Ind., Eng. Chem. Res. 1991,30, 196-201
196
from Composite Poly(acrylonitri1e) Fiber and Its Adsorption Ability for Uranium. Bull. SOC.Sea Water Sci. Jpn. 1989,42,
Chem. Eng. Sci. 1978,33, 1375-1384.
Received for review March 5, 1990 Revised manuscript received July 9, 1990 Accepted July 18, 1990
279-203. Wakao, N.; Funazkri, T. Effect of Fluid Dispersion Coefficients on Particle-to-Fluid Mass Transfer Coefficient in Packed Beds.
Evaluation of Crystal Size Distribution Enlargement in a Multistage Column Crystallizer Robert W.Farmer and J a m e s R. Beckman" Chemical and Bio Engineering Department, Arizona State University, Tempe, Arizona 85787
Enlargement of product size in a staged column crystallizer was investigated for precipitation of sparingly soluble salts. Precipitation of barium sulfate was employed to examine the effects of the number of stages, production rate, and impeller speed. The number of stages exerted the most dramatic effect on product size; increases in the volume-average size by a factor of 2 were observed in a five-stage column, as compared to product from an equal-volume MSMPR crystallizer. Population balance analysis indicated that this size increase coincided with an order of magnitude increase in calculated agglomeration efficiency. Introduction For many crystallization processes, enlargement of product crystal size distribution (CSD) can improve process economics and product value. In this study, a staged column crystallizer was investigated as a means of CSD enlargement in the production of sparingly soluble salts (i.e., class I1 systems). This arrangement is analogous to cascades of mixed-suspension, mixed-product-removal (MSMPR) crystallizers (e.g., Larson and Wolff, 1971; Randolph and Tan, 1978) but incorporates a number of well-mixed stages within a single vessel. Precipitation of barium sulfate, or barite, was selected to evaluate the effects of key process variables: number of stages, solids production rate, and impeller speed. Precipitation of barium sulfate offers two features of interest. First, at high precipitation rates, barite growth involves both deposition and agglomeration mechanisms. Examination by scanning electron microscopy of barite crystals produced in these experiments confirmed the predominance of agglomerated morphologies. These agglomerates were composed of many planar, dendritic crystallites, identical with those observed by Liu et al. (1976) and others. Second, due to the small sizes of the product crystals, barite exhibits relatively minor secondary nucleation effects. In the staged column, interaction of slurry upflow and relatively smaller backmixing flow induced a pronounced concentration profile of solids, with the highest concentration in the top stage. Experimentally, it was observed that particle growth was promoted by agglomerationwithin this concentrated slurry. As the number of stages increased, the concentration of solids in the top stage was further amplified, which promoted greater agglomeration and CSD enlargement. A population balance analysis of CSD effects in column crystallization of sparingly soluble salts was presented by Beckman and Farmer (1984) for systems exhibiting only deposition or primary growth. This work examined the influence of interstage backmixing and secondary nucleation kinetics upon product size. Later, these findings
* Address correspondence to this author. 0888-5885191/ 2630-0196$02.50/0
were extended to theoretically predict the effects of feed/product stream location along the column (Farmer and Beckman, 1987). If secondary nucleation predominates, it was shown that enlargement of a given salt can be controlled by the relative position of the feed and product streams. In this study, the stage population balance technique was extended to include agglomerative growth terms. The resulting model was used to empirically fit experimental CSD data and to quantify agglomeration efficiency. An empirical expression, developed for barite precipitation1 agglomeration in an MSMPR crystallizer (Beckman and Farmer, 1987), was used to calculate size-dependent agglomeration efficiencies for each data set. Process Description As shown in Figure 1, the staged column crystallizer can be envisioned as a backmixed cascade of individual MSMPR crystallizers. In general, aqueous cation and anion feed streams can be introduced and product withdrawn, in any desired profile. However, the experiments described here mostly involved single cation and anion streams located at the top and bottom stages, respectively. The product flow rate, Q , is small compared to the anion feed stream, QF,so t i a t the net convective flow direction is upward. Solid product is transported down the column via backmixing flows QB,i only, countercurrent to the bulk upflow. These conditions induce a concentration profile of solids and the column inventory of solids is larger than in an MSMPR crystallizer of equal total volume. Interstage concentration differences are amplified as the number of stages,N , is increased or backmixing flow reduced. From a design viewpoint, stage residence times and solid concentrations can be specified by varying stage geometry and volume, agitator characteristics, and feedlproduct stream locations. Thus, a column crystallizer permits greater flexibility in establishing favorable growth and nucleation conditions than a conventional MSMPR-type crystallizer. The overhead stream clarifier reduces product loss via the effluent, Q1, exiting the top stage. Concentrated underflow suspension, Qcl,rtn,is recycled to the top stage, and 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 1,1991 197 Table I. Staged Column Crystallizer Experimental Summary impeller av prod prod rate, speed, Ba2+feed size, t, std dev test - g/min rpm concn, M pm o f t , pm MSMPR, Single Stage
'
L
1
1122 1123O 1221 1311 1321 1322O 1331 1421 1511 1521 1531
0.27 0.27 0.42 0.53 0.51 0.50 0.51 0.80 0.98 0.99 1.03
3121 3221 3311 3324 3325O 3331 3422 3423 3521 3531
0.29 0.36 0.52 0.55 0.49 0.50 0.84 0.85 0.94 0.98
202 203 202 144 204 209 349 200 140 206 352
'
0%59 0.055 0.086 0.113 0.112 0.109 0.109 0.159 0.193 0.196 0.201
16.84 18.53 15.73 16.36 10.45 11.51 9.53 6.93 5.76 6.04 7.57
0.20 0.96 0.22 0.20 0.23 0.53 0.23 0.39 0.06 0.17 0.03
Three Stage, Countercurrent Feed
Product slurry
I
OQ
Figure 1. Schematic and nomenclature for the multistage column crystallizer.
relatively solid-free overflow, Qohd, represents a clear-liquor-advance stream. Experimental Section Three crystallizer configurations were examined: three-stage and five-stage columns and an equal-volume MSMPR crystallizer. The total active crystallizer volume (excludingclarifier) was 2.4 L, which was divided into equal portions for the multistage tests. A baffled crystallizer with 10.2-cm (4-in.) inside diameter was constructed from clear acrylic, with feed and sampling ports located at the column wall, perpendicular to the column axis. A more detailed description of the apparatus and procedure has been given by Farmer (1986). The multistage columns differed from the MSMPR unit only by the inclusion of stage separator disks. For all cases, the ratio of column inside diameter to separator port diameter was set at 8.0. Simple two-bladed paddle impellers, 7.6 cm wide, were centered within each stage: three impellers were evenly spaced along the column for the MSMPR arrangement. The total impeller blade area was held constant for all column confiiations. Measurements of backmixing rate, by tracking the decay of an acid spike, indicated that individual single-pass stage residence times ranged from 0.5 to 2.2 min (Farmer, 1982, 1986). The product stream was withdrawn through a plungetype solenoid valve (3.2-mm diameter) opened at 1-min intervals by an electronic repeat-cycle timer. Product flow rate setting yielded a total residence time of solids of 53 min for the column. The Ba2+/S0f ion feed ratio was maintained at 0.5, since this ratio can affect the morphology of barite at high precipitation rates (Gunn and Murthy, 1972; Liu et al., 1976). A conical glass clarifier, 15.2-cm diameter and 1.2-L volume, was used to separate solids from the top stage effluent stream to provide clear liquor advance. An air lift pump carried the recycle suspension back to the top stage; this method caused no detectable changes in CSD of the recycled material. The primary experimental data were the steady-state CSD and concentration profile of solids within the crystallizer. Barite CSD was determined by using a Particle Data Inc. Model 80XY analyzer, fitted with a 150-pm
205 204 143 199 203 357 202 205 204 353
0.059 0.082 0.112 0.112 0.106 0.110 0.173 0.174 0.205 0.206
18.74 16.58 17.74 13.24 13.68 13.02 10.75 10.58 11.37 8.75
0.30 0.20 0.99 0.77 0.41 0.52 0.26 0.21 0.20 0.19
Three Stage, Feed Equally Distributed to Three Stages 3122 3312 3322 3323O 3332 3522
0.28 0.55 0.53 0.54 0.55 0.89
5111 5121 5122' 5131 5321 5331 5332b 5333'
0.25 0.26 0.26 0.27 0.49 0.55 0.54 0.55
201 145 203 204 358 202
0.057 0.113 0.111 0.112 0.114 0.119
16.51 6.82 7.27 7.69 11.21 5.63
0.30 0.27 0.22 0.34 0.27 0.35
Five Stage, Countercurrent Feed 146 201 203 345 201 346 347 345
0.056 0.057 0.056 0.053 0.110 0.117 0.119 0.109
20.88 23.84 23.60 25.00 16.06 14.81 13.86 10.69
0.25 0.14 0.90 0.35 0.42 0.30 0.06 0.43
'Replicate test. Single cation feed located at middle column stage. Single cation feed located at bottom column stage. orifice and calibrated to span a 3.01-46.63-pm size range. Concentrations of solids were measured gravimetrically; aliquot samples were vacuum filtered on preweighed 0.45-pm porosity membranes (Gelman DM450), washed with methanol, and weighed after drying. The volume-average size was calculated by using the appropriate moment ratio for the differential size distribution, n(L),
t = m3/m2
(1)
with the kth moment of the distribution defined as
mk = JmLk n(L)dL
(2)
and calculated numerically from the discrete n(L)data obtained from the 80XY particle size analyzer. The experimental conditions summarized in Table I cover three levels of impeller speed (nominal values = 145, 200, and 350 rpm) and five levels of barite production rate (from 0.2s to 1.0 g/min). In most cases, a single cation stream was fed to the top stage, although some three-stage runs examined equally distributed cation feed. For two five-stage runs (5332 and 53331, a single cation stream was fed to the middle or bottom stage. The volume average product size was calculated by using eqs 1 and 2, and the
198 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 Nominal lmpellor Speed: 200 R P M
1 \
21
5 Stage,
Counter-current
3 Stage,
Counter-current Particle Size, L (micron) Figure 2. Typical barite size distribution data for product stream and internal stages.
;i
5
)
0.2
I
I
I
I
I
0.6
0.4
I
I
0.8
I
1
Product rate (g/min) Figure 4. Effect of product rate on average product size. Nominal Product Rate: 0.5 g/min 17
\< \h Counter-current
\
C
3
'
9
\
\
\
'
'
15 ' Particle Size,
..__ 21
I2
27
3:
L (microns)
$
Counter-current
5
f
Figure 3. Effect of product rate on cumulative barite size distribution.
2
standard deviation covers three to six CSD determinations.
-s
al
Discussion of Experimental Results Figure 2 depicts a typical barite differential population density function, n(L).Barite CSD consistently exhibited this bimodal shape as is generally observed in strongly agglomerating systems. The relative affect of agglomeration on product size could be qualitatively compared based on significant differences in magnitude and location of agglomerate "peak" on a In n(L)plot. Individually measured CSDs of each stage and of the product were identical for all test conditions, which indicated that interstage size classification did not occur. While not examined here, it might be possible to take advantage of interstage classification to further increase product size. Significant particle breakage or attrition could also account for the observed bimodal CSD. To evaluate the integrity of the barite CSD, aging tests of solids were conducted by extended agitation of the column after stopping feed and product streams. Measurements of CSD during the agitation period indicated that breakage effects were minor and involved only a small fraction of the largest crystals (Beckman and Farmer, 1987). The typical effect of barite product rate on CSD is illustrated in Figure 3 for the three-stage, single cation feed column crystallizer. Higher product rate resulted in smaller product size for all impeller speeds and crystallizer configurations. The population balancb analysis indicated that agglomeration efficiency improved at higher product
-
14
13-
12-
11
-
10
-
9 :
100
I
I
300
lmpellor Speed (RPM) Figure 5. Effect of impeller speed on average product size.
rates, but these data suggest that the nucleation rate must have also increased. Figures 4-6 show the separate effects of the number of stages, product rate, and impeller speed on the average product particle size. On these plots, the error bars represent a span of two standard deviations around the observed average. Figure 4 shows that larger product was obtained with greater number of stages over the entire range of product rates, and the relative enlargement becomes more pronounced at higher product rates. As shown in Figure 5, an increase in impeller speed generally resulted in a smaller average size product. However, both the solid concentration profile and the kinematics of particle collision depend on impeller speed, so agglomeration can be promoted or inhibited by specific changes in agitation. The relative position and shape of the curves shown in Figure 6 suggest that the product rate had a larger effect than impeller speed for a given column configuration. Recently, it has been reported that the supersaturation
Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 199 Product Rate lmpellor Speed (g/min) (RPW -
Product Rate lmpellor Speed fo/minl (RPW 25
0.08
0 23
g
‘E
0
h 0
0.25 0.5 1.0 0.5
-. E
21
+
0
0.49 0.55
0 A
0.26 0.26
346 201 203
X
0.27
345
20 1
5 Stage Counter-current
0.06
19
Y
.-$
IJ
$
l7
cn
15
8
0.04
-
13
s$
0.03
-
0.02
-
0.01
-
C
E
a$
-
al
-5 P
-
F +
5 al
0.05
I
11
$
9
z
g
7 5
I
!
,
Stage 5 Stage Counter-current Figure 6. Increase in average product size with increasing number of stages. MSMPR
0
3
’
I
I
I
1
2
3
Agglomeration Efficiency Effects A population balance analysis was applied to quantify the effects of experimental variables on agglomeration
I
I
5
Prod
(TOP)
Stage Number Figure 7. Concentration profiles of solids for the five-stage crystallizer.
Table 11. Average Interstage Concentration Ratios of Solids M T , i - l / M ~(stage ,i index, i = 2, .., IV) nominal impeller meed. rDm 3 stages 5 stages 140 1.7 1.9 200 1.4 1.7 1.3 345 1.2
distribution even within a mixed magma can influence the barite precipitate CSD (Tosun, 1988). Such an affect would further complicate the interactions of impeller speed and product rate (i.e., solute feed concentration) upon barite product size. Representative concentration profiles of solids are shown in Figure 7 for a five-stage column crystallizer. Such profiles are typically dependent on impeller speed,product rate, and geometry of column internals (e.g., separator port diameter) (Haug, 1971; Viadurri and Sherk, 1985; Farmer, 1982,1986). It was found that the ratios of solid concentrations in adjacent stages, as listed in Table 11, were constant at all product rates for a given column configuration and impeller speed. These ratios were larger for the five-stage column compared to the three-stage unit. The effect of varying the concentration of solids at constant product rate and impeller speed was examined by shifting a single cation feed along a five-stage column or by equally distributing the total cation feed to a three-stage unit. In Figure 8, the concentration profile of solids in the five-stage tests remained constant (lower curve) while the average product size decreased as the feed was moved to more dilute stages. Similarly, the average product sizes shown in Table I for three-stage columns having equally distributed cation feed ranged from 12% to 62% smaller than those from corresponding single-feed experiments. This indicates that for a given product rate, agglomeration was most promoted when all precipitation occurred in the stage containing the highest concentration of solids.
I
4
Nominal Product Rate, 0 5 g/”? Nominal lmpellor Speed 345 RPM
5 Stage Counter-current
0.06
-. c
E cn
- 0.05
k .-$
-E6
0.04
L
s 5 s
al p 5
0.03
0.02
0.01
0
1 (TOP)
i
3
4
5
Prod
Column Stage or Feed Location Figure 8. Effect of solute feed location on average product size.
efficiency. In ita current form, the model does not predict product CSD but provides a tool for separating the influences of deposition and agglomeration growth modes within a specified particle size range. Based on the population density function, n(L),agglomerative terms corresponding to crystal “birth” and “death” at size L can be calculated for colliding particles of arbitrary sizes, L’ and L” (e.g., Hulbert and Katz, 1964; Drake 1972) birth (3)
death
200 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991
where conservation of the total solid volume is imposed: L3
= (L’)3
+ (L”)3
Nominal Product Rate: 0.5 gimin Nominal impellor Speed: 200 RPM
Prefactors @b and @d represent the efficiency of agglomeration, that is, the probability that a particle collision will result in formation of a cemented agglomerate. Various expressions for this efficiency have been presented for crystallizing systems (Halfon and Kaliaguine, 1976a,b; Remillard et al., 1978; Hartel et al., 1986) and general coalescing systems (e.g., Friedlander and Wang, 1966; Ramabhadran et al., 1976; Lee and Chen, 1984). In this study, agglomeration was empirically described by using a three-parameter, size-dependent function (Beckman and Farmer, 1987):
P=
k (L”)2L’e-xL’ [(~”)3
+ (~’)3)p
5 Stage,
When agglomeration terms are included in a steady-state population balance for an MSMPR crystallizer, the resulting expression in
Counter-current -------____ --,-----/
/ /
10-71
(6)
= 25pm
Capturing size. L“
(5)
0
I
,
~
4
8
, 12
,
,
,
16
,
20
,
1
24
Captured size, L’ (microns)
Figure 9. Representative agglomeration efficiency, ob, for MSMPR and multistage crystallizers. Capturing size, L“ = 25 p m
Product rate
lmpellor speed
1/2JL@b(L’,L”)n(L”) n(L’) dL‘357
0.50 0.94
impd(L,L’)n(L)n(L’) dL’-n(L)Qp = GUT dn(L)/dL
204
(7) Analogous expressions can be written for the ith stage of a column crystallizer, based on the nomenclature of Figure 1:
10-5 4J. 0
An additional variable, 8, accounts for interstage concentration effects; however, experimental values for 0 did not exceed f0.05. For seeded precipitation of barium sulfate (a class I1 system), it was apparent that all precipitation occurred in the fed stage. It follows that the growth and agglomeration terms for the nonfed stages can be set equal to zero. For a column crystallizer with single solute feed, the algebraic flow terms for the nonfed stages can be combined into a calculable term, aN,and the set of stage population balances collapses to a single equation:
Equation 9 was numerically fitted to n(L)experimental data, such as that shown in Figure 2, to calculate the values of agglomeration efficiency, ob, as defined by eq 6. An optimal set of constants for eq 6 describing a given data set was determined by using a Hooke and Jeeves numerical search. For each search iteration, numerical values of the agglomeration integrals given by eqs 3 and 4 were compiled and a fourth-order Runge-Kutta algorithm was used to integrate eq 9. The growth rate, Gi, was estimated in the large-size region where agglomeration effects were presumed minor. A satisfactory agglomeration efficiency function was obtained when a logarithmic sum of squares objective
4
8
12
Captured Size, L’
16
20
(microns)
Figure 10. Representative agglomeration efficiency, &, for threestage crystallizers at various operating conditions.
function, based on n(L)within the experimentally measured size range (3-46 pm), approached a minimum: j=J
SSQ = C (log n(Lj)calc - log n(Lj),qJ2 j=1
(10)
where J corresponds to the largest size analyzer channel in which a significant number of particles was detected. Figure 9 compares calculated @b values for MSMPR and multistage column crystallizers at one set of operating conditions for a representative “capturing” particle size, L ” = 25 pm. An increase in agglomeration efficiency, of about an order of magnitude for the multistage cases, coincided with experimental increases in average particle size by as much as a factor of 2. In general, differences in agglomeration efficiency values between multistage and MSMPR crystallizers became smaller as the impeller speed increased. Representative @b values shown in Figure 10 for a three-stage column at differing product rates and impeller speeds show that agglomeration efficiency increased at higher product rate. This suggests a synergisticinteraction between higher precipitation rate and greater agglomeration efficiency for barite, probably due to more rapid crystallite intergrowth. At constant product rate, the effect of impeller speed upon agglomeration efficiency apparently depends on the relative size of the particles.
Ind. Eng. Chem. Res., Vol. 30, NO. 1, 1991 201
Conclusions It has been demonstrated that a staged column crystallizer can deliver significantly enlarged product CSD in precipitation of sparingly soluble salts. For the barite system, a multistage arrangement with solute fed to the top stage, which had the highest concentration of solids, resulted in average product size enlargement by u p to a factor of 2 compared t o equivalent MSMPR cases. The greatest CSD enlargement was experimentally observed with a larger number of stages, higher product rates, and lower impeller speeds. By numerically integrating a particle population balance including agglomeration terms, it was possible to quantify the role of agglomeration in enlarging product size. Compared to MSMPR crystallizer tests, size-dependent agglomeration efficiency values increased by about an order of magnitude for multistage operation. In design of a multistage column crystallizer, the relative influence of deposition growth, secondary nucleation, and agglomeration will suggest the appropriate strategy to obtain enlarged product. In this study, barite enlargement was promoted by introducing solute feed to a stage containing a higher concentration of solid than would exist in an equivalent MSMPR crystallizer. In general, control of the concentration profile of solids provides improved flexibility in establishing a favorable balance between growth and nucleation. Acknowledgment We acknowledge the financial assistance of NSF Grant CPE-8211671-01 in support of this work. Nomenclature B,, = particle birth rate into size L by agglomeration (no./ (min pm))
CSD = crystal size distribution particle death rate from size L by agglomeration $no./(min pm)) G = primary growth rate (pm/min) J = size analyzer channel of largest detected size k = agglomeration efficiency model constant L = particle size (pm) t = volume-average particle size = m3/m2(pm) L’ = size of captured particle in agglomeration (pm) L” = size of capturing particle in agglomeration (pm) MT = concentration of solids (g/mL) mk = kth moment of size distribution MSMPR = mixed-suspension, mixed-product removal N = number of column stages n(L) = population density function (no./(mL pm)) Q = bulk liquor flow rate (mL/min) Q B = interstage backmixing flow rate (mL/min) S(L,) = clarifier separation efficiency unit step function, S = 1 for L < L, SSQ = sum of squares objective function V = total volume (mL) u = volume of ith column stage (mL) UT = volume of MSMPR crystallizer (mL) fi = agglomeration efficiency X = agglomeration efficiency model constant y = agglomeration efficiency model constant (mL-’) 9 = stage washout coefficient function B = interstage solids concentration parameter
D, =
Subscripts b = agglomerative birth
calc = calculated value c1,rtn = clarifier return stream d = agglomeration death exp = experimental data value F = cation or anion feed stream i = stage index j = channel index for particle size analyzer data k = distribution moment index ohd = overhead clarifier stream P = product stream 0 = cation feed stream
Literature Cited Beckman, J. R.; Farmer, R. W. Improved CSD by a Countercurrent Tower Crystallizer. Presented at the Spring National Meeting of the American Institute of Chemical Engineers, Atlanta, GA; American Institute of Chemical Engineers: New York, 1984;paper 56d. Beckman, J. R.; Farmer, R. W. Bimodal CSD Barite due to Agglomeration in an MSMPR Crystallizer. AZChE Symp.Ser. 1987,83 (NO. 2531,85-94. Drake, R. L. A General Mathematical Survey of the Coagulation Equation. In Topics in Current Aerosol Research; Hidy, G. M., Brock, J. R., Eds.; Pergamon: Oxford, England, 1972;Vol. 3, Chapter 4. Farmer, R. W. Characterization of a Countercurrent Staged-Feed Crystallizer. M.S. Thesis, Arizona State University,Tempe, 1982. Farmer, R. W. Experimental and Modeling Studies of Product Size Enlargement in a Staged Column Crystallizer. Ph.D. Disaertation, Arizona State University, Tempe, 1986. Farmer, R. W.; Beckman, J. R. Particle Size Improvement by a Countercurrent Tower Crystallizer. AZChE J. 1986, 32, 1099-1106. Friedlander, S. K.; Wang, C. S. The Self-preserving Particle Size Distribution for Coagulation by Brownian Motion. J . Colloid Interface Sci. 1966,22,126-132. Gunn, D. J.; Murthy, M. S. Kinetics and Mechanisms of Precipitation. Chem. Eng. Sci. 1972,27,1293-1313. Halfon, A.; Kaliaguine, S. Alumina Trihydrate Crystallization, Part 1: Secondary Nucleation and Growth Rate Kinetics. Can. J. Chem. Eng. 1976a,54, 160-167. Halfon, A.; Kaliaguine, S. Alumina Trihydrate Crystallization, Part 2: A Model of Agglomeration. Can. J. Chem. Eng. 1976b,54, 168-172. Hartel, R. W.; et al. Mechanisms and Kinetic Modeling of Calcium Oxalate Crystal Aggregation in a Urinelike Liquor. AZChE J. 1986,32,1176-1185. Haug, H. F. Backmixing in Multistage Agitated Contactors - A Correlation. AZChE J . 1971,17,585-589. Hulbert, H. M.; Katz, S. Some Problems in Particle Technology. Chem. Eng. Sci. 1964,19,555-574. Larson, M. A.; Wolff, P. R. Crystal Size Distributions from MultiStage Crystallizers. Chem. Eng. Prog. Symp.Ser. 1971,67(No. 1101,97-107. Lee, K. W.; Chen, H. Coagulation Rate of Polydisperse Particles. Aerosol Sci. Technol. 1984,3, 327-334. Liu, S.T.; et al. Scanning Electron Microscopic and Kinetic Studies of the Crystallization and Dissolution of Barium Sulfate Crystals. J . Crystal Growth 1976,33,11-20. Ramabhadran, T. E.;et al. Dynamics of Aerosol Coagulation and Condensation. AIChE J . 1976,22,840-851. Randolph, A. D.; Tan,C. S. Numerical Design Techniques for Staged Classification Recycle Crystallizers: Example of Continuous Alumina and Sucrose Crystallizers. Znd. Eng. Chem. Process. Des. Deu. 1978,17, 189-200. Remillard, M.; et al. Cristallisation du Trihydrate dAlumine: un Modele pour 1’Attrition. Can. J . Chem. Eng. 1978,56,230-235. Tosun, G. An Experimental Study of the Effects of Mixing on the Particle Size Distribution in BaSOI Precipitation Reaction. h o c . 6th Eur. Conf. Mixing, Pauia, Italy 1988,161. Viadurri, T. C.; Sherk, F. T. Low Backmixing in Multistage Agitated Contactors Used as Reactors. AIChE J. 1985,31,705-710.
Received for reuiew November 20, 1989 Reuised manuscript received June 29, 1990 Accepted July 9, 1990
