Modeling
of CSD: J. Mydlarz
and D. Briedis
Rousseau and Parkd8 for continuous cooling crystallization of magnesium sulfate heptahydrate in a drafttube MSMPR crystallizer. The population density data exhibit a significant curvature particularly in the small size range (Figure 8). The magnesium sulfate log population density data are approximately linear with size for crystals larger than about 400 pm. Here the growth rate is practically size independent and is easily estimated from the slope of the population density data plot and the mean residence time ( = - l/G,,r). Linear regression of the log-population density data for crystal sizes greater than about 400 pm results in a slope equal to - 0.0075 pm-‘. With T = 900 s, the maximum growth rate, G,,,,, approaches a value of about 0.15 pm/s. The fit of the ASL population density curve is also presented in this figure. The curve is based on the values of parameters b, no, and G,, which best fit the experimental data. 38Use of the ASL steady-state population density function (see Table I) with parameter values reported by Rousseau and Parks (1981) (see Table 2) reproduces the experimental population density data with excellent agreement. Direct fitting of the population density and cumulative number oversize distribution data to the M-2 model is also presented in Figure 8. This results in values of parameters that are presented in Table 2. The best fit In n(L) and In N(L) curves represented by Equations 5 and 9, respectively, are also shown in Figure 8. These lines smooth the experimental data points very well for all crystal sizes. Comparison of the variances of fit (see Table 2) indicates that the M-2 steady-state population and cumulative oversize distribution functions offer a slightly better fit than the MJ-2 and MJ-3 and a significantly better fit to experimental data than the ASL steady-state function. A comparison of predicted growth rates for the magnesium sulfate heptahydrate CSD data is shown in Figure 9. The magnesium sulfate growth rates evaluated from the White, Bendig, and Larson (Equation 13) are also shown in the figure. Again, as for potassium sulfate, the ASL equation predicts values of G(L) that
3 2 1.6
- Gm.x=0.15
pm/s
3
_----______-_--l
I
I
200
I
Figure g Predicted overall growth in MSMPR crystallizer.
218
Sep. Technol.,
I
400 Crystal
MJ-i!,MJ-3 ASL WBL method
600 size,
I 800
L (pm)
rate of magnesium
1993, vol. 3, October
sulfate
- - - - -__________
p6
n
Jan%
M-2 MJ-Z,MJ-3 ASL
& Garside
data
20 Crystal
40 size,
5 0
74 s z
ai
G L
2
f S e W
0
0
60
80
L (pm)
Figure 10 Overall crystal growth kinetics of potash alum crystals at 30°C reported by Garside and Jani%.”
continue to increase to infinity with increasing crystal size, which is physically unrealistic. Figure 9 also shows the values of G(L) as given by the MJ-2, MJ-3, and M-2 models, estimated by fitting the magnesium sulfate CSD data to the steady-state population density functions of the MJ-2, MJ-3, M-2, and ASL models. The values of G(L) estimated from the M-2 model are consistent with the values of G(L) estimated from the WBL equation for the entire crystal size range. These predictions are also consistent with experimentally determined magnesium sulfate heptahydrate crystal growth rates measured by other independent techniques. 39-42Furthermore, predictions of the M-2, MJ-2, and MJ-3 size-dependent growth models are consistent with the maximum growth rate, G,,, obtained from the slope of the log-population density data plot for larger crystals. Potash alum batch data Garside and Jancic” reported the results of direct measurements of potash alum crystal growth kinetics in the small size range (3-71 pm). The measurements were made in an agitated seeded crystallizer operated in a batch mode. The growth kinetics were calculated from a transient crystal size distribution represented in terms of cumulative number oversize.” The CSD was measured by a Coulter Counter. Figure IO shows the overall crystal growth rate of potash alum at 30°C versus crystal size for constant supersaturation equal to 0.415 x lo-* kg hydrate/kg solution. These potash alum crystal growth rate data were fitted to size-dependent growth models. The values of the kinetic parameters obtained, as well as variance of fit I?*, are reported in reference 21. The plot of each correlation is presented in Figure 10. Again, the lowest value of variance was obtained by using the proposed hyperbolic M-2 model.*’
Conclusion This work has been primarily concerned application
of the hyperbolic
two-parameter
with the size-de-
Modeling
pendent growth model for direct estimation of crystal growth rates from MSMPR crystallizer population density or cumulative number oversize distribution data. Inherent in this model is prediction of size-dependent behavior in which the growth rate continues to increase with increasing size asymptotically to a maximum. This behavior has been observed for many systems including potash alum4v8; magnesium sulfate heptahydrate38; and potassium sulfate.3*7*20 The proposed size-dependent growth model (Equation 3) results in a relatively simple form of both the population density (Equation 5) and the cumulative number oversize (Equation 9) distributions and also facilitates direct curve fitting of experimental population density (or cumulative number oversize) data to determine the coefficients of the model. In the case of simulated data obeying McCabe’s AL law, the best estimate of growth rate is obtained by direct fitting of population density data to the simple steady-state MSMPR population density distribution function given by Equation 5. When simulated data violating McCabe’s AL law are used, a slightly better fit is offered by both the MJ-2 and MJ-3 models than by the M-2 model. The kinetics data reported in the literature for magnesium sulfate growth rate are consistent with the prediction of the proposed size-dependent growth model, at least for larger crystals. Application of these size-dependent growth models to the batch potash alum growth kinetics data reported by Garside and JanEic (1976) shows that the best fit is obtained using the hyperbolic M-2 model. When the experimental log-population density (or log-cumulative number oversize) plot exhibits a significant curvature over the whole crystal size range, the M-2 model offers a slightly better fit to the experimental data than the MJ-2 and MJ-3 models. Very often, however, the log-population density (or log-cumulative number oversize) plot becomes linear at larger sizes. In this case both the MJ-2 and MJ-3 models give a more realiable value of the effective size-dependent growth rates than the M-2 model.21
References 1. 2. 3. 4. 5. 6. 7.
Randolph, A.D. and Larson, M.A. Theory ofParticulate Processes, 2nd ed New York: Academic Press, 1988 JanCiC, S.J. and Garside, J. On the determination of crystallization kinetics from crystal size distribution data. Chem. Eng. Sci. 1975,30, 1299-1301 White, E.T., Bending, L.L. and Larson, M.A. The effect of size on the growth rate of potassium sulfate crystals. AZChE Symp. Ser., No. 153, 1976, 72,41-47 Garside, J. and JanEiC, S.J. Prediction and measurement of crystal size distribution for size-dependent growth. Chem. Eng. Sci. 1978,33, 1623-1630 Garside, J. and JanEE, S.J. Measurement and scale-up of secondary nucleation kinetics for the notash alum-water svstern. AZCE J. 1979, 25, 948-958 Jones. A.G.. Budz. J. and Mullin. J.W. Crvstallization kinetics of potassium sulfate in an MSMPR agitated vessel. AZChE J. 1986,32,2002-2009 Jones, A.G. and Mydlarz, J. Continuous crystallization and
8.
9. 10. 11.
12. 13. 14. 15.
16.
17. 18. 19. 20. 21. 22. 23.
24. 25. 26. 27.
28. 29. 30. 31.
of CSD: J. Mydlatz
and 0. Briedis
subsequent solid-liquid separation of potassium sulphate: I. MSMPR kinetics. Chem. Eng. Res. Des. 1989, 67, 283-293 Jones, A.G. and Mydlarz, J. Continuous crystallization of potash alum: MSMPR kinetics. Can. J. Chem. Eng. 1990,68, 250-259 Jones, A.G. and Mullin, J.W. Programmed cooling crystallization of wtassium sulfate solutions. Chem. Eng. Sci. 1974,29, lOS-118 Garside, J., Mullin, J.W. and Das, S.N. Growth and dissolution of potassium sulfate in an agitated vessel. Znd. Eng. Chem. Fundam. 1974, l3, 299-305 Garside, J. and JanEP, S.J. Growth and dissolution of potash alum crystals in the subsieve size range. AZChE J. 1976, 22, 887-894 Mydlarz, J. and Jones, A.G. Growth and dissolution kinetics of potassium sulfate crystals in aqueous 2-propanol solutions. Chem. Eng. Sci. 1989,44, 1391-1402 Molnar, I., Halasz, S. and Blickle, T. Determination of sizedependent crystal growth characteristics from batch experiments. Chem. Eng. Sci. 1990,45, 1243-1251 Garside, J., Gaska, C. and Mullin, J.W. Crystal growth rate studies with potassium sulfate in a fluidized bed crystallizer. J. Cryst. G&h 1972, W/14, 510-516 Garside. J. and Mullin. J.W. The crystallization of aluminum potassium sulfate: A study in the assessment of crystallizer design data: III. Growth and dissolution rates. Trans. Inst. Chem. Eng. 1968, 46, 11-18 Mullin, J.W. and Garside, J. The crystallization of aluminum potassium sulfate: A study in the assessment of crystallizer design data: II. Growth in a fluidized bed crystallizer. Trans. Inst. Chem. Eng. 1%7,45, 291-295 Mullin, J.W. and Gaska, C. Potassium sulfate crystal growth rates in aqueous solution. Can. J. Chem. Eng. 1969, 47, 483-489 Hemdon, R.C. and Kirwan, D.J. Size-dependent singlecrystal growth kinetics. AZChE Symp. Ser., No. 215,1982,78,19-25 Abeam. _-. G.F.. Stevens. J.D. and Larson, M.A. Crystal size distribution in continuous crystallizer when growth rate is sizedependent. AZChE J. 1%8,14, 118-122 Mydlarz, J. and Jones, A.G. On modeling the size-dependent growth rate of potassium sulphate in an MSMPR crystallizer. Chem. Eng. Commun. 1990,98,47-56. Mydlarz, J. Kinetics of crystal growth in an MSMPR crystallizer and a new crystal growth rate models. Habilitation thesis, 1990. Wroclaw Technical University, Wroclaw Mydlarz, J. and Jones, A.G. On numerical computation of size-dependent growth rates. Comput. Chem. Eng. 1989,l3, 959-965 Mydlarz, J. and Jones, A.G. On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizer. Chem. kng. J., in press Janse, A.H. and deJong, E.J. The occurrence of growth dispersion and its consequences. In Industrial Crystallization ‘78, J. Mullin, ed. New York: Plenum Press, 1978, p. 195 Randolph, A.D. and White, E.T. Modelling size dispersion in the prediction of crystal size distribution. Chem. Eng. Sci. 1977,32, 1067-1076 Tavare, N.S. and Garside, J. The characterization of growth dispersion. In Industrial Crystallization ‘81. S.J. JanEic and E.J. deJong, eds. North Holland: Amsterdam, 1982, p. 21 Larson, M.A., White, E.T., Ramanarayanan, K.A. and Berglund, K.A. Growth rate dispersion in MSMPR crystallizers. AZChE J. 1985,31,90-94 Ulrich, J. Growth rate dispersion: A review. Cryst. Res. Technol. 1989,24, 249 Zumstein, R.C. and Rousseau, R.W. Growth rate dispersion by initial growth rate distributions and growth rate fluctuations. AZChE J. 1987.33. 121-129 Wang, S. and Mersmann, A. Initial size-dependent growth rate disnersion of attrition fragments and secondary nuclei. Chem. Eng. Sci. 1992,47, 1365 Mydlarz, J. and Briedis, D. Growth rate dispersion vs. sizedependent growth rate for MSMPR crystallizer data. Comp. and Chem. Eng. 1992,16,917-922
Sep. Technol.,
1993, vol. 3, October
219
Modeling 32.
33.
34. 35.
36. 37.
220
of CSD: J. Mydlarz
and D. Briedis
Mydlarz, J. The calculation of crystal growth kinetics from linear In n(L) vs. L steady-state data including growth rate dispersion. Paper presented at the annual AIChE meeting, 2-5 November 1992, Miami, FL, USA Mydlarz, J. On crystallization kinetics estimation in an MSMPR crystallizer in the presence of growth rate dispersion. Accepted for presentation at the 12th Symposium on Industrial Crystallization ‘93, 21-23 September 1993, Warsaw, Poland Mullin, J.W. Crystallisation, 2nd ed. London: Butterworths, 1972 JanEif, S.J. and Garside, J. A new technique for crystal size distribution analysis in an MSMPR crystallizer. In Industrial Crystallization ‘76. J.W. Mullin, ed. New York: Plenum Press, 1976, p. 363 Mydlarz, J. and Briedis, D. Calculation of crystal growth rate for MSMPR crystallizer data. In Industrial Crystallization ‘90. A. Mersmann, ed. GVC-VDI, Dusseldorf, 1990, pp. 577-588 Sikdar, SK. Size-dependent growth rate from curved In n(L)
Sep. Technol.,
1993, vol. 3, October
38. 39. 40. 41.
42.
vs. L steady state data. Ind. Eng. Chem. Fundam. 1977,16, 390-391 Rousseau, R.W. and Parks, R.M. Size dependent growth of magnesium sulfate heptahydrate. Ind. Eng. Chem. Fundam. 1981,20,71-76 Clontz, N.A., Johnson, R.T., McCabe, W.L. and Rousseau, R.W. Growth of magnesium sulfate heptahydrate crystals from solution. Ind. Eng. Chem. Fundam. 1972,11,368-373 Karpinski, P.H. Mass crystallization in a fluidized bed. Monograph 40 (22). Technical University of Wroclaw, Wroclaw, 1981 Karpinski, P.H., Budz, J. and Larson, M.A. Influence of cationic admixtures on kinetics of crystal growth from aqueous solutions. Industrial Crystallization ‘84. S.J. JanCD and E.J. deJong, eds. North Holland: Amsterdam, 1984, p. 85 Mydlarz, J. The effect of Ni(I1) on the growth of magnesium sulfate from aqueous solution. In Industrial Crystallization ‘90. A. Mersmann, ed. GVC-VDI, Dusseldorf, 1990,615-620
of Chemical Engineering,
Michigan State University, East Lansing,
MI, USA
A revised empirical two parameter size-dependent growth rate model proposed recently is discussed in detail and used to illustrate the analysis of both simulated and real population density and cumulative number oversize distribution data from a mixed suspension, mixed product removal crystallizer. The two-parameter model has superior curve-fitting properties compared with other models tested and shows improved consistency in the prediction of the growth rate in the case in which log population density plot exhibits curvature across a large crystal size range.
Keywords: mixed suspension, gr&vth rate
mixed product
removal crystallizer;
Introduction In recent years, the population balance technique developed by Randolph and Larson’ for the continuous mixed suspension, mixed product removal (MSMPR) crystallizer has been used in an effort to understand the mechanism of crystallization from solution. This model predicts that, for an ideal MSMPR crystallizer, the plot of log population density vs. size will result in a straight line if growth rate is constant for all crystals and nuclei are near zero in size. In several systems, nonlinearity in log population density vs. size curves is observed over a wide range of crystal sizes. Various causes for this deviation have been suggested in the literature including (1) secondary nucleation in a small, finite size range; (2) size-dependent growth rate; and (3) growth rate dispersion. Several MSMPR crystallizer studies support the idea of a size-dependent growth rate.‘-’ Further support for size-dependent growth rates was obtained using batch suspension crystallizers.9-‘3 Measurements of growth kinetics in fluidized bed crystallizers also supported the idea of increasing growth rate with crystal size.i4-” Finally, measurements of single crystal growth kinetics for potash alum and nickel ammonium
Address reprint requests to Dr. J. Mydlarz at his present address: Eastman Kodak Company, Rochester, NY 14650-1708, USA. Received 25 February 1992; accepted 26 May 1993
212
Sep. Technol.,
1993, vol. 3, October
modeling of growth rate; size-dependent
sulfatei indicated an increase in the growth rate with the size of crystals. Size-dependent growth rate was successfully modeled for several MSMPR crystallizer systems.2*6*8,12+19-21 The effect of size-dependent growth on the crystal size distribution (CSD) obtained in MSMPR crystallizers has been recently analyzed in the literature.20.22-23 The proposed models discussed in these references, the MJ-2, and MJ-3 as well as the ASL model are summarized in Table 1. Recent crystallization studies have confirmed that growth rate dispersion may also be a source of nonlinearity in the log population density plot.24-33 Such curvature can also arise by several mechanisms including variant crystal growth, crystal size classification in the withdrawal stream, and secondary processes such as crystal breakage and agglomeration. Researchers in crystallization still debate about sizedependent growth and growth rate dispersion. As cited previously, experimental evidence exists for both phenomena. These two models have recently been compared for real MSMPR crystallizer data by Mydlarz and Briedis.31 This comparison has shown that in the case of curvature in log population density vs. size data plots, use of the current growth rate dispersion models leads to significant underestimation of growth rate whereas using recently proposed size-dependent growth models gives good estimation of the growth rate especially for large crystal size. In this article, we will discuss these size-dependent growth models. 0
1993 Buttenvorth-Heinemann
Modeling Table 1
The ASL, M-2, MJ-2, and MJ-3 size-dependent
Model
Abbrevi&on
G(L) = G, (1 + aLjb
ASL
+ aL)F%Jxp
Cumulative
function
1 - (1 +
G(L)
=
a+(1
G(L) = G,,,Il
oversize
_ (1 +
sL)‘-b
~(1 - b)rG,
expt-al)1
n(L) = n+exp[a(L
MJ-2
G(L) = G,,,(l - expl-a(L
is a simpler
+ ~11)
exponential
-
L’)l
/v(L) =/v
1 (-,-b),b - 1)
n(L) = Kexp(eLl(A
MJ-3t
form of the Rojkowski
-
model
(,$+b
exp(aLl
-
N(L) = K,&A c+ - l)-“b
I)‘-‘-@‘*
(private communication,
Mydlan
Jones
&
(19931
(2)
density. In thi.iase a semilog plot of population density versus size results in a straight line of slope - l/(e) and intercept In (no), and growth and nucleation kinetics are easily calculated.
= 0
Proposed hyperbolic growth rate model M-2 In this work, use of recently proposed empirical two-parameter hyperbolic growth rate model (M-2)*l is discussed in detail. The model satisfies most of the essential model requirements3:
(1)
where G(L) is the crystal growth rate, n(L) is the population density, L is the crystal size, and 7 is the mean
r--
Jones
where no is the lim n(L), i.e., the zero-size population-
balance
+ n(L)/7
Mydlan & (1999.1990)
n(L) = no exp( - LIGr)
For an MSMPR crystallizer operating at steady state, Randolph and Larson’ have shown that the general population balance equation reduces to d[G(L)n(L)]ldL
This work
retention time of suspension within the crystallizer. For size-independent growth, G(L) = G = constant, and Equation 1 can be integrated to give
Theory population
t-l* = n(L =.L”l W = N(L = L9 G* = G(L = L”) d = rG* oaas1 n+ = n(L = L*l N* = NIL = L”1 b = erG,,, a>O,b>O A=& b=ar(i, a>O,b>O.c>O K = #(A - l)‘+b”
1999).
The aim of this article is to examine recently proposed empirical size-dependent growth rate models for growth rate estimation from MSMPR crystallizer data. An improved form is proposed that permits direct estimation of model parameters from either the population density or cumulative number oversize distributions.
MSMPR
Abegg, Stevens and Larson (19681
)
-a)LIL*
-
a>0 O
- 1
M-2
exp(aL) exp(aL*L
tMJ-3
number
N(L) = NOewp
sLpb
1 -b (
G* L/L+
and D. Briedis
growth models
Population-density
n(L) = nw
of CSD: J. Mydlatz
Notation
a, c parameters
of size-dependent growth models constant of the MJ-3 model (= exp(ac)), constant of the MJ-2 and MJ-3 models (= aTG,J, B” nucleation rate, #/(s L) constant of the M-2 model (= 7G*), pm d ET dimensionless growth rate (= G(L)IG*), G linear growth rate, pm/s, m/s GWI limiting growth rate for large crystals, pm/s GO growth rate of crystals at zero site, pm/s G* growth rate of crystals at size L*, pm/s K constant of the MJ-3 model (= n”(A - l)“‘),#/(pm L) constant of the MJ-3 model KN (= N”(A - I)l’b),#/L L crystal size, pm nuclei size, pm LN
A b
n(L) no nN
N(L) N” X
Y
population density, #/(pm L) zero-size population density, #/km L) nuclei population density, #/(pm L) cumulative number oversize distribution, #IL zero-size cumulative number oversize distribution, #IL dimensionless crystal size (= L/L*), dimensionless population density (= n(L)ln*), -
Greek letters 7
mean residence time, s variance of fit, pm-* variance of fit in log n(L) (or log N(L)) plane, pm-*L-*, L-*
Sep. Technol.,
1993, vol. 3, October
213
Modeling G(L)
of CSD: J. Mydlam and 0. Briedis = G*(LIL*)I[a
+ (1 - a)LIL*];
0 I a 5 1 (3)
where L is the crystal size, L* is the chosen crystal size, G* is the growth rate at crystal size L*, and a is a coefficient. G(L) approaches values of G* as L j L* and zero for L = 0. The latter condition can present conceptual difficulties, however, in estimation of the zero-size nucleation rate. Because only positive sizedependent crystal growth rates are observed in practice, the value of parameter a should also be positive. For a = 0, G(L) = G* = constant demonstrating that this size-dependent growth model is also capable of describing systems for which McCabe’s AL law holds. To examine the influence of the parameter a on the predicted G(L), Equation 3 can be written in dimensionless form as g(x) = G(L)IG*
= xl[u + (1 - a)~]
(4)
where x = L/L*. Figure I shows the dimensionless growth rate curves with dimensionless size x for different values of parameter a. For any value of parameter u (except a = 0 or a = l), the growth rate increases with increasing size to an asymptotic maximum. The “critical” size for which the growth rate becomes size independent (G(L) reaches its maximum value) is strongly related to the value of the parameter a. The smaller the value of parameter a, the smaller the value of the “critical” size. For a = 0, however, the g(x) results in horizontal straight line, indicating size-independent growth rate, as discussed earlier. There is a large body of evidence in the literature that, in continuous crystallization, the secondary nuclei are a substantial source of new particles. Conversely, these secondary nuclei can only survive in a supersaturated solution when they are larger than the so-called critical size, Lcrit, which, of course, has a positive value. It is well known that the Lcrit is related to the actual supersaturation; the lower is the supersat-
a=0.90
uration, the larger the Lctit.34Thus in continuous crystallizer, the Lcritis fairly large, because continuous crystallizers operate with a small supersaturation. Logically, no = n(L = 0) is equal exactly to zero, because there are no particles that have a size equal to zero, or there are no crystals in a real MSMPR crystallizer that have size smaller than the Lcrit. The proposed M-2 models suffers from an inability to estimate zero-size crystal growth rate (G,) directly; however, there are no zero-size crystals in any real crystallizer, and the zero-size growth rate, G,, only has hypothetical meaning. In view of the preceding, the zero-size growth rate, G,, is only a mathematical concept that is frequently used in deducing a “hypothetical” nucleation rate at zero size, B”. Nevertheless, Jancic and Garside35 showed that the nucleation rate can also be calculated from the mean residence time and population density data only. MSMPR crystallizer population density distribution. Using the proposed hyperbolic size-dependent model (Equation 3), one can integrate the MSMPR population balance (Equation 1) between sizes L* and L (L > L* > 0) to give the simple steady-state population density distribution21 n(L) = n* a + (1 - (I)$
I(L1
ew
(-oL*-d)ld
L*
[
Y(L
- L*)
1
(5)
where the superscript * has the same meaning as in Equation 3, and TG* = d. Although this equation does not predict the population density at zero size, it is possible to choose a value of L* arbitrarily close to zero. In the practical application of Equation 5, L* is the smallest measured crystal size that is dependent on the size measurement technique used. Taking logarithms of both sides of this equation leads to an expression that is a very convenient form for fitting of experimental log population density data. Equation 5 contains only two parameters (a and d); thus this equation permits direct estimation of the parameters of the hyperbolic size-dependent growth rate model (Equation 3) from MSMPR crystallizer CSD in a simple way. This procedure was recently applied to different crystallization systems for which CSD data are available in the literature.21 Introducing the dimensionless variables y = n(L)ln*;
x = LIL*
(6)
into Equation 5 leads to a=0.60 Y a=030 La=0 0
(size-indepandant 50
100
Dimensionless
200
x (-)
Figure 1 Variation of dimensionless growth rate with dimensionless crystal size for different values of parameter a.
214
Sep. Technol.,
(7)
growth)
150 size.
y = [a + (1 - u)x](x)‘-“L’-d”d exp[(u - l)L* (x - 1)/d]
1993, vol. 3, October
Figure 2 presents the variation of the dimensionless population density distribution with dimensionless size for different values of parameter a and a constant value of parameter d (= 4 pm). When growth rate increases with size, a increases, and semilog plots that are con-
Modeling
of CSD: J. Mydlatz
and D. Briedis
Application of the proposed hyperbolic model for growth rate estimation Simulated
data
It has been shown that use of simulated data of known characteristics is very useful in making an absolute comparison between the alternative methods of CSD analysis. u-23,36 Two sets of such simulated log population density vs. crystal size data are again considered in the present work. The first set is derived from the linear relation In n(L) = -0.0125L
+ 13.28
(10)
The second set exhibits a significant curvature calculated from the following equation: p-10
L 0
In n(L) = P, exp(P,L”* 100 Dimensionless
200 size, x (-)
300
Figure 2 Dependence of dimensionless population density distribution defined by Equation 7 on dimensionless size for different values of parameter a.
cave upward are obtained. As growth rate becomes less size dependent, a decreases to a value of zero for which the semilog population density plot results in a straight line. For parameter u equal to zero, Equation 7 reduces to a form similar to Equation 2, thus McCabe’s AL law is obeyed. Figure 3 shows the dependence of the log-dimensionless population density distribution versus dimensionless size for different values of parameter d and a constant value of parameter u (= 0.90). As can be seen from this figure, the curvature of the log-population density plot can be considerable, particularly for higher values of parameter d and in the small crystal size range. For a larger size range, however, the steadystate population density obtained for the M-2 model (Equation 7) produces an approximately straight line when plotted on a semilogarithmic scale.
n(L) = -dN(L)ldL
+ P,L) + P,L + Ps (11)
with P, = 29.31, P, = -0.135 pm-“*, P, = 1.82 x 10m3 pm-‘, P4 = -0.0125 pm-’ and Ps = 3.73. The simulated log population data are presented in Figure 4. Direct regression fitting of the first set of log population density data (Equation 10) to Equation 5 results in the following parameters of the size-dependent growth model: a = 1.705 x 10-9, d = 80 pm. Because the value of a is close to zero, Equation 5 can be reduced to a similar form of Equation 2. Assuming that the mean retention time within the crystallizer is 1,080 s, from the value of parameter d = 80 pm = TG*, we can obtain G = G* = 0.08 pm/s. In the case in which the log population density exhibits a significant curvature (Equation ll), direct fitting of Equation 5 gives values of a = 0.179 and d = 88.362 pm. The actual growth rate for the CSD represented by Equation 11 can be calculated from the Sikdar equation3’:
L’=
1 /M-l7
0
0.90
=
7=
MSMPR crystallizercumulative number oversize distribution. The cumulative number oversize distribution, N(L), is defined as N(L) = 1x n(L)dL:
and is
1000
s
(8)
0
Substituting Equations 3 and 8 into the MSMPR population balance Equation 1 leads to N(L) = N*(LIL*)-“L”d exp[(u - l)(L* - L)ld]
(9)
where N* = N(L = L*). For a = 0, the cumulative number oversize distribution (Equation 9) reduces to a form that is linear when plotted in semilogarithmic scale; again, McCabe’s AL law is obeyed. As did Equation 5, this equation contains only two parameters (a and d) and thus also permits direct estimation of the parameters of the hyperbolic growth rate model (Equation 3) from MSMPR crystallizer CSD data.
0
100 DImensionless
200 size, x (-)
300
Figure 3
Variation of dimensionless population-density distribution defined by Equation 7 with dimensionless size for different values of parameter c/.
Sep. Technol.,
1993, vol. 3, October
215
Modeling of CSD: J. Mydlarz and D. Briedis mn(L)dL G(L) = IL 7 n(L)
l
(12)
The integral in Equation 12 was calculated numerically by Simpson’s rule. A comparison of the actual (Equation 12) and predicted (Equation 3) growth rates for the CSD represented by Equation 11 is shown in Figure 5. It is apparent from this figure that the proposed twoparameter hyperbolic size-dependent growth model predicts very closely the exact values of growth rate over the entire size range. It is also interesting to note that use of Equation 5 with parameters a = 0.179 and d = 88.362 pm reproduces the population density data represented by Equation 11 with excellent agreement (see Figure 4). The next section will describe the application of the hyperbolic model to several sets of experimental data.
l
5 0
I 10 Crystal
‘3‘
IdN(L)l
inIn
\
-
205 =:
I size,
20 L (pm)
3o”
Figure 6 CSD of potassium sulfate precipitated with acetone in MSMPR crystallizer.’ o
Equation
(10)
q
Equation
(11)
Experimental MSMPR crystallizer CSD data Eauotion
(51
-6
-12
_
--1Fl ,_ 0
Figure 4
3
The simulated
i a00
1200
600
Crystal
size,
L
(pm)
number population-density
data.
8-
\ 3 “o r 5. x
6-
4d
- n
5 f
-
-
-
MJ-2,MJ-3
G(L)=N(L)/[rn(L)I
2-
e 0 0
400 Crystal
1200
800 size,
L (pm)
Figure 5 Comparison of real growth rate with growth rate estimated by the hyperbolic size-dependent model for population density represented by Equation 11.
216
Sep. Technol.,
1993, vol. 3, October
Potassium sulfate data. The crystallization kinetics of potassium sulfate from aqueous solutions in a continuous MSMPR crystallizer were recently reported by Jones and Mydlarz.’ Figure 6 shows a curved log-population density and log-cumulative number distributions typical of those obtained in the case of drowning-out precipitation of potassium sulfate with pure and slightly diluted acetone. The size distribution presented in this figure was determined with the Malvem Size Analyzer. In the small crystal size range, an increase in population density is exhibited that is greater than what would be expected on the basis of Equation 2. The log population density and log cumulative number data were fitted to the proposed M-2 model (Equations 5 and 9). The values of parameters obtained are included in Table 2. Also included are the values of the MJ-2, MJ-3, and ASL parameters.21 The use of the M-2 population density and cumulative number oversize functions (Equations 5 and 9) gives a better fit to the experimental data (gives the smallest value of variance) than other model tested. Reproduction of logpopulation and log-cumulative number density data by the M-2 model is excellent as shown in Figure 6. Also shown is the best fit In n(L) curve of the continuous ASL population density equation. This curve also smooths the experimental data points very well. The estimated size-dependent growth kinetics of potassium sulfate precipitated from aqueous solution with acetone is presented in Figure 7. The predictions of potassium sulfate growth by fitting the experimental log-population density data to the steady-state population density functions of the MJ-2, MJ-3, M-2, and ASL models (see Table I) are shown in this figure. The predictions of the MJ-2, MJ-3, and M-2 models are quite similar; however, for the larger crystal sizes, the M-2 model predicts higher values of G(L) than both
of CSD: J. Mydlatz
Modeling Table 2
Comparison
of fit of different
model steady-state
a
Model
System
K2S04
M-2 MgSD,j
X 7H,D
MgS04
X 7ti,O
K2=34
MJ-3 MgSO.,
X 7ti,O
K2s04
ASL
MgS04
*Dimensionless
b
j.km-’
-
0.841* 0.882* 0.846’ 0.2211
-
0.154 0.144 5 x 10-s 4.1 x 10-S 0.148 0.078 4.4 x 10-S 3.6 x 1O-3 28.386 0.037
K2s04
MJ-2
X 7H2D
CSD functions to potassium
d
A
In(nO)
sulfate MSMR data In(NO)
pm-lL-l
-
pm
&g
-
35.181 -
24.837
-
1.0071 1.0954 1.0296 1.0775 -
9.592 -
16.768
-
7.435 1.854 1.220 0.203 7.525 3.314 1.301 0.579 9.251 1.756 1.487 0.354 13.600 -
-
-
31.657 12.495
x 102
L-2 prnm2Le2
L-1
-
0.219 0.421 33.196 163.38 -
0.378 0.388 0.712 0.649 0.364 0.248 0.854 0.566 0.711 0.51
and magnesium
and D. Briedis
value.
the MJ-2 and MJ-3 models. Moreover, the values of G(L) estimated using the M-2 model are consistent with the value of G(L) calculated using the WhiteBending-Larson (WBL) equation3: (13)
It has been shown that the WBL equation gives a good estimate of growth rate, particularly for the small crystal size range.i2 Figure 7 also shows the estimated size-dependent growth rate of potassium sulfate predicted by the ASL model. The ASL prediction is different from the MJ2, MJ-3, and M-2 models and from the prediction of the WBL equation (Equation 13). Use of the ASL model gives rise to overestimation of G(L) for larger
crystals and underestimation of G(L) for small crystals. The ASL equation predicts values of G(L) that continue to increase to infinity with increasing crystal size, because lim G(L) = Fi
L-b=
[GO(I + uL)~] + CO
(14)
which is physically unrealistic. This is a main disadvantage of the ASL model. Note that in the case of M-2, MJ-2, and MJ-3 models, G(L) approaches limited values as L + w. This behavior has been often observed.2-8.20”’ Magnesium sulfate beptehydrate data. Figure 8 shows both the population density and cumulative number oversize data that are typical of those reported by
. In[N(L)l . In[n@)l M-2 MJ-2
_---__________ ---
0
MJ_3
ASL 8 WBL method
5 Crystal
10 size,
15
20
L (pm)
Figure 7 Predicted overall growth rate of potassium precipitated with acetone in MSMPR crystallizer.
0
300 Crystal
sulfate
size.
600 L
900 (pm)
Figure 8 CSD of magnesium sulfate produced cooling crystallization in MSMPR crystallizer.”
Sep. Technol.,
by continuous
1993, vol. 3, October
217
Modeling
of CSD: J. Mydlarz
and D. Briedis
Rousseau and Parkd8 for continuous cooling crystallization of magnesium sulfate heptahydrate in a drafttube MSMPR crystallizer. The population density data exhibit a significant curvature particularly in the small size range (Figure 8). The magnesium sulfate log population density data are approximately linear with size for crystals larger than about 400 pm. Here the growth rate is practically size independent and is easily estimated from the slope of the population density data plot and the mean residence time ( = - l/G,,r). Linear regression of the log-population density data for crystal sizes greater than about 400 pm results in a slope equal to - 0.0075 pm-‘. With T = 900 s, the maximum growth rate, G,,,,, approaches a value of about 0.15 pm/s. The fit of the ASL population density curve is also presented in this figure. The curve is based on the values of parameters b, no, and G,, which best fit the experimental data. 38Use of the ASL steady-state population density function (see Table I) with parameter values reported by Rousseau and Parks (1981) (see Table 2) reproduces the experimental population density data with excellent agreement. Direct fitting of the population density and cumulative number oversize distribution data to the M-2 model is also presented in Figure 8. This results in values of parameters that are presented in Table 2. The best fit In n(L) and In N(L) curves represented by Equations 5 and 9, respectively, are also shown in Figure 8. These lines smooth the experimental data points very well for all crystal sizes. Comparison of the variances of fit (see Table 2) indicates that the M-2 steady-state population and cumulative oversize distribution functions offer a slightly better fit than the MJ-2 and MJ-3 and a significantly better fit to experimental data than the ASL steady-state function. A comparison of predicted growth rates for the magnesium sulfate heptahydrate CSD data is shown in Figure 9. The magnesium sulfate growth rates evaluated from the White, Bendig, and Larson (Equation 13) are also shown in the figure. Again, as for potassium sulfate, the ASL equation predicts values of G(L) that
3 2 1.6
- Gm.x=0.15
pm/s
3
_----______-_--l
I
I
200
I
Figure g Predicted overall growth in MSMPR crystallizer.
218
Sep. Technol.,
I
400 Crystal
MJ-i!,MJ-3 ASL WBL method
600 size,
I 800
L (pm)
rate of magnesium
1993, vol. 3, October
sulfate
- - - - -__________
p6
n
Jan%
M-2 MJ-Z,MJ-3 ASL
& Garside
data
20 Crystal
40 size,
5 0
74 s z
ai
G L
2
f S e W
0
0
60
80
L (pm)
Figure 10 Overall crystal growth kinetics of potash alum crystals at 30°C reported by Garside and Jani%.”
continue to increase to infinity with increasing crystal size, which is physically unrealistic. Figure 9 also shows the values of G(L) as given by the MJ-2, MJ-3, and M-2 models, estimated by fitting the magnesium sulfate CSD data to the steady-state population density functions of the MJ-2, MJ-3, M-2, and ASL models. The values of G(L) estimated from the M-2 model are consistent with the values of G(L) estimated from the WBL equation for the entire crystal size range. These predictions are also consistent with experimentally determined magnesium sulfate heptahydrate crystal growth rates measured by other independent techniques. 39-42Furthermore, predictions of the M-2, MJ-2, and MJ-3 size-dependent growth models are consistent with the maximum growth rate, G,,, obtained from the slope of the log-population density data plot for larger crystals. Potash alum batch data Garside and Jancic” reported the results of direct measurements of potash alum crystal growth kinetics in the small size range (3-71 pm). The measurements were made in an agitated seeded crystallizer operated in a batch mode. The growth kinetics were calculated from a transient crystal size distribution represented in terms of cumulative number oversize.” The CSD was measured by a Coulter Counter. Figure IO shows the overall crystal growth rate of potash alum at 30°C versus crystal size for constant supersaturation equal to 0.415 x lo-* kg hydrate/kg solution. These potash alum crystal growth rate data were fitted to size-dependent growth models. The values of the kinetic parameters obtained, as well as variance of fit I?*, are reported in reference 21. The plot of each correlation is presented in Figure 10. Again, the lowest value of variance was obtained by using the proposed hyperbolic M-2 model.*’
Conclusion This work has been primarily concerned application
of the hyperbolic
two-parameter
with the size-de-
Modeling
pendent growth model for direct estimation of crystal growth rates from MSMPR crystallizer population density or cumulative number oversize distribution data. Inherent in this model is prediction of size-dependent behavior in which the growth rate continues to increase with increasing size asymptotically to a maximum. This behavior has been observed for many systems including potash alum4v8; magnesium sulfate heptahydrate38; and potassium sulfate.3*7*20 The proposed size-dependent growth model (Equation 3) results in a relatively simple form of both the population density (Equation 5) and the cumulative number oversize (Equation 9) distributions and also facilitates direct curve fitting of experimental population density (or cumulative number oversize) data to determine the coefficients of the model. In the case of simulated data obeying McCabe’s AL law, the best estimate of growth rate is obtained by direct fitting of population density data to the simple steady-state MSMPR population density distribution function given by Equation 5. When simulated data violating McCabe’s AL law are used, a slightly better fit is offered by both the MJ-2 and MJ-3 models than by the M-2 model. The kinetics data reported in the literature for magnesium sulfate growth rate are consistent with the prediction of the proposed size-dependent growth model, at least for larger crystals. Application of these size-dependent growth models to the batch potash alum growth kinetics data reported by Garside and JanEic (1976) shows that the best fit is obtained using the hyperbolic M-2 model. When the experimental log-population density (or log-cumulative number oversize) plot exhibits a significant curvature over the whole crystal size range, the M-2 model offers a slightly better fit to the experimental data than the MJ-2 and MJ-3 models. Very often, however, the log-population density (or log-cumulative number oversize) plot becomes linear at larger sizes. In this case both the MJ-2 and MJ-3 models give a more realiable value of the effective size-dependent growth rates than the M-2 model.21
References 1. 2. 3. 4. 5. 6. 7.
Randolph, A.D. and Larson, M.A. Theory ofParticulate Processes, 2nd ed New York: Academic Press, 1988 JanCiC, S.J. and Garside, J. On the determination of crystallization kinetics from crystal size distribution data. Chem. Eng. Sci. 1975,30, 1299-1301 White, E.T., Bending, L.L. and Larson, M.A. The effect of size on the growth rate of potassium sulfate crystals. AZChE Symp. Ser., No. 153, 1976, 72,41-47 Garside, J. and JanEiC, S.J. Prediction and measurement of crystal size distribution for size-dependent growth. Chem. Eng. Sci. 1978,33, 1623-1630 Garside, J. and JanEE, S.J. Measurement and scale-up of secondary nucleation kinetics for the notash alum-water svstern. AZCE J. 1979, 25, 948-958 Jones. A.G.. Budz. J. and Mullin. J.W. Crvstallization kinetics of potassium sulfate in an MSMPR agitated vessel. AZChE J. 1986,32,2002-2009 Jones, A.G. and Mydlarz, J. Continuous crystallization and
8.
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subsequent solid-liquid separation of potassium sulphate: I. MSMPR kinetics. Chem. Eng. Res. Des. 1989, 67, 283-293 Jones, A.G. and Mydlarz, J. Continuous crystallization of potash alum: MSMPR kinetics. Can. J. Chem. Eng. 1990,68, 250-259 Jones, A.G. and Mullin, J.W. Programmed cooling crystallization of wtassium sulfate solutions. Chem. Eng. Sci. 1974,29, lOS-118 Garside, J., Mullin, J.W. and Das, S.N. Growth and dissolution of potassium sulfate in an agitated vessel. Znd. Eng. Chem. Fundam. 1974, l3, 299-305 Garside, J. and JanEP, S.J. Growth and dissolution of potash alum crystals in the subsieve size range. AZChE J. 1976, 22, 887-894 Mydlarz, J. and Jones, A.G. Growth and dissolution kinetics of potassium sulfate crystals in aqueous 2-propanol solutions. Chem. Eng. Sci. 1989,44, 1391-1402 Molnar, I., Halasz, S. and Blickle, T. Determination of sizedependent crystal growth characteristics from batch experiments. Chem. Eng. Sci. 1990,45, 1243-1251 Garside, J., Gaska, C. and Mullin, J.W. Crystal growth rate studies with potassium sulfate in a fluidized bed crystallizer. J. Cryst. G&h 1972, W/14, 510-516 Garside. J. and Mullin. J.W. The crystallization of aluminum potassium sulfate: A study in the assessment of crystallizer design data: III. Growth and dissolution rates. Trans. Inst. Chem. Eng. 1968, 46, 11-18 Mullin, J.W. and Garside, J. The crystallization of aluminum potassium sulfate: A study in the assessment of crystallizer design data: II. Growth in a fluidized bed crystallizer. Trans. Inst. Chem. Eng. 1%7,45, 291-295 Mullin, J.W. and Gaska, C. Potassium sulfate crystal growth rates in aqueous solution. Can. J. Chem. Eng. 1969, 47, 483-489 Hemdon, R.C. and Kirwan, D.J. Size-dependent singlecrystal growth kinetics. AZChE Symp. Ser., No. 215,1982,78,19-25 Abeam. _-. G.F.. Stevens. J.D. and Larson, M.A. Crystal size distribution in continuous crystallizer when growth rate is sizedependent. AZChE J. 1%8,14, 118-122 Mydlarz, J. and Jones, A.G. On modeling the size-dependent growth rate of potassium sulphate in an MSMPR crystallizer. Chem. Eng. Commun. 1990,98,47-56. Mydlarz, J. Kinetics of crystal growth in an MSMPR crystallizer and a new crystal growth rate models. Habilitation thesis, 1990. Wroclaw Technical University, Wroclaw Mydlarz, J. and Jones, A.G. On numerical computation of size-dependent growth rates. Comput. Chem. Eng. 1989,l3, 959-965 Mydlarz, J. and Jones, A.G. On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizer. Chem. kng. J., in press Janse, A.H. and deJong, E.J. The occurrence of growth dispersion and its consequences. In Industrial Crystallization ‘78, J. Mullin, ed. New York: Plenum Press, 1978, p. 195 Randolph, A.D. and White, E.T. Modelling size dispersion in the prediction of crystal size distribution. Chem. Eng. Sci. 1977,32, 1067-1076 Tavare, N.S. and Garside, J. The characterization of growth dispersion. In Industrial Crystallization ‘81. S.J. JanEic and E.J. deJong, eds. North Holland: Amsterdam, 1982, p. 21 Larson, M.A., White, E.T., Ramanarayanan, K.A. and Berglund, K.A. Growth rate dispersion in MSMPR crystallizers. AZChE J. 1985,31,90-94 Ulrich, J. Growth rate dispersion: A review. Cryst. Res. Technol. 1989,24, 249 Zumstein, R.C. and Rousseau, R.W. Growth rate dispersion by initial growth rate distributions and growth rate fluctuations. AZChE J. 1987.33. 121-129 Wang, S. and Mersmann, A. Initial size-dependent growth rate disnersion of attrition fragments and secondary nuclei. Chem. Eng. Sci. 1992,47, 1365 Mydlarz, J. and Briedis, D. Growth rate dispersion vs. sizedependent growth rate for MSMPR crystallizer data. Comp. and Chem. Eng. 1992,16,917-922
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Mydlarz, J. The calculation of crystal growth kinetics from linear In n(L) vs. L steady-state data including growth rate dispersion. Paper presented at the annual AIChE meeting, 2-5 November 1992, Miami, FL, USA Mydlarz, J. On crystallization kinetics estimation in an MSMPR crystallizer in the presence of growth rate dispersion. Accepted for presentation at the 12th Symposium on Industrial Crystallization ‘93, 21-23 September 1993, Warsaw, Poland Mullin, J.W. Crystallisation, 2nd ed. London: Butterworths, 1972 JanEif, S.J. and Garside, J. A new technique for crystal size distribution analysis in an MSMPR crystallizer. In Industrial Crystallization ‘76. J.W. Mullin, ed. New York: Plenum Press, 1976, p. 363 Mydlarz, J. and Briedis, D. Calculation of crystal growth rate for MSMPR crystallizer data. In Industrial Crystallization ‘90. A. Mersmann, ed. GVC-VDI, Dusseldorf, 1990, pp. 577-588 Sikdar, SK. Size-dependent growth rate from curved In n(L)
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42.
vs. L steady state data. Ind. Eng. Chem. Fundam. 1977,16, 390-391 Rousseau, R.W. and Parks, R.M. Size dependent growth of magnesium sulfate heptahydrate. Ind. Eng. Chem. Fundam. 1981,20,71-76 Clontz, N.A., Johnson, R.T., McCabe, W.L. and Rousseau, R.W. Growth of magnesium sulfate heptahydrate crystals from solution. Ind. Eng. Chem. Fundam. 1972,11,368-373 Karpinski, P.H. Mass crystallization in a fluidized bed. Monograph 40 (22). Technical University of Wroclaw, Wroclaw, 1981 Karpinski, P.H., Budz, J. and Larson, M.A. Influence of cationic admixtures on kinetics of crystal growth from aqueous solutions. Industrial Crystallization ‘84. S.J. JanCD and E.J. deJong, eds. North Holland: Amsterdam, 1984, p. 85 Mydlarz, J. The effect of Ni(I1) on the growth of magnesium sulfate from aqueous solution. In Industrial Crystallization ‘90. A. Mersmann, ed. GVC-VDI, Dusseldorf, 1990,615-620