Ind. Eng. Chem. Fundam. 1981, 20, 71-76
ug = geometric standard deviation in ideal log-normal dis-
tribution, dimensionless Literature Cited
71
Furnas, C. c. Bur. Mines Rept. Invest. 1928,2894, 7. Furnas, C. C. Bur. Mines Bull. 1929,307, 74. Kawamura. J.: Aoki. E.: Okusawa. K. Kaoaku K m k u 1971. . 35.. 777. Manegold, E.; Hofman, R.; Solf, K: Kollo6Z. 56, 143. McGeary, R. K. J. Am. Ceram. SOC.1961,44, 513. Ouchiyama, N.; Tanaka, T. Ind. Eng. Chem. Fundam. 1960, 79, 338. Smith, W. 0.; Foot, P. D.; Busang, P. F. fhys. Rev. 1929,34, 1271. Sohn, H. Y.; Moreland, C. Can. J . Chem. Eng. 1988,46, 162.
lsfi,
Anderegg, F. 0. Ind. Eng. Chem. 1931,23, 1058. Arakawa, M.; Nlshino, M. Zaityo 1973,22, 858. Bernal, J. D.; Mason, J. Nature (London) 1980, 788, 910. Dallavalle. J. M. “Micromeritics”, Pltman Publishing Corp., 1948; p 135. Eastwood, J.; Matzen, E. J. P.; Young, M. J.; Epstein, N. Br. Chem. Eng. 1969, 74, 1542. Epstein, N.; Young, M. J. Nature (London) 1962, 796, 885.
Received f o r review J a n u a r y 15, 1980 Accepted September 30, 1980
Size-Dependent Growth of Magnesium Sulfate Heptahydrate Ronald W. Rousseau’ and Robert M. Parks Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27650
This study examines nucleation and growth behavior of magnesium sulfate heptahydrate crystals in a perfectly mixed continuous crystallizer. Growth kinetics were found to depend on crystal size and were correlated with the Abegg-Stevens-Larson equation. Previous studies indicating that this crystalline system followed site-independent growth kinetics are believed to have examined size distributions over a range too narrow for proper analysis. Kinetic equations for nucleation rate were evaluated from the experimental data and show dependence on supersaturation and magma density.
Introduction The crystal size distribution (CSD)of a product from a continuous crystallizer is determined by a relationship between nucleation and growth kinetics and by magma residence time distributions. Since a priori predictions of crystallization kinetics are not possible, experimentally evaluated and statistically correlated nucleation and growth rates are needed for design and analysis of commercial crystallizers. Perfectly mixed (MSMPR), small scale crystallizers have a well-defined residence time distribution and are easily analyzed to evaluate nucleation and growth kinetics. This approach has been successfully used by a number of researchers and is described in detail by Randolph and Larson (1971). The basic concepts are outlined in the remainder of this section. A population density function n is defined so that ndL is the number of crystals per unit volume in the size range L to L + dL. A balance on the number of crystals per unit volume in a perfectly mixed crystallizer at steady state leads to the equation d(nG) n +-=o (1) d L T Solutions to eq 1 require information regarding the relationship of growth rate to crystal size. Most systems have been observed to follow size-independent growth kinetics, which means that eq 1 can be integrated to give n = no exp(-L/G7) (2) The nucleation rate in the crystallizer is given by
Bo = noG (3) which serves as a system boundary condition. Clearly, if growth rate depends upon the size of the growing crystal, the population density function will not follow the simple exponential relationship given by eq 2. Furthermore, Canning (1971) pointed out several other factors besides size-dependent crystal growth that can 0196-4313/81/1020-007 l$Ol .OO/O
cause deviation of a population from eq 2. The importance of these additional factors can often be evaluated through an examination of the crystalline product or simple tests on the crystallizer. For example, significant crystal breakage invalidates eq 1but can be detected through an inspection of the crystalline product. Size-Dependent Growth If it is determined that crystals of a particular solutesolvent system exhibit growth rates dependent on the size of the growing crystal, then this fact must be incorporated into the population balance given by eq 1. All other restrictions used in deriving this equation still apply. Because the nature of size-dependent growth is poorly understood, an empirical expression is used to relate growth rate and crystal size. White et al. (1976) list most of the proposed equations for size-dependent growth and discuss requirements to which growth models must conform. The equation proposed by Abegg et al. (1968) satisfies all these requirements and has been used most frequently in expressing size-dependent growth kinetics. The AbeggStevens-Larson (ASL) model is
G = G o ( l + T L ) ~ ;b < 1 (4) The number of parameters in eq 4 is often reduced by setting y = 1/G07 (5) Experience has shown that the reduced ASL model (i.e., using eq 5 to define y) fits observed growth kinetics as well as the three-parameter version. Accordingly, the reduced version will be used in the remainder of this discussion. Substitution of eq 4 and 5 into eq 1 and solving the resulting differential equation gives n = nO(l+ L/GOT)-~exp
[
1 - (1
f/bG07)’-*
]
(6)
It can also be shown that the nucleation rate is given by 0 1981
American Chemical Society
72
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
R” =
n O G O
(7)
--
Moments of the distribution given by eq 6 may be evaluated using techniques described by O’Dell (1978). Suspension density may be related to the ASL parameters using an equation derived by O’Dell and Rousseau (1978)
MT = Ci(b)pk,no(GoT)J
Cooling B a t h
li
( 8)
where C,(b) is a quantity whose value depends upon b. Evaluation of C,(b) is accomplished by a complex scheme of numerical integration; O’Dell and Rousseau (1978) determined C,(b) for b ranging from 0.9 to -10.0. Rousseau and Woo (1980) confirmed the validity of this approach by comparing experimental data on potassium alum with predictions from eq 8. Research described in this paper focuses on the growth and nucleation kinetics of MgSO4.7Hz0. Although this system has been studied previously, conflicting descriptions of its growth kinetics are found in the literature. Analysis of experimental data obtained in the crystallization of MgS04-7H20in an MSMPR crystallizer utilized model equations presented in this section to develop correlations of nucleation and growth kinetics. These correlations should be useful in the design and analysis of MgS0,-7H20 crystallizers. Furthermore, the data and resulting correlations are significant in that they provide additional information on the general field of crystallizer operations in which size-dependent growth kinetics are found. MgS04.7H20. Crystallization of this solute from aqueous solutions has been examined in a variety of contexts. For example, it was used in mechanistic studies of contact nucleation by Clontz and McCabe (1971),Johnson e t al. (1972), Bauer et al. (1974), Shah et al. (1973), and Tai et al. (1975) among others. In all of these experimentally based efforts the numbers of crystals found by a controlled contact of an object with a seed crystal were determined. Much of the early understanding of contact nucleation resulted from this collection of work. Clontz et al. (1972) examined mechanisms of crystal growth by measuring the linear advance of individual faces on crystals suspended in flowing supersaturated solutions. Few studies of the crystallization of MgS04.7H20 in multicrystal stirred tank crystallizers have been conducted. Those that have been done often produced unusual or complex results. Ottens and de Jong (1973) and Ottens (1973) had difficulty with agglomeration of crystals in magma samples; several washings and extended periods of sieving were required to eliminate these problems so that CSD data could be obtained. These researchers also observed instabilities in crystal size distributions. They attributed such behavior to operation of their crystallizer in a critical range of supersaturation that caused competing nucleation mechanisms, contact nucleation and needle breeding, to occur. Small variations of supersaturation resulted in a mechanistic change and, hence, oscillations in CSD. Ottens (1973) found that reducing supersaturation eliminated cycling behavior. Wood (1975) and Canning (1971) reported CSD data on MgSO4-7H20that did not follow the expected exponential relationship of eq 2 . These data were obtained using perfectly mixed crystallizers and, apparently, it was not possible in either study to identify causes for noncompliance with eq 2 , other than size-dependent crystal growth. Wood used the ASL growth equation to analyze system behavior but found large unexplained variations in model parameters. His population density data were taken on crystals ranging in size from 44 to 1000 km. Sikdar and Randolph (1976) and Bauer et al. (1974) reported size-independent growth kinetics for this system.
Cooling Coi I s
Rotameter Filter Crystallizer
Heaters
I n
Return Tank
n r
feed Tank
feed Pump
Figure 1. Schematic diagram of experimental crystallizer.
However, their data were taken on systems having much shorter residence times than those of Wood and Canning. Furthermore, Sikdar and Randolph obtained kinetic data in a “micro” system that had a very short residence time and was equipped with a particle counter that measured population densities over a size range from 3 to 60 pm. Clearly, some of the discrepancy in statements concerning MgS04-7H20growth kinetics is related to large variations in the range of crystal sizes examined in the various studies: the smaller the crystals and the narrower their size range, the more likely are growth kinetics to appear independent of size. Experimental Section A schematic diagram of the system used for these studies is shown in Figure 1. The crystallizer was a cylindrical vessel with an i.d. of 24.1 cm and height of 47 cm. Four vertical baffles 38.1 cm long and 1.9 cm wide were oriented 90” apart in the vessel. A draft tube was formed by a tightly wound coil of 0.64-cm stainless steel tubing. The tubing was 12.5 m in length and was wound so as to have an 0.d. of 15.1 cm. The bottom of the draft tube was postioned 3.5 cm above the bottom of the crystallizer and had a height of 24.0 cm. A 7.6-cm diameter, three-blade marine propeller was positioned just inside the bottom of the draft tube. It was operated to pump fluid down the center of the draft tube and up the annular space between the coils and the crystallizer walls. The impeller speed was controlled at 1000 rpm during all experiments reported in this paper. The product withdrawal port was located 14.5 cm from the bottom of the crystallizer; it consisted of a piece of 0.95-cm stainless steel tubing bent so that magma flow impinged directly on the entrance to the port. The purpose of this arrangement was to ensure isokinetic removal of the well-mixed magma. Removal of crystals was shown to be isokinetic by experiments involving sampling from various points in the crystallizer and the crystallizer exit. These tests are described by Rousseau and Woo (1980). Magma was removed from the crystallizer with a centrifugal pump activated intermittently by a liquid level
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981 73 Run 9-19 Residence T i m e Data
Set
Run
= 15 mi-
7-18
Residence l i m a
=
29.5 min
MT ( g / m l )
5
e
1
0.50
a
2
0.51
4
3
1
C
E
1
0
1
-1
-. 200
400
400
800
1000
L 200
400
600
800
L Figure 2. Typical CSD sieve data for MgS0,.7H20.
controller. Approximately 500 mL was removed during each pumping cycle. The level was held essentially constant so that the crystallizer volume in all runs was 13 L. A typical run with the system began by heating the feed and product tanks to about 50 OC. When all crystals in these tanks had dissolved, solute concentration and saturation temperature were determined. Flow through the system was begun with the temperature of the feed reduced to a few degrees above saturation. After conditions in the system had stabilized, flow of cooling water through the draft tube-cooling coil was begun and the crystallizer magma was brought to the desired temperature. Experience showed that following this procedure minimized fouling of the coil during both startup and operation. The system was operated with all process variables held constant for 10 residence times; this ensured steady-state conditions as had been shown by Wood (1975). Solute concentrations were restricted to eliminate cycling that had been observed by Ottens (1973) and no evidence of such behavior was noted in these experiments. Samples of the crystals to be sieved were withdrawn rapidly from the magma in the crystallizer using a vacuum flask. Slurry was always taken from the same point in the vessel, even though experiments showed the magma was perfectly mixed. Care was taken to prevent further crystallization during filtration of crystals from the sample. Collected crystals were washed with saturated acetone, dried, and sieved for a fixed period of time. Experimental Results A series of runs were performed in which MgS04.7H20 was crystallized using the equipment and procedure described in the previous section. Samples obtained in each of these runs were sieved and population densities were determined from the mass distribution of crystals. Data are shown for two typical runs in Figures 2 and 3. Each data set on a figure represents the sieve analysis of a sample obtained under a given set of steady-state condi-
Figure 3. Typical CSD sieve data for MgS04.7H20.
tions and, therefore, is indicative of the reproducibility of the experiment. Each figure corresponds to a different set of steady-state conditions. All data obtained with this system were of the form shown in Figures 2 and 3. Curvature of the population density plots means that eq 2 cannot be used to describe crystal size distribution. Furthermore, a series of experiments confirmed the homogeneity of the circulating magma, isokinetic product removal, and insignificant crystal breakage and/or agglomeration. It was determined, therefore, that anomalous crystal growth was the cause of the curvature of data illustrated in Figure 2 and 3. Sizedependent crystal growth was assumed responsible for this anomolous kinetic behavior. Analysis of system behavior was then undertaken using the Abegg-Stevens-Larson equation to describe growth rates. Assuming the ASL growth expression correctly describes MgS04.7H20growth kinetics, model parameters no, G o , and b can be determined by fitting eq 6 (population density function obtained by solving eq 1, 4, and 5 ) to experimentally evaluated population density data. It had been recognized that evaluation of parameters in a population density model should utilize experimentally measured values of magma density rather than a simple fit to population density data. This is straightforward for systems exhibiting size-independent growth kinetics as the magma density is related simply to the model parameters by the third moment of the population density function.
MT = pkvJmnL3 dL = 6 p k , n O ( G ~ ) ~
(9)
It is difficult to use measured magma density as a constraint for systems following size-dependent growth kinetics because an explicit relationship between the kinetic parameters and magma density does not exist. Accordingly, a three parameter nonlinear least-squares search procedure was used to evaluate no, G , and b from each set of experimental data. This approach did not use measured magma densities as a constraint and resulted in values of no that were inordinately large. Also, calculating magma
74
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981 1
I
Run
9-19
Residence
Time
=
15min
C o r r e l 0 t ion C o e f f ic ien t
= 0.9 7
5To
In
ne
A 4-
? 2-
C
-
1-
C
C
C
t
0 b
-1-
b
b
‘\ -2.
-3.
L -
Figure 4. Hypothetical data showing difficulty in evaluating kinetic parameters for size-dependent growth.
densities by substituting evaluated parameters into eq 8 resulted in values that were much larger than had been observed experimentally. The source of these difficulties can be described as follows. As shown in Figure 4, size distribution data are generally obtained over a limited size range, but they must be used to fit a distribution over a range of sizes from 0 to m. Without the magma density constraint, particularly with a system following size-dependent growth kinetics, considerable uncertainty exists in any model giving crystal size distribution. An alternative method utilizing measured values of magma density was developed to evaluate the kinetic parameters no,Go, and b from experimental data. In this method eq 8 is substituted into eq 6 to eliminate no
n=
200
400
Run
600
800
1000
7- 1 8
Residence T i m e
=
29.5min
Correlation Coefficient
= 0.98
MT Ci(b)~k(G~7)*
Equation 10 has two adjustable parameters, Go and b, which may evaluated by fitting to experimental data. Such a procedure, however, requires a lengthy numerical integration to evaluate Cl(b) for each tested value of b. For this work a polynomial was fit to the results of O’Dell and Rousseau (1978) making it possible to evaluate Cl(b) directly. Coefficients of b were determined for values of b between 0.1 and 0.8 resulting in the equation Ci(b) = 1.79 + 4.27b - 14.97b2 71.96b3 - 121.25b4 - 79.05b5
+
(11)
Figure 6. Comparison of experimental data to ASL growth rate model.
Equations 10 and 11 were then fit to the population density data using a least-squares search routine to evaluate b and Go for each data set. The fit obtained is illustrated in Figures 5 and 6, which show smoothed data
(bars indicate variations among samples taken a t a given steady-state condition) and model predictions. Values of system variables and model parameters are summarized in Table I.
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981 75
Table I. Values of System and Model Parameters for Various Residence Times residence time, run MT G", n o3 min no. g/mL b pm/min no./pm mL 9
15.0
8-12
15.0
9-06
15.0
9-19
20.0
7-16
22.0 23.0
9-27 9-21
29.5
7-18
42.0
8-13
0.0550 0.0560 0.0250 0.0240 0.0172 0.0227 0.0198 0.0900 0.0700 0,0690 0.1280 0.0469 0.0564 0.0820 0.0770 0.0840 0.0790
0.62 0.65 0.51 0.39 0.41 0.50 0.51 0.62 0.62 0.58 0.68 0.53 0.52 0.62 0.59 0.58 0.53
Kinetic Correlations Systems exhibiting size-dependent growth rates present some difficulty in developing correlations for nucleation kinetics. This problem stems from the common practice of using growth rate as an independent variable in power law correlations for nucleation rates. Rousseau and Woo (1980) showed that Go, the growth rate of zero-size crystals as determined from the ASL growth expression (eq 4), could be used successfully in these correlations. Accordingly, kinetics evaluated in this study were correlated by the equation
Bo = kNGoiMd
(12) where k N is recognized to be a complex function of temperature, agitation, material of construction of the crystallizer intern&, impurity concentration and a variety of other undetermined factors. These factors were held constant in the experimental program of this research and, therefore, kN was considered invarient. A nonlinear least-squares procedure was used to fit eq 12 to data obtained in this study. The resulting power law kinetic expression is B O = 2-49 X 103G00.51MT0.68 (13) where Bo is in no./(min)(mL), Go is in pm/min, and MT is in g/mL of slurry. Discussion Sikdar and Randolph (1976) evaluated nucleation kinetics of MgS04.7H20in a quasi-steady-state crystallizer. Their experiments were conducted in a seeded system in which retention time of newly formed crystals and liquor was of short duration, while the seed crystals remained permanently in the crystallizer. This arrangement allowed direct measurement of CSD (excluding seed crystals) and left the exiting liquor with a measurable supersaturation. Growth and nucleation kinetics were then fit by the power law equations G = e x p ( 8 . 9 8 9 ) ~ ~(SR-7) .~~
Combining the above equations gives
Although the operation of the crystallizers used by Sikdar and Randolph was different from that used in this study, nucleation in both systems was by the same
1.41 1.54 2.10 3.31 3.20 1.95 1.82 1.48 1.40 1.59 1.00 1.52 1.24 0.92 0.96 0.94 0.81
941.0 958.0 241.0 82.0 60.0 250.0 267.0 401.0 388.0 352.0 703.0 268.2 688.0 517.0 565.0 181.0 478.0
Bo no./d
process temp, "C
1327 705 505 276 191 487 485 593 543 560 703 408 854 436 655 170 387
30.9
9
30.8 26.4 31.0 31.0 32.0 34.3 31.0 31.0
mechanism. In the Sikdar and Randolph crystallizer contact nucleation resulted from impeller impacts with seed crystals selectively held in the system; in this study, contact nucleation resulted from impeller impacts with crystals in the mixed magma typical of continuous stirred-tank crystallizers. Accordingly, it is not surprising that the exponents on magma density in eq 13 and 14 are essentially identical. However, the Sikdar and Randolph crystallizer differed substantially from that of this study in two respects: first, the crystal size distributions in the two studies were measured differently and covered different size ranges; second, the two systems operated over significantly different ranges of supersaturation. Sikdar and Randolph used a Coulter Counter to determine crystal population densities in the size range 10 to 70 pm. Accordingly, the growth rates they determined were for crystals in this size range. In this study population densities were determined by sieve analysis of crystals in the size range from 44 to 1000 pm. These data showed growth rates to depend strongly on crystal size and a quantity Go that was evaluated by fitting the ASL equation to the measured population density. Since the quantities Go in eq 13 and G in eq 14 are fundamentally different, it is not unexpected to find that they effect nucleation rates differently. The Sikdar and Randolph crystallizer was operated at measurable levels of supersaturation; in the present study, the crystallizer was operated to approach Class I1 behavior, which means that negligible supersaturation remained in the exit liquor. It has been demonstrated by a number of researchers that variations in effective nucleation rates with supersaturation can be significant. (Garside and Davey (1980) give an excellent review of this subject.) Hence, the differences noted in the previous paragraph are not surprising. Because of the noted inconsistencies in eq 13 and 14 it is suggested that eq 13 provides a guide for nucleation rates of MgS04.7H20 in Class I1 operations. Equation 14 is useful in estimating the effect of temperature on nucleation rates and for operations at high supersaturations (Class I behavior). It is interesting to note that values of b given in Table I are not constant from one set of experimental conditions to the next. Garside and JanEie (1978) observed such variations when analyzing experimental data for potassium alum obtained from three crystallizers of different sizes. They suggested b was related to specific power input to the magma. Rousseau and Woo (1980) also used ASL growth kinetics to model potassium alum crystal size
76
Ind. Eng.
Chem. Fundam., Vol. 20, No.
1, 1981
distributions. Their values of b, however, were a factor of 2 or 3 greater than those obtained by Garside and JancEiE. Clearly, the value of b depends on the crystalline system and the process conditions. A mechanism explaining this observation has not been developed. It was noted, however, that values of b obtained in this study seemed to vary with Go and 7. A power-law correlation was examined to test a statistical relationship between b and these variables; i.e., data were fit to the expression b = aGO"Tfl (15) Parameters from runs 8-12,946, and 9-19 were correlated first because 7 had the same value in each of these experiments; the correlation gave CY = -0.55 with a correlation coefficient of 0.933. Parameters for the remaining runs were then correlated to give fl = -0.46 and a = 2.83 with a correlation coefficient of 0.903. The resulting correlation is b = 2.33G04.55T-0.46 (16)
with Go in Fm/min and 7 in min. A mechanistic interpretation of eq 16 is not possible without more experimental data on a variety of systems. However, since both Go and 7 are indirectly related to supersaturation, eq 16 may be indicating an effect of supersaturation on size-dependent growth. Garside (1979) observed such behavior by potassium alum crystals in a size range below 30 ym. His data showed that a fraction of these crystals would not grow and that this fraction depended upon system supersaturation. Perhaps larger crystals also exhibit an unusual and complex relationship where the magnitude of the effect of crystal size on growth rate is determined by supersaturation Conclusions Magnesium sulfate heptahydrate was found to follow size-dependent growth kinetics; earlier studies had examined limited size ranges and had led incorrectly to the conclusion that this system followed size-independent kinetics. Population density data were fit to evaluate nucleation and growth rate parameters, including those in the two-parameter form of the Abegg-Stevens-Larson equation. These parameters were evaluated using measured values of magma density and the relationship of magma density to the kinetic parameters as a constraint in the search procedure. Nucleation rates were fit by the power-law expression B O = 2.49 x 103G00.51M 0.68 T The dependence on magma density corresponds closely to that observed in earlier studies. A significant deviation exists between effects of supersaturation (as reflected through the parameter Go) observed in this study and those in previous studies. Reasons for this discrepancy and suggestions for utilization of the nucleation rate equations are given in the paper.
The parameter b in the ASL equation incorporates the effect of crystal size on its growth rate; experimental conditions investigated caused variation in b from 0.39 to 0.68. An empirical fit of these results gave eq 16, which shows a relationship of b to nucleus growth rate and residence time, and introduces evidence useful in determining reasons for size-dependent growth. Acknowledgment The financial support of the National Science Foundation through grants numbered ENG 73-08315 A02 and ENG 7822656 is gratefully acknowledged. Nomenclature Bo = nucleation rate, no./min mL b =. kinetic . parameter in ASL growth rate equation, dimensionless C,(b) = moment coefficient for two parameter ASL model, dimensionless G = crystal growth rate, pm/min Go = nuclei growth rate, pm/min i, j, kN = parameters in power law model for nucleation rate-see eq 11 k , = volume shape factor, cm3/pm L = characteristic crystal dimension, Fm MT = suspension density, g/mL n = population density, no./mL ym no = population density of zero-size crystals, no./mL pm s = supersaturation, g of solute/g of H20 Greek Letters y = constant in ASL growth rate equation,set equal to l / G O r in two-parameter ASL equation p = crystal density, g/cm3 T = residence time, min Literature Cited Abegg, C. F.; Stevens, J. D.; Larson, M. A. AIChE J. 1988 74, 118. Bauer, L. G.: Larson, M. A.; Dallons, V. L. Chem. Eng. Sci. 1974, 29, 1253. Canning, T. F. Chem. Eng. Prog. Symp. Ser. No. 770 1971, 67, 74. Clontz, N. A.; McCabe, W. L. Chem. Eng. Prog. Symp. Ser. No. 770 1971, 67,6. Clontz, N. A.; Johnson, R. T.; McCabe, W. L.; Rousseau, R. W. Ind. Eng. Chem. Fundam. 1972, 7 1 , 368. Garside, J. Ind. Crysf. 78 1979, 143. Garside, J.; Davey, R. J. Chem. Eng. Commun. 1980, 4, 393. Garside, J.; Jancic, S. J. Chem. Eng. Sci. 1978, 33. 1623. Johnson, R. T.; Rousseau, R. W.; McCabe, W. L. AIChE Symp. Ser. No. 727 1972, 68,31. O'Dell, F. P. Ph.D. Thesis, North Carolina State University, Raleigh, NC, 1978. O'Dell, F. P.; Rousseau, R. W. AIChE J. 1978, 24, 738. Ottens, E. P. K. Ph.D. Dissertation, Technoioglcal University of Delft, The Netherlands, 1973. Ottens, E. P. K.; de Jong, E. J. Ind. Eng. Chem. Fundam. 1973, 72, 179. Randolph, A. D.; Larson, M. A. "Theory of Particulate Processes,'' Academic Press: New York, 1971; Chapter 4. Rousseau, R. W.; Woo, R. AIChE Symp. Ser. No. 793 1980, 76, 27. Shah, 9. C.; Rousseau, R. W.: McCabe, W. L. AIChE J. 1973, 79, 194. Sikdar, S . K.; Randolph, A. D. AIChE J. 1976, 22, 110. Tai, C. Y.; McCabe, W. L.; Rousseau, R. W. AIChE J . 1975, 27, 351. White, E. T.; Bendig. L. L.;Larson, M. A. AIChE Symp. Ser. No. 753 1978, 72, 41. Wood, L. M., M.S. Thesis, North Carolina State University, Raleigh, NC, 1975.
Receiued for reuiew February 19, 1980 Resubmitted September 29, 1980 Accepted November 7 , 1980
