Ind. Eng. Chem. Res. 1989,28, 1718-1725
1718
In this paper evidence is presented to show that latexbased inks are more paper independent than dyed-fluidbased inks. Literature Cited Ashley, C. T.; Edds, K. E.; Elbert, D.L. ZBM J. Res. Deu. 1977,21, 69. Bailey, W. D.;Beach, B. L.; Edds, K. E.; Elbert, D. L. U.S.Patent 4,197,135,1980. Barrett, K.E. J. Dispersion Polymerization in Organic Media; Academic Press: London, 1975. Croucher, M. D.; Lok, K.; Wong, R. W.; Hair, M. L. J.Imaging Sci. 1988,14, 129. Gamble, R. C.; Hair, M. L.; Lukac, S. R.; Taylor, M. G. U.S.Patent 4,783,220,1988. Goldin, M.; Yerushalmi, J.; Pfeffer, R.; Sinnar, R. J. Fluid Mech. 1969,38,689. Hackleman, D. E.; Johnson, L. E.; Norton, K. A. US. Patent 4,670,059,1987). Hiemenz, P. C. Principles of Colloid and Surface Chemistry, Marcel Dekker: New York, 1977. Hotchkiss-Brandt Company. European Patent CP-36790-D41,1981.
Kang, H. R. US. Patent 4,659,382,1987. Napper, D. H. Polymeric Stabilkation of Colloidal Dispersions; Academic Press: New York, 1983. Ober, C. K.;Lok, K.; Wong, R. W.; Hair, M. L.; Croucher, M. D. unpublished results, 1989. Ohta, T.; Yano, Y.; Matsufuji, Y.; Haruta, M.; Eida, T. U S . Patent 4,597,794,1986. Oliver, J. Surface and Colloid Science in Computer Technology; Plenum Press: New York, 1987;p 409. Piirma, I., Ed. Emulsion Polymerisation; Academic Press: New York, 1982. Rayleigh, Lord. Proc. Lond. Math. SOC.1879,10,4;Proc. R. SOC. London 1879,29,71. Solodar, W . E.; Kang, H. R.; Weber, J. R. U.S.Patent 4,789,400, 1988. Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colliods; Elsevier: Amsterdam, 1948. Visser, J. Adv. Colloid Interface Sci. 1972,3, 331. Webber, C.Z.Angew. Math. Mech. 1931,11,136. Yao, Y. U.S. Patent 4,246,154,1978. Received for review August 26, 1988 Accepted May 25, 1989
The Probability Distribution of Growth Rates of Anhydrous Sodium Sulfate Crystals Diana L. Klug* and Robert L. Pigford Department of Chemical Engineering, University o f Delaware, Newark, Delaware 19716
Based on single-crystal studies and batch crystallization experiments of the growth of anhydrous sodium sulfate, it has been found that individual crystals have intrinsic growth activities that do not fluctuate as the crystals grow or as supersaturation levels change. It is proposed, based on an analysis of the Burton, Cabrera, and Frank (BCF) growth model, that the growth kinetics of the crystals can be expressed as the product of an intrinsic level of growth activity and a driving force function. By the use of this stochastic model for crystal growth, a graphical procedure is presented for determining the growth kinetics of crystals grown from batch crystallizers for the general case where supersaturation is time-dependent, and growth dispersion is occurring. The results of the batch crystallization study are compared to single-crystal growth data and are found to be in good agreement.
A number of studies on batch crystallization processes have shown that narrowly sized seed crystals become increasingly broad distributions as growth progresses (White and Wright, 1971; Janse and de Jong, 1976; Tavare and Garside, 1982; JanZii. et al., 1984). Several explanations for this behavior have been postulated, including size-dependent growth and growth rate dispersion. The latter phenomenon has been described in two distinct ways: (a) each crystal’s growth rate fluctuates randomly with time, even though exposed to a constant environment (Randolph and White, 1977),and (b) each crystal inherits an intrinsic growth rate that remains constant as growth proceeds, termed the “CCG” model (Ramanarayanan et al., 1982; Larson et al., 1985). The results of. many single-crystal growth experiments (Davey et al., 1979; Berglund et al., 1983; Shanks and Berglund, 1985; Mathis-Lilley and Berglund, 1985) have tended to support the applicability of the CCG model. There have been few studies to date, however, that have coupled observations of single-growth behavior with
* Current address: Experimental Station, E304, E. I. du Pont de Nemours & Co., Inc., Wilmington, DE 19880-0304.
measurements of crystals grown in agitated crystallizers. As a result, there is little quantitative evidence that the behavior of crystals grown under a microscope describes the behavior observed in an agitated crystallizer. Moreover, the extent to which the performance of a crystallizer can be predicted from single-crystal measurements alone is uncertain. An objective of this study is to provide quantitative information regarding this comparison. Two types of experiments were performed to study the growth behavior of anhydrous sodium sulfate crystals. In the first, single crystals were observed and photographed under magnification with a flow cell similar to that used in other studies (Davey et al., 1979; Garside and RistiE, 1983; Berglund et al., 1983). By the use of this approach, measurements were made over a range of supersaturation and temperatures with prepared seed crystals as well as crystals nucleated in the flow cell. In the second set of experiments, crystal size distributions of sodium sulfate were measured using an isothermal, batch crystallizer. A stochastic model was developed that enabled growth kinetics to be computed from size distribution measurements taken under time-varying supersaturation levels. The results of this study were then com-
0888-5885/89/2628-171~~01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1719
A
J
Stngle Crystol Growth R o t e M e o s u r e m e n t s
Plus One Standard Deviotion c 0
a
4 -
C o r r e l o t l o n of Averoge Growth R a t e s f r o m Unseeded Batch Crystallizer, Eq. ( 2 8 )
2 -
0 ' 0
S t a n d a r d Deviation
5
IO 15 20 T i m e , minutes
25
I
0 -
30
0
Figure 1. Crystal size vs time for several single crystals nucleated and grown in a flow cell at constant temperature and supersaturation level. Supersaturation (a) = 0.088, temperature = 61 OC.
pared with data obtained from the single-crystal growth rate measurements.
0.02
0.06
0.04
S u p e r s o t u r o t i o n Rotio,
0.08
0.10
~7
Figure 2. Comparison of the growth rates of single crystals nucleated and grown in the flow cell (triangles) with the moment analysis results from unseeded batch crystallizer. Solid curve represents the correlation of average growth rates from unseeded batch crystallizer (eq 28). Dashed curves are computed from the spread (standard deviation) of the growth rate activity distribution of crystals grown in the unseeded batch crystallizer.
Experimental Procedure Single-Crystal Study. Aqueous solutions of sodium sulfate were prepared from deionized water and reagentgrade sodium sulfate. To ensure that the salt solution was saturated, it was agitated with excess sodium sulfate crystals at a known constant temperature for a minimum of 24 h prior to the start of an experiment. The salt concentration was determined from temperature measurements with a correlation of solubility vs temperature (Solubilities, 1965; International Critical Tables, 1933). During an experiment, saturated solution was pumped through an inline, 10-pm particle filter and a heat exchanger before reaching the flow cell. Because anhydrous sodium sulfate exhibits relatively temperature-insensitive solubility behavior, a large temperature increase was needed (- 10-30 "C) to produce even a small level of supersaturation. Even so, temperature fluctuations in the flow cell were generally less than *O.l "C. The growth rates were studied over a range of temperatures from 60 to 69 "C and at supersaturation levels, u, from 0.02 to 0.10. Under these conditions, sodium sulfate crystallized with a dipyramidal habit composed of eight equal [lll]faces. To determine the crystal growth rate, the dimension of the longest crystallographic axis was measured from photographs taken at timed intervals over a total period of 20-60 min. Growth experiments using the flow cell were conducted with two groups of crystals: (a) "seed" crystals produced from separate batch crystallization experiments, and (b) crystals nucleated within the flow cell. The latter group of crystals was formed by heterogeneous nucleation on the glass cover slip of the flow cell. In all, as many as 40 crystals could be simultaneously observed and photographed during an experiment. Results Single-Crystal Study. Shown in Figure 1 are measurements of size vs time for several crystals grown at identical conditions of temperature and supersaturation. The linearity of the data indicate that individual crystals maintained constant but independent growth velocities. The results of these tests confirm that sodium sulfate crystals, like many other crystalline materials, exhibit a
range of growth velocities even when exposed to the same solution environment. The growth experiment of Figure 1was repeated at other conditions of temperature and supersaturation using over 200 different single crystals. Analysis of the resulting data indicated that the growth rate was temperature dependent, in agreement with an earlier study of sodium sulfate crystallization (Rosenblatt et al., 1984). To account for the temperature effect, the data were normalized with the following expression, using the value of activation energy determined by Rosenblatt et al. (1984):
G, =
G eXp(-13700/RT)
The resulting temperature-normalized data are plotted as a function of supersaturation in Figure 2. From the spread of the measurements, it is evident that significant growth rate dispersion occurred over the entire range of supersaturation studied. It should be noted that no completely satisfactory means exists for correlating the growth rate data shown in Figure 2. However, a number of investigators have suggested the applicability of the Burton, Cabrera, and Frank (1951) (BCF) model of crystal growth to systems in which growth rate dispersion occurs (Davey et al., 1979; Garside and RistiE, 1983; Zumstein and Rousseau, 1987). In the following section, a general, stochastic growth expression based on this model will be developed to allow a quantitative description of growth rate dispersion.
Derivation of a Stochastic Crystal Growth Model In early attempts to explain finite crystal growth velocities at exceedingly small supersaturation levels, Burton, Carbrera and Frank (BCF) postulated that the crystal faces of some materials contain self-propagating spiral defects that provide a continuous source of steplike dislocations. The dislocations act as growth sites on the crystal because of the relatively low activation energy for incorporation of growth units. A significant amount of experimental and theoretical evidence for this type of growth behavior has been found and is reviewed by Bennema (1976).
1720 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 Table I. Definitions of Statistical Properties of Growth and Population Distributions I. Properties of the Population Density Distribution, n(L) a. jth moment mi = J;LJ n(L) dL
&
,
I
11. Properties of the Growth Rate Probability Distribution, P(g*) I
l
l
-
Figure 3. Distribution of growth rate activities, P(g*),of sodium sulfate crystals as determined from the single-crystal growth study.
The BCF model further assumes that groups of adjacent defects on the crystal face interact in a way to produce growth velocities that can be greater (or less) than if the defects acted independently. The growth behavior is therefore predicted to depend, in a complex way, on the number and orientation of defects on the crystal face. It thus appears likely that the randomness of defect interactions during crystal growth may be a cause of growth rate dispersion. According to the BCF theory, the linear growth rate of a crystal is related to supersaturation by the formula
where a1and C are temperature-dependent constants. The interaction of defects and the resulting "growth rate activity" of the crystal face is described by the parameter 6, which is estimated to vary from 1to 5 (Bennema, 1976). The application of the BCF theory to crystallization experiments and mathematical modeling is somewhat limited by the complexity of the parameters C and ul,both of which contain factors that are difficult to estimate even to an order of magnitude (Elwell, 1975). However, if it is assumed that, as a first approximation, E is a random quantity that varies among different crystals but not with supersaturation and temperature, then the BCF model can be generalized as follows:
G = g*D(a,T) (3) where G is the measured growth rate, g* represents a structure-sensitive variable characterizing a crystal's individual growth rate activity, and D(a,T) is the driving force for crystal growth which is considered to be the same for all the crystals. This model thus assumes that a crystal's growth activity, g*, is fixed at "birth" and will not change with size, temperature, or supersaturation. The validity of these assumptions will be tested by application to single-crystal studies and batch crystallization experiments.
Determination of the Growth Rate Activity Distribution Single-Crystal Study. The probability distribution of growth rate activities (defined hereafter as P(g*))can be determined from single-crystalgrowth data by recording the frequency (or number) of Occurrences of specific values of g*. From any given experimental measurement of
a.
j,"P@*)dg* = 1 b. j t h moment (Pk*))j= I;@*)' Pk*)dg* c. mean value of g* g* = J,"g* P@*) dg* d. jth central moment about the mean wJ = J;@* - P * )j P@*)dg* ru2 = (P@*))Z- (P@*))I* w3 = ( P @ * ) h- 3(Pk*))l(p@*))2+ 2(Pk*))13 e. standard deviation SD = +li2 f. coefficient of variation CV = SD/g* g. skewness K = r3/SD3
growth rate (C ), the value of g* can be calculated from the ratio C / D ( a , T ) . The denominator of this ratio was determined by correlating the measured growth rates of the single crystals (denoted by the triangles in Figure 2) with an expression of the form
D ( a , T ) = 1 X 101'a2 exp(-13700/RT) tanh (O.l/u) (4) Shown in Figure 3 is the resulting growth rate activity distribution of single crystals of sodium sulfate, computed by plotting the number of occurrences of g* vs g*. The distribution is normalized such that the area under the curve is equal to unity. (The definitions of the statistical properties used in this figure are given in Table I). The most notable feature of the distribution is the asymmetry; greater numbers of crystals are found in the high growth activity region. It is interesting to note that similar behavior has been reported by Garside and RistiE (1983) for the ammonium dihydrogen phosphate system.
Batch Crystallization Experiments Population Balance Analysis. To assess the extent of growth dispersion during batch crystallization, a stochastic model was developed based on the principal assumption that growth kinetics in the crystallizer can be described by eq 3 and 4. In accordance with the singlecrystal investigation, g* is assumed constant for each crystal and independent of supersaturation, temperature, and crystal size. It will be shown that the moments of the unknown growth rate probability distribution, P(g*),can be calculated from crystal size distribution data measured from either seeded or unseeded batch crystallization experiments. It is first assumed that the entire population of crystals in the crystallizer is represented by the size distribution, n(L,t). The total distribution is furthermore considered to consist of subpopulations, in which every crystal has the same value of growth rate activity, g*. Letting one of these subpopulations be denoted as rt(L,t;g*),it follows that (Larson et al., 1985) n(L,t) = JErt(L,tg*) dg*
(5)
The population balance equation (Hulburt and Katz, 1964; Randolph and Larson, 1971) for a subpopulation of crystals
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1721 with growth rate activity, g*, can be written for the batch crystallizer as
If it is further assumed that g* is determined randomly when a crystal is nucleated or introduced as a seed, then the following expression holds:
The equation can be simplified greatly by applying the following variable transformation (Hulburt, 1976; Tavare et al., 1980)
Substituting this expression into eq 14 yields
i i ( X ; g * ) = P(g*)n(X)
8 = S ' D ( a ( t ) )dt or d e / d t = D ( a ( t ) ) 0
(7)
such that
The population distribution of the crystal size, n(L,8),can be determined from size analyses of suspension samples taken from the batch crystallizer at timed intervals. The value of 8 corresponding to the sampling time, t , can be calculated by numerical integration of the area under the experimental supersaturation vs time curve, per eq 7. The primary objective, then, is to relate the time-dependent moments of the crystal size distribution to the moments of the unknown growth rate activity distribution, P(g*). This relationship is developed as follows. The jth moment of the population density is defined as mj(e) = J - L ~ n(L,e)
(9)
By the use of eq 5, this expression can be written in terms of the subpopulations, ii(L,B;g*)
dnj(8)/d8 =
(15)
jjm(h + g*e)j-ln(X) dX Jmg* P(g*) dg* 0
(16) From this result, it can be seen that the moments of the unknown growth rate activity distribution, P(g*), can be found directly from the moments of experimentally measured crystal size distributions. For example, setting j equal to one gives dm1(8)/d8 = Smn(X)dX S m g *P(g*) dg* 0
0
(17)
The first integral in eq 17 is equal to the zeroth moment of the population density distribution, n(L,8=0). The second integral is, by definition, the first moment of the growth rate activity distribution. Thus, eq 17 can be written more simply as dml(8)/de = mo(P(g*)), (18) Integrating both sides with respect to 8 yields the result m1(8)/mo - ml(0)/mo = e ( W * ) ) ,
(19)
After eq 16 is rearranged, the time derivative of the second moment is found to be dmZ(8)/d8 = 2JmXn(X) dXSmg*P(g*) dg* 0
+
and taking the time derivative yields which can be expressed more simply as d m d e ) / d e = 2h(0)(P(g*))1+ emo(P(g*)),l With the population balance, eq 8, this equation becomes
and integrating with respect to 8 yields
and integrating by parts yields
In a similar fashion, the third moment is found: m3(W mdO) ----
l ii(L,B;g*) dg* dmj(8)/d8 = j S m d L S m L j -g* 0
0
(13)
Before this expression can be integrated with respect to 8,the time-dependent behavior of the crystal size distribution, ii(L,Bg*),must be determined. Derivation of the Moment Equations. Solving the population balance equation, eq 8, by the method of characteristics ( h i s and Amundson, 1973) reveals that the number density of crystals, ii(L,B;g*),is constant along characteristic lines of slope g* in the 8,L plane. The crysbl size distribution of each subpopulation simply translates along the size axis, L , unchanged in shape or proportion. It follows that after a certain increment of transformed time, 8,all the crystals with growth rate activity, g*, will have experienced an increase in size equal to the product of g* and the increment in transformed time, 8. Let X be equal to the initial value of L at time equal to zero such that ii(L,B;g*)= $Ax*). A t any time thereafter, the size, L, will be given as X + g*8. By the use of this relationship, eq 13 can be written as dmj(e)/d8 = j l m d X l m g *(A 0
0
+ g*e)'-l
ii(X;g*) dg*
(14)
(21)
Equations 19, 22, and 23 can be algebraically solved to yield explicit expressions for the first three moments of the growth rate activity distribution as a function of experimentally measured moments. The final result is thus ml(8)/mo = 8(P(g*))1 + ml(0)/mo (24)
1722 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 Table 11. Statistical Properties of the Growth Rate Activity Distribution of Crystals from Seeded Batch Crystallization Exaeriments
Pk*))3,
1%
1 2 3 4 5 6
7
av
430 (0.999) 790 (0.996) 540 (0.979) 380 (0.986) 430 (0.987) 610 (0.931) 460 (0.996) 520
20 (0.999)U 26 (0.998) 22 (0.986) 19 (0.994) 19 (0.979) 24 (0.995) 21 (0.997) 22
10 (0.981) 24 (0.995) 15 (0.976) 10 (0.977) 10 (0.980) 36 (0.760) 11 (0.990) 17
31 116 52 38 50 28 20
SD,
1%
( ~ m / m i n ) ~ (pm/min)2
cv,
( ~ m / m i n ) ~ pm/min 130 -1980 340 1410 -750 20 000 720
5.6 10.8 7.2 6.1 7.0 5.3 4.5 7
T,
%
K
"C
28 42 32 33 36 22 21 31
0.8 -1.6 0.9 6.1 -2.1 13gb 8.2 2
60.7 65.7 72.2 60.1 61.2 61.0 67.1
Correlation coefficients shown in parentheses. *Value not included in average because of poor correlation.
..
.
~
~
~
..~..Ttme .. . . .
.. .
~
___
3 3 3 ,In', Tlme T 2 = I 2 OBmin , Tlme T 3 = 21 6 7 m i n Tlme T 4 = 3 6 6 7 m i n Time T 5 = 5 6 6 7 m i n
1
TI =
t
I 400
t
1
I
loo 00
0
100
200
C r y s t a l Size,
300
L
400
500
(pm)
Figure 4. Population density distributions, n(L,t),measured at various times from a seeded batch crystallizer. Temperature = 67 OC, initial supersaturation level ( u ) = 0.068, impeller speed = 600 rpm.
In each of the three equations above, a plot of the left-hand side of the equation vs the appropriate j t h power of 8 is predicted to be a straight line with a slope equal to the jth moment of the growth rate activity distribution. Equations 24-26 thus form the basis of a graphical method for determining growth kinetics from batch crystallizers for the general case where supersaturation is time dependent and growth rate dispersion is occurring.
Experimental Procedure Batch Crystallization Experiments. Crystals of anhydrous sodium sulfate were grown in a 5-L, agitated glass crystallizer. A flat-bladed turbine impeller was connected to a variable-speed motor and positioned one-third of the way from the vessel bottom. The aqueous solutions used in these experiments were prepared in the same manner as the single-crystal experiments. Both seeded and unseeded batch crystallization experiments were conducted. In the former, a measured mass of crystals between 125 and 150 pm was added at the start of the experiment. In the unseeded experiments, crystals were allowed to nucleate spontaneously after the agitator was started. During each experiment, a total of five to seven measurements of the crystal size distribution were made. The size distribution of the samples was determined by sieving the crystals with 3-in. standard sieves. Further details of the experimental procedure are given elsewhere (Klug and Pigford, 1984; Klug, 1985).
1
2
3
4
5
6
7
8
9
1
0
T i m e V o r i o b l e , 8, min
Figure 5. Calculation of the first moment of the growth rate activity distribution from batch crystallization data. Moments of crystal size distributions measured from the batch crystallizer are plotted as a function of the time variable, 8. The slope of the line gives directly the first moment (or average) of the growth rate activity distribution,
(Pk*))l.
The moments, mi,were determined from experimental crystal size distributions measured from the batch crystallizer. In both the seeded and unseeded experiments, the value of the time-dependent supersaturation was determined from a material balance based on the known starting concentration and suspension density measurements made at timed intervals. Values of the concentration driving force, D ( t ) , defined by eq 4, were then computed. The time variable, 8,was determined by numerical integration of these D ( t ) vs time curves.
Results and Discussion Batch Crystallization Studies. Shown in Figure 4 are typical crystal size distributions (on a population basis) measured from seeded batch crystallization experiments. Like many other crystalline systems, the seed distribution is observed to become progressively broader as the batch time lapses. Crystal breakage was ruled out as the cause of this behavior by microscopically examining individual crystals for evidence of fracture. As a further check, the zeroth moment of the seed distribution (equal to the total number of crystals) was computed for each sample measured and found to remain approximately constant throughout the experiment. By applying the moment analysis (eq 24-26), it was found that the broadening behavior can be described by the stochastic growth rate model developed earlier. In accordance with the model, the moments, mj,of the crystal
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1723 Table 111. Statistical Properties of t h e Growth Rate Activity Distribution of Crystals from Unseeded Batch Crystallization ExDeriments ~~~
(Pes*)
expt
1,
pm/min
4.7 6.8 7.7 7.1 6.6
1 2 3 4
av
(0.983)' (1.000) (0.921) (0.976)
(P@*))2,
(pm/min)2
(Pk*))3, (pm/min)3
27 (0.982) 52 (0.999) 101 (0.980) 59 (0.957) 60
210 (0.964) 510 (0.998) 940 (0.995) 570 (0.969) 560
k2t
(pm/minI2
k3?
30 67 -490 38
~
cv, %
K
"C
2.3 2.4 6.5 2.9 4.0
48 35 85 40 52
3 5 -2 2 2
70.2 70.8 60.3 60.2
( ~ m / m i n ) ~ pm/min
5.1 5.9 42.8 8.1
~~
T,
SD,
Correlation coefficienb shown in parentheses. Loborotory Time, min
Loborotory Time, min
II
0 5
17
31
0 5 II
47 62
17
47
62
I
28
NE
c
24 20 16 12
1
q L x
E
4 0
0
TIE.
-Y
-4 -8
-E"
20 -24l
0
' IO
' ' ' ' ' ' ' J 20 30 40 50 60 70 80 90 100 Time Vorioble,
e2
0
200
(min)2
400
600
T i m e V o r i o b l e , 83 (min)
800
1000
3
Figure 6. Calculation of the second moment of the growth rate activity distribution from batch crystallization data. The slope of the line gives directly (P(g*)),.
Figure 7. Calculation of the third moment of the growth rate activity distribution from batch crystallization data. The slope of the line gives directly (P(g*))%
size distributions were computed and graphed as a function of time, 8. A total of 11batch crystallization experiments were conducted, and from each one, a set of graphs like those shown in Figures 5-7 was generated. From the slopes of the lines, the first three moments of the growth rate activity distribution were computed and tabulated in Table I1 (seeded experiments) and Table I11 (unseeded experiments), along with other statistical properties of the distribution. The fit of the data to the model was found to be good as evidenced by the linearity of the data and the high correlation coefficients. From examining the third moments of the growth rate activity distribution, most of the experiments also indicated that the growth rate activity distribution is skewed to higher growth activities, in agreement with the single-crystal study. Upon comparing the first moments (or average) of the growth activity distributions of seeded vs nonseeded batch crystallization experiments, it was unexpectedly found that seed crystals grew on average 3 times faster than did crystals nucleated in the batch crystallizer. Moreover, by comparing the coefficient of variation (CV) for each experiment (listed in Tables I1 and 111),it was also discovered that the seed crystals grew with less growth rate dispersion. It is believed that surface roughness and damage to the seed crystals during their preparation (growth, separation, sieving, and drying) may be responsible for both of these findings. The phenomenon of damaged crystals exhibiting accelerated growth kinetics has also been documented for single crystals of orthorhombic sulfur (Hampton et al., 1974). Based on the results of Tables I1 and 111, it is concluded that the high degree of linearity of the experimental data verify the method of analysis and the following major
model assumptions: (a) individual crystals exhibit different but constant levels of growth rate activity, g*, in the agitated crystallizer, and (b) the value of g* is independent of supersaturation level, crystal collisions, and crystal size. It would be expected that different crystallization behavior (i.e., size-dependent growth, crystal fracture, etc.) would result in nonlinear plots.
Comparison of Batch Crystallization Results to Single-Crystal Study In order to make a direct comparison of the results of the single-crystal study to the batch crystallization results, correlations of the average growth rate from the batch crystallizer were computed with the following expression
G
= g*D(u,T)
(27)
where g* = (P(g*)) 1. For the unseeded and seeded batch crystallization experiments, the best correlations were, respectively, G = 6.6
X
lo1' exp(-13700/RT)u2 tanh (O.l/a)
(28)
10l2 exp(-13700/RT)u2 tanh (O.l/u)
(29)
and
G = 2.16
X
In Figure 2, the results of the single-crystal growth measurements (represented by the triangles) are compared with the correlation of average growth rates, measured from the unseeded batch crystallizer, eq 28. The upper and lower dashed lines are computed from the second central moment of the growth rate activity distribution, and enclose a region of growth rates that are plus and minus one standard deviation from the average growth value.
1724 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
In general, there is good agreement between the batch crystallizer and single-crystal study, although somewhat less growth rate dispersion is found among the single crystals. The most striking feature of this comparison, though, is the evidence that the spread of growth rate activities (as manifested in the second moment of P(g*)) was found to remain constant over the range of supersaturation studied. This further substantiates the hypothesis that the growth rate activity of these crystals, as a first approximation, is independent of supersaturation and crystal size.
G, = temperature-normalized growth rate, pm/min L = crystal size, pm m . = jth moment of the population density n(L,t) = population density, number/ (pmsg of suspension) ii(Lg*)= population density of subpopulation, number/(pmg suspension) P@*) = growth rate activity distribution ( P @ * ) ) i = jth moment, growth rate activity distribution, (pm/ min)j R = gas constant, cal/(mol-K) t = time, min T = temperature, K
Conclusions Experiments have been conducted that demonstrate that anhydrous sodium sulfate crystals exhibit a range of growth rates, even when grown in the same environment. Moreover, there is direct experimental evidence from both single-crystal studies and batch crystallization experiments that individual crystah have intrinsic growth rate activities that do not fluctuate even as supersaturation levels change. It has been proposed, based on an analysis of the BCF growth model, that the growth kinetics of the crystals can be described as the product of a random level of growth activity, g*, and a supersaturation and temperature-dependent term, D(a,T). An objective of this study was to determine whether the magnitude of growth rate dispersion measured among single crystals in a flow cell was comparable to that found in crystals grown from an agitated crystallizer. To accomplish this comparison, a stochastic model for crystal growth in a batch crystallizer was developed for the general case in which supersaturation is time dependent. The model was used to compute moments of the growth rate activity distribution, P(g*),for crystals grown from both a seeded and unseeded batch crystallizer. Based on this study, it was found that growth rate dispersion among single crystals grown in the flow cell is somewhat less than that found among crystals grown in the batch crystallizer. It is believed that the crystal collisions occurring in the batch crystallizer produced greater numbers of faster growing nuclei, thus accounting for the difference. Experimentation with the batch crystallizer showed that seed crystals added to the batch crystallizer grew with less growth dispersion, and higher average growth rates, than did crystals nucleated in the crystallizer. It is postulated that surface roughness of the seed crystals, produced during their preparation and screening, was responsible for this effect. A question of debate is whether the effect of crystal colliiions in agitated crystallizers can cause sudden changes in intrinsic growth rate. The successful application of the constant growth activity model indicates that, if this phenomenon is occurring, it is on a level too small to detect through monitoring of crystal size distribution changes. It is not known whether this result may be generalized to include other crystallizing substances.
Greek Symbols t = constant in BCF model, eq 2 h = particle size (pm) at time, 8,equal to zero pj = jth central moment of growth rate activity distribution, (pm/min)J 8 = time variable defined in eq 7, min u = supersaturation ratio, [(c - c*)/c*], dimensionless CJ~ = constant in BCF model, eq 2, dimensionless
Nomenclature c = concentration of dissolved salt in solution, g of salt/g of
HZO
c* = equilibrium concentration of g of d t / g of HzO
dissolved salt in solution,
C = constant in BCF model, eq 2, pm/min D ( u , T ) = supersaturation function, dimensionless g* = growth rate activity, pm/min g* = average growth rate activity, pm/min G = growth rate, pm/min G = average growth rate, pm/min
Registry No. Sodium sulfate, 7757-82-6.
Literature Cited Aris, R.; Amundson, N. R. Mathematical Methods i n Chemical Engineering; Prentice-Hall: Englewood Cliffs, NJ, 1973. Bennema, P. Theory and Experiment for Crystal Growth from Solution: Implications for Industrial Crystallizers. In Industrial crystallization; Mullin, J. W., Ed.; Plenum: New York, 1976. Berglund, K. A.; Kaufman, E. L.; Larson, M. A. Growth of Contact Nuclei of Potassium Nitrate. A I C h E J. 1983,29, 867-868. Burton, W. K.; Cabrera, N.; Frank, F. C. The Growth of Crystals and the Equilibrium Structure of Their Surfaces. Phil. Trans. R. SOC. (London) 1951,243, 299-358. Davey, R. J.; RistiE, R. I.; ZiiiE, B. The Role of Dislocations in the Growth of Ammonium Dihydrogen Phosphate Crystals from Aqueous Solutions. J. Crystal Growth 1979,47, 1-4. Elwell, D. Flux Growth. In Crystal Growth; Pamplin, B. R., Ed.; Pergamon Press: New York, 1975. Garside, J.; RistiE, R. I. Growth Rate Dispersion Among ADP Crystals Formed by Primary Nucleation. J. Crystal Growth 1983, 61, 215-220. Hampton, E. M.; Shah, B. S.; Sherwood, J. N. The Growth and Perfection of Orthorhombic (a)Sulfur Single Crystals. J. Crystal Growth 1974,22, 22-28. Hulburt, H. M. Population Balance Models for Batch Crystallization. In Industrial Crystallization; Mullin, J. W., Ed.; Plenum: New York, 1976. Hulburt, H. M.; Katz, S. Some Problems in Particle Technology, A Statistical Mechanical Formulation. Chem. Eng. Sei. 1964, 19,
555-574.
Internotional Critical Tables; McGraw-Hill: New York, 1933; Vol. n
d.
JanEiE, S. J.; Van Rosmalen, G. M.; Peeters, J. P. Growth Dispersion in Nearly Monosize Crystal Populations. In Industrial Crystallization 84; Jancic, s. J., de Jong, E. J., Eds.; Elsevier Science: Amsterdam, 1984. Janse, A. H.; de Jong, E. J. The Occurrence of Growth Dispersion and Its Consequences. In Industrial Crystallization; Mullin, J. W., Ed.; Plenum: New York, 1976. Klug, D. L. Crystallization of Anhydrous Sodium Sulfate. Ph.D. Dissertation, The University of Delaware, Newark, DE, 1985. Klug, D. L.; Pigford, R. L. Growth and Nucleation of Anhydrous Sodium Sulfate in a Batch Crystallizer. Presented at the Winter National Meeting of the AIChE, Atlanta, GA; American Institute of Chemicai Engineers: New York, 1984. Larson, M. A.; White, E. T.; Ramanarayanan, K. A.; Berglund, K. A. Growth Rate Dispersion in MSMPR Crystallizers. AIChE J . 1985, 31, 90-94. Mathis-Lilley, J. J.; Berglund, K. A. Contact Nucleation from Aqueous Potash Alum Solutions. AIChE J. 1985, 31, 865-867. Ramanarayanan, K. A.; Berglund, K. A.; Larson, M. A. Growth Kinetics in the Presence of Growth Rate Dispersion from Batch Crystallizers. Presented at the 75th Annual Meeting AIChE, Los Angeles, CA; American Institute of Chemical Engineers: New York, 1982.
I n d . Eng. C h e m . Res. 1989,28, 1725-1730 Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Academic: New York, 1971. Randolph, A. D.; White, E. T. Modelling Size Dispersion in the Prediction of Crystal Size Distribution. Chem. Eng. Sci. 1977,32, 1067-1076. Rosenblatt, D.; Marks, S. B.; Pigford, R. L. Kinetics of Phase Transitions in the System Sodium Sulfate-Water. Znd. Eng. Chem. Fundam. 1984,23, 143-147. Shanks, B. H.; Berglund, K. A. Contact Nucleation from Aqueous Sucrose Solutions. AIChE J . 1985, 31, 152-154. Solubilities; Linke, W.F., Ed; American Chemical Society: Washington, DC, 1965;Vol. 11, pp 1121-1122. Tavare, N. S.;Garside, J. The Characterization of Growth Disper-
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sion. In Industrial Crystallization 81; Jancic, S. J., de Jong, E. J., Eds.; North Holland: Amsterdam, 1982. Tavare, N. S.; Garside, J.; Chivate, M. R. Analysis of Batch Crystallizers. Znd. Eng. Chem. Process Des. Deu. 1980, 19, 653-665. White, E.T.; Wright, P. G. Magnitude of Size Dispersion Effects in Crystallization. Chem. Eng. Prog., Symp. Ser. 1971,67 (No. 110), 81-87. Zumstein, R. C.; Rousseau, R. W. Growth Rate Dispersion by Initial Growth Rate Distributions and Growth Rate Fluctuations. AZChE J. 1987, 33, 121-129. Received for review May 6 , 1988 Accepted July 3, 1989
Aging of Precipitated Magnesium Hydroxide John W. Mullin,* John D. Murphy, Otakar Sohnel,?and Gerald Spoors3 Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London W C l E 7JE, England
Freshly precipitated magnesium hydroxide, prepared by mixing aqueous solutions of MgC1, and NaOH in a stoichiometric ratio a t 25 “C, consists of primary crystals (-40 nm) that are agglomerated into larger particles ( 20 pm). The specific surface area of the precipitate, measured in situ by a dye adsorption technique, increases with the concentration of the reacting solutions but decreases with increasing aging time in an agitated suspension. The aging process is caused by a combination of several effects, including Ostwald ripening and precipitate dehydration. The agglomerate size decreases with increasing agitation time, until it reaches a reasonably constant value independent of the solution concentration, but decreases with increasing agitation intensity. N
1. Introduction The formation of a solid phase by chemical reaction in a liquid medium is the combined result of nucleation and growth processes that may proceed simultaneously or sequentially. In principle, these two primary processes should determine the chemical and physical properties of the precipitate. Due to secondary processes (mainly aging and agglomeration) taking place in the suspension, however, these properties can often be substantially modified, although this takes a much longer time than that required for the initial solid-phase formation stage. The potential influence of secondary processes should always be considered when basing industrial designs on laboratory or pilot-plant data, because delays or hold-up stages can usually be expected during the large-scale production of precipitates. The present paper describes a quantitative study of the influence of these secondary processes on some physical properties of precipitated magnesium hydroxide produced from the reaction MgC12 + 2NaOH
-% Mg(OH)2 + 2NaC1
2. Theoretical Section 2.1. Aging. The particle size distribution of a precipitate kept in contact with its mother liquor can change as a result of the dependence of equilibrium solubility on particle size, e.g., as described by the Gibbs-Thomson relation (Enustun and Turkevich, 1960)
where r is a characteristic dimension (radius) of a crystal. The activity of the solute is defined by a = Qy+”cY
Because the equilibrium solubility increases with decreasing crystal size, the solution can become undersaturated with respect to small crystals during the course of a batch precipitation while still remaining supersaturated with respect to the large. Under these conditions, therefore, small crystals begin to dissolve and the resulting solute is transported from the solution to the larger crystals which continue to grow. The crystal size distribution of the precipitate thus shifts toward the larger sizes as time progresses. This process is generally called “Ostwald ripening”. The rate of change of crystal size due to ripening depends on the mechanism controlling the crystal growth process. For diffusion-controlled growth, the maximum rate of ripening is given by (Kahlweit, 1975; Nielsen, 1964) d?/dt = 8DVm2ysce,/9uRT
0888-5885/89/2628-1725$01.50/0
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Equation 3 also holds for growth controlled by a surface reaction of the first-order. For a second-order surface reaction, the maximum rate of ripening is given by Kahlweit (1975): dT3/dt = 16k,Vm3(ys)2~e~/9v2R2T2 (4) According to both eq 3 and 4, therefore, the average crystal size in a system undergoing Ostwald ripening should be expected to follow a relation of the form (5) where the maximum achievable value of the time constant, A , is given by the cube root of the right-hand side of eq 3 or 4, depending on the growth mechanism. A more general form of eq 5 was used by Hanitzsch and Kahlweit (1969): p = At’/3
* Correspondence to this author. ‘Present address: Research Institute of Inorganic Chemistry, RevoluEni 86, 400 60 Usfi nad Labem, Czechoslovakia. Present address: Steetley Quarry Products, Ltd., Hartlepool, U.K.
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0 1989 American Chemical Society
