and Limestone Scrubbers", a paper presented at the EPA Flue Gas Desulfurization Symposium, Atlanta, Ga.. Nov 4-7, 1974c. Chen, B. H., Douglas, W. J. M., Can. J. Chem. Eng.,46, 245 (1968). Epstein. M.. "EPA Alkali Scrubbing Testing Facility: Sodium Carbonate and Limestone Test Results", report prepared by Bechtel for the EPA (Aug 1973a). Epstein. M., "EPA Alkali Scrubbing Test Facility at the TVA Shawnee Power Plant", Bechtel progress report prepared for the EPA for July 1, 1973 to Aug 1, 1973 (Aug 31, 1973b). Epstein, M., progress report for Oct 1, 1973 to Nov 1, 1973 (Nov 30, 1 9 7 3 ~ ) . Epstein. M., "EPA Alkali Scrubbing Test Facility: Limestone Wet Scrubbing Test Results", report prepared by Bechtel for the EPA (Jan 1974a). Epstein, M., "EPA Alkali Scrubbing Test Facility at the TVA Shawnee Power Plant", Bechtel progress report prepared for the EPA for Dec 1, 1973 to Jan 1, 1974 (Jan 31, 1974b). Epstein. M., progress report for Jan 1, 1974 to Feb 1, 1974 (Feb 28. 1 9 7 4 ~ ) . Epstein, M.. progress report for May 1. 1974 to June 1, 1974 (June 30, 1974d). Epstein. M.. progress report for June 1, 1974 to July 1, 1974 (July 31. 1974e). Epstein, M., progress report for July 1, 1974 to Aug 1, 1974 (Aug 31, 19741).
Epstein, M., Sybert, L., Wang. S.C.. Leivo, C. C.. Princiotta, F. T., "Scrubbing Test Facility at the TVA Shawnee Power Plant", a paper presented at the 66th Annual Meeting of the A.I.Ch.E., Philadelphia, Pa.. Nov 1973. Fan, L. S.,Ph.D. Dissertation, West Virginia University, 1975. Gleason, R. J. "Limestone Scrubbing Efficiency of Sulfur Dioxide in a Wetted Film Packed Tower in Series with a Venturi Scrubber", paper presented at the Second International Lime/Limestone Wet Scrubbing Symposium, New Orleans, La., Nov 8-12, 1971. Nannen, L. W., West, R. E., Kreith, F., J. Air. Pollut. ControlAssoc., 24, 29 (1974).
Received for review August 4 , 1975 Accepted March 3, 1976 T h e authors wish to express their gratitude to the Environmental Protection Agency for the support of this work under grant number EHS-D-71-20 and to J. Orndorff and C. Y. Lin for help in preparing the manuscript.
COMMUNICATIONS
Design Considerations for a Multistage Cascade Crystallizer
Experimental approaches were examined for determining the maximum allowable crystal growth rate that can be achieved in the absence of nucleation. This information was considered essential for the design of a multistage continuous suspension crystallizer in which nucleation occurs only in the first stage. A single-stage crystallizer seeded continuously with monodisperse crystals was shown to be better suited than a multistage system for obtaining useful growth rate data.
Much work has been done over the past decade in understanding the crystallization process. The work of Randolph and Larson (1962) and Hulburt and Katz (1964) has provided useful techniques for interpreting crystal size distributions (CSD) from suspension crystallization processes. Many experiments have since made use of these techniques in the design and analysis of the continuous mixed-suspension, mixed-product-removal (CMSMPR) crystallizer. Steady-state experiments with a single-stage CMSMPR crystallizer provide size distribution data that are readily interpreted to give meaningful estimates of the growth and nucleation kinetics (Randolph and Larson, 1971). Crystallization is sometimes performed continuously in staged vessels such that the entire magma of one vessel discharges as feed to a subsequent vessel. The resulting CSD can be altered significantly in this cascade system under different operating strategies. Variations of this problem have been treated previously (Robinson and Roberts, 1957; Randolph and Larson, 1962, 1971; Randolph, 1965; Larson and Wolff, 1971; Njvlt, 1971). Randolph (1965) considered a two-stage continuous cascade system with allocated production and retention times and indicated that the two-stage system produces a considerably smaller and somewhat more uniform CSD than a single stage of equal total volume. Larson and Wolff (1971) studied mathematically the case where no nucleation occurs in the second stage and defined operating conditions necessary to obtain a given maximum allowable supersaturation (and hence, a given maximum permissible growth rate) without nucleation. These investigators considered an arbitrary choice of the limit of the metastable region where growth can occur without nucleation but suggested no method for determining this limit for a given system.
Larson and Garside (1973) discussed the need for knowing such a limit in the design of both batch and continuous crystallizers in which the suppression of nucleation is desired. Their analysis considered techniques for determining the maximum allowable supersaturation for a Class I (low-yield) crystallization system. However, for a Class I1 (high-yield) system where the supersaturation generally cannot be measured, determination of the maximum growth rate is not nearly as straightforward. The purpose of this communication is to suggest an experimental approach for determining the maximum permissible growth rate (in the absence of nucleation) for all but the first stage of a multistage cascade system. The following discussion is especially useful f i r Class I1 systems in which the supersaturation cannot be determined experimentally.
Two-Stage Cascade System Consider a cascade of two well-stirred stages operating with no seeding in the first stage. Under the assumptions of a size-independent growth rate (McCabe's AL.law), zero-size nuclei, and constant and equal input and output flowrates, the steady-state CSD for each of the two stages can be given (Randolph, 1965) as nl =
nlOe-L/GiTi
(1)
The suspension density for each stage can be used as a constraint on the above two equations. Thus
MI =
som
pkvnlL3dL = 6phVnl0(G1T1)*
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
(3) 467
l 5 - - - - - - l
.I5
-25
t 0
2
4
6
8
1 0 1 2
Dimensionless size ( x )
Figure 1. Crystal size distribution for the second stage of a two-stage cascade system ((3 = 3).
M Z=
Jm
pkVnzL3dL = 6 p k , n z " ( G ~ T z ) ~
For a Class I1 system, the suspension density can be calculated from a simple mass balance and serves as one of the experimental parameters. The nucleation and growth kinetics for the unseeded case can be determined in a straightforward manner (Randolph and Larson, 1971) from an analysis of the CSD in the first stage. However, analysis of the CSD in the second stage is more complicated since the final product distribution is influenced by the growth and nucleation processes taking place in both stages. Also, it is possible to operate the second stage with growth only. The effect of growth rate in the second stage on the final CSD will now be considered. Introducing the following dimensionless variables and parameters
the CSD and constraint for the second stage can be rewritten as
would be somewhat smaller than the maximum value. The dashed curve represents the CSD when nucleation and growth occur simultaneously. The increase in the density function in the small size region is due to the presence of nuclei. It should be noted that for a given system, there may be a maximum 0 (related to a maximum supersaturation) and a corresponding maximum growth rate allowable to avoid nucleation in the second stage. Experimentally, it is difficult to determine from photomicrographs of the product stream whether nucleation occurs in the second stage since the input to this stage includes small nuclei generated in the first stage. Furthermore, size distribution data obtained from experiments will be mostly in the straight-line portion of the curves ( x > 1).The slope of CSD curves in this region is fairly insensitive to the value of the growth rate in the second stage. Attempts to fit the CSD curves with experimental data to determine growth rate would be extremely difficult owing to the usual scarcity of data in the small size region. It is obvious that a two-stage system is not suitable for determining the maximum allowable growth rate. We therefore suggest a continuously seeded, single-stage system (Larson and Garside, 1973; Desai et al., 1974) to simulate the performance of the second stage in a two-stage cascade system. This simulation should be fairly reasonable for systems in which the crystallization processes are not seriously influenced by the size distribution in suspension.
Continuously Seeded System Consider now a single-stage CMSMPR crystallizer that is continuously seeded with crystals of size L,. Again, under the same assumptions made earlier for the two-stage cascade system, the steady-state CSD and the constraint for this system can be given as n = noe-L/GT (0 IL n = [noe-Ls/GT + n,]e-(L-Ls)/GT
+
< L,) ( L 2 L,)
+ 3G2T2Ls2 + 6G3T3Ls+ 6 G 4 T 4 )
(7) (8)
M = 6 ~ k , n " ( G T ) ~ pk,n,(GTLS3
(9)
where n, is the seed number density. Equations similar to the above have been derived previously (Larson and Garside, 1973; Desai et al., 1974). By defining the following dimensionless variables and parameters
eq 7 , 8 , and 9 can be rewritten as Here, y o represents the ratio of nuclei density in the second stage to that in the first stage so that when y o = 0, no nucleation occurs in the second stage. ct is equivalent to the ratio of the average amount of growth in the two stages, and /3 represents the ratio of suspension densities. In practice, /3 must be greater than 1to provide crystallization in the second stage. These three parameters ( y o , a, and p) are related by the constraint, eq 6. Figure 1 shows the dimensionless CSD in the second stage for a given suspension density ratio (6= 3 ) . The solid curve represents the predicted CSD when no nucleation occurs ( y o = 0). Owing to the absence of nucleation, the distribution curve yields a maximum in the small size region and then drops rapidly. The corresponding growth rate obtained here is equivalent to the "theoretical" maximum rate for this particular suspension density ratio. If nucleation occurs, there would be a competition between nucleation and growth for available supersaturation so that the growth rate obtained 468
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 3, 1976
Y = Yoe-aX
(0 IX
< 1)
+ lle-a(x-1) ( X 2 1) a4b = 6 Y 0 + a3 + 3a2 + 6a + 6
Y = [YOe-U
(10) (11) (12)
where Y o is the number density ratio of nuclei to seeds; a represents the ratio of seed size to an average crystal size due to the growth in the crystallizer, and b is equivalent to the ratio of suspension density to the mass concentration of seeds. Again, these three parameters are related by the constraint, eq 12. Figure 2 shows the dimensionless CSD for a given ratio of output suspension density to mass of seeds ( b = 3). For a = 1.82, the distribution represents the case when no nucleation occurs ( Y o = 0). Because of the absence of nucleation, the number density vanishes in the size region where X < 1 ( L < L,). If nucleation occurs, the plot of In Y vs. X would show two straight lines having the same slope but divided at X = 1. This is illustrated by the dashed curves in Figure 2.
monodisperse crystals. This approach is particularly useful for Class I1 systems in which the measurement of supersaturation is difficult. The analysis can be combined with that of Larson and Wolff (1971) to determine an optimal operating policy which satisfies given criteria based on a desired final crystal size distribution.
Acknowledgment The authors wish to thank R. W. Strong for assisting in the numerical computations.
Dimensionless size ( X )
F i g u r e 2. Crystal size distribution for a continuously seeded, single. stage system ( b = 3).
The comments made earlier concerning the maximum growth rate allowable to avoid nucleation in the second stage of a two-stage cascade system should also hold for this case. However, there are distinct advantages in analyzing the single-stage system seeded with monodisperse crystals. With seeding, one can experimentally choose relatively large seeds so that the seeds and nuclei are discernible in size and the occurrence of nucleation can be detected in the output. Unlike the two-stage system, the size distributions here are rather sensitive to growth rate. Furthermore, as indicated in Figure 2 , the plot of In Y vs. X yields straight lines with slope proportional to -l/GT. Therefore, a reasonably accurate growth rate can be determined from the slope of this plot. Knowing the growth rate and the occurrence of nucleation, one can determine from a series of experiments the maximum ratio of suspension density to seed mass and the corresponding maximum growth rate permissible to avoid nucleation. The information for this maximum growth rate is important in the design of multistage cascade crystallization systems where nucleation in the subsequent stages is undesirable.
Summary The design of a multistage continuous cascade crystallizer, operating with nucleation in the first stage only, requires a knowledge of the maximum crystal growth rate that can be achieved without nucleation. This information can be obtained readily from experiments carried out using a singlestage CMSMPR crystallizer seeded continuously with
Nomenclature a = dimensionless parameter, L,/GT b = dimensionless parameter, M/pk,n,LS4 G = linear crystal growth rate k , = crystal volume shape factor L = linear crystal size M = crystal suspension density n = population density function n o = nuclei population density T = crystallizer mean residence time X = dimensionless variable, L/L, x = dimensionless variable, L / G I T 1 Y = dimensionless variable, n/n, Y o = dimensionless parameter, n o / n s y = dimensionless variable, n2/nIo y o = dimensionless parameter, n2O/n1' (Y = dimensionless parameter, G ~ T ~ / G ~ T I 6 = dimensionless parameter, M2/M1 p = crystal density Subscripts 1 = first crystallizer 2 = second crystallizer s = seeds Literature Citqd Desai, R. M., Rachow, J. W., Timm, D. C., AlChE J., 20, 43 (1974). Hulburt. H. M., Katz, S.,Chem. Eng. Sci., 19,555 (1964). Larson, M. A., Garside, J., Chem. Eng., (London), No. 274, 318 (1973). Larson. M. A.. Wolff, P. R., Chem. Eng. Prog. Symp. Ser., 67, 97 (1971). NjYlt, J., "Industrial Crystallization from Solutions," Butterworths, London, 1971 Randolph, A. D., AlChE J., 11, 424 (1965). Randolph, A. D., Larson, M. A., AlChE J.. 8,639 (1962). Randolph, A. D., Larson. M. A.. "Theory of Particulate Processes," Academic Press, New York, N.Y., 1971. Robinson, J. N., Roberts, J. E.. Can. J. Chem. Eflg., 35, 105 (1957).
Research Laboratories E a s t m a n Kodak Company Rochester, N e w York 14650
Jong-Shinn Wey* J a m e s P. T e r w i l l i g e r
Received for review September 5, 1975 Accepted M a r c h 1, 1976
Mass Transfer and Power Consumption in Reciprocating Plate Extractors
Mass transfer data and power consumption in two types of reciprocating plate extractors are compared. Extractors having plates with large holes and large free area consume more energy to achieve the same performance than those having plates with small holes and small free area.
Introduction Two designs of reciprocating plate extractors, developed by Karr (1959) and Prochazka and Landau (19641, are in common use. The main difference between the two types lies
in the design and function of the plates. In the former design the plates are of open structure with large holes and large free area. The plates do not restrict the flow of the two phases but considerable agitation is required to disperse one phase in the Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
469
