Ind. Eng. Chem. Fundam. 1986, 25, 174-176
174
s =
slave or secondary loop
Shinskey, F. G. Chem. Eng. Progr. 1978, 7 2 , 73. Yu, C. C.; Luyben, W. L. Ind. Eng. Chem. Fundam.. in press.
Process Modeling and Control Center Cheng-Ching Yu Department of Chemical William L. Luyben* Engineering Lehigh University Bethlehem, Pennsylvania 18015
Superscript
set = set point
Literature Cited Fuentes, C.; Luyben, W. L. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 361. Luyben, W. L. Ind. Eng. Chem. Fundam. 1973a, 12, 463. Luyben, W. L. “Process Modeling, Simulation, and Control for Chemical Engineers”; McGraw-Hill: New York. 1973b; p 420.
Received for review October 23, 1984 Revised manuscript received July 18, 1985 Accepted August 28, 1985
Modeling Growth Rate Dispersion in a Batch Sucrose Crystallizer Growth rate data from experiments on sucrose crystals are analyzed for growth rate dispersion (the variation of crystal growth rates within a population). Two competing models are used in this analysis. Experimental results suggest that carefully designed experiments must be performed to distinguish between the two.
Introduction The phenomenon of growth rate dispersion (GRD), sometimes called “size dispersion”, is a significant factor in the control of the crystal size distribution (CSD) in batch crystallizers. This phenomenon describes the variation in growth rate among crystals of the same size while they grow under constant microscopic conditions. Two methods of modeling GRD, based upon different mechanisms, have been presented in the literature. Randolph and White (1977) proposed one in which it is assumed that the growth rate of an individual crystal fluctuates in the course of time. A second mechanism based on the work of Berglund (1981), Berglund and Larson (1982), and Ramanarayanan et al. (1982) has been proposed in which it is assumed that an individual crystal has an inherent growth rate that is constant, but different crystals have different inherent growth rates. The purpose of this work is to analyze batch crystallization data for sucrose in the context of these two possibilities. Growth Rate Dispersion Experiments White and Wright (1971) conducted batch experiments in which they grew an initially monosized distribution of sucrose crystals. By sampling their crystallizer at subsequent times, they observed that the CSD widened with time, indicating that some type of growth rate dispersion was present. Valcic (1975) performed photomicroscopic experiments on sucrose crystals and observed a difference in growth rates for various faces as well as differences from crystal to crystal. These results were explained in light of the Burton-Cabrera-Frank (BCF) dislocation model of crystal growth. Valcic (1975) asserted that variations in the number of dislocations from crystal to crystal would be responsible for these observations. Shanks and Berglund (1985) conducted experiments with a photomicroscopic cell similar to that of Valcic (1975). In these experiments, sucrose crystals formed by contact nucleation were grown for a period of about an hour. It was found that individual crystals had a constant growth rate with time but the variation in growth rate from crystal to crystal was very large. In fact, the variation from the smallest to the largest growth rate was an order of magnitude and constituted a growth rate distribution. Modeling of Growth Rate Dispersion The first to attempt to quantify GRD was White and Wright (1971), using the concept of a dispersion coefficient, p , defined by 0196-4313/86/1025-0174$01.50/0
p = AaL2/af;
(1)
The value p is a measure of how much the CSD widens as mean size is increased. It is found by determining the slope of a plot of the variance of the CSD vs. mean size. In order to develop a modeling procedure based upon the population balance, Randolph and White (1977) introduced the concept of growth rate diffusivity, DG. This term was developed as an analogue to Taylor dispersion and introduced intothe population balance. Upon solution of the population balance for a batch crystallizer, the following results were obtained.
(4) From these results, we may conclude that a plot of the variance of the CSD vs. time should yield a straight line with a slope equal to twice the growth rate diffusivity and that the latter quantity is equal to the product of the dispersion coefficient and the mean growth rate. An inherent assumption in the model is that the mean growth rate is not a function of size. This is evident from eq 4 in which growth rate is estimated from the overall change in average size. The model presented above will be referred to in subsequent discussion as the ”random fluctuation” (RF) model. This designation is deemed appropriate since the mechanism through which GRD is assumed to occur involves the fluctuation of a crystal’s growth rate through various values. Randolph and White (1977) applied the RF model to the data of White and Wright (1971). From their calculations they determined values for DG and p . When a plot of DG vs. G was made, a very large amount of scatter was present, so calculation of p from this plot was not very precise using eq 3. An alternate approach to that of Randolph and White has been suggested by Ramanarayanan et al. (1982). The basis of this second model is the assumption that each crystal has i b own inherent growth rate. This model will subsequently be referred to as the “constant crystal growth” (CCG) model. As in the RF model, the CCG model assumes no size dependence of the mean growth @
1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
rate. The CCG model results from assuming that the size of any crystal may be approximated by L = Lo Gt (5)
0
70°C 6OoC A 5OoC 0 4OoC 0
+
0
After taking expected values of eq 5 and assuming that growth rate and initial size are independent, the following two equations result L = t o at (6)
+ aL2 = ob2 + t 2 a G 2
0 0
E
?- 30-
=
t2oG2
0
p
0
0 O .
0 0
fa-
(7)
e
0
O
O
0
0
CLmg =278 -
f
0
0
O20-
Upon assuming a monosized seed distribution, i.e., u L t = 0, eq 7 becomes .L2
175
0
/
pm
-
-
0 0
0 O O
O
A
A
0
10-
(8)
A
Ramanarayanan et al. (1982) showed that, upon using eq 6-8, the following results can also be obtained.
I00
50
150
200
250
300
G , pm/hr
Figure 1. Dispersion coefficient vs. mean growth rate for sucrose crystals [data from Berglund (1980)l. 5000 70'C 60°C A 50°C 0 4OoC 0
At least two conclusions can be drawn from the above analysis. (1)Equation 7 indicates that a plot of the variance of the CSD vs. time squared should yield a straight line with a slope equal to the variance of the growth rate distribution. This is in contrast to the RF model which predicts a linear relation between the variance of the CSD and time to the first power. (2) Equations 9 and 10 indicate that, in the context of the CCG model, the dispersion coefficient and the growth rate diffusivity are not constants. Consequently, these parameters should not be used to characterize GRD. Rather, the variance of the growth rate distribution should be used. In the remainder of this paper the use of these two models will be discussed. Experimental Section In the analysis that follows the data of Berglund (1980) will be used. A full description of the experimental apparatus and techniques is presented in that work. The experiments were conducted by seeding a batch sucrose crystallizer with nearly monodisperse crystals (250-297 pm). The experiments were conducted such that no subsequent nucleation and no depletion of supersaturation occurred (initial rate experiments). The 350-mL batch cooling crystallizer was equipped with a draft tube to ensure a well-mixed vessel. During each run, the CSD was measured at timed intervals by a Coulter Counter Model TA I1 equipped with a 2000-pm aperture tube. The temperature range covered in these experiments was 40-70 "C.
Results and Discussion The raw data obtained from the 39 experiments that were performed consisted of a series of CSD measurements at various times. From these data the mean size and variance of the CSD were calculated. The mean growth rate for each experiment was determined by plotting mean size vs. time and finding the slope of the linear regression line. In all cases the plot was linear, substantiating the assumption of no size-dependent growth. The variance of the size distribution was also plotted in several different ways to ascertain which model for GRD was most appropriate. The dispersion coefficient, p , for each experiment was calculated by plotting the variance of the size distribution against the mean size and finding the slope of the linear regression line. Figure 1is a plot of p vs. mean growth rate for all experiments. From this plot (in which the data are
0
4000-
L
3000 N
E
0 "
2000
'Oo0l
"0
50
100
150
200
250
300
G, p m / h r
Figure 2. Growth rate diffusivity vs. mean growth rate for sucrose crystals [data from Berglund (1980)l.
quite scattered) no apparent trend could be discerned; however, an average value of 27.8 pm2/pm was calculated. The scatter seen in this plot is similar to that observed by White and Wright (1971) when p was plotted vs. growth rate. The growth rate diffusivity, D G , was calculated for each experiment from the slope of the linear regression of the variance of the size distribution against time. Figure 2 is a plot of D G vs. growth rate. A straight line is predicted for this plot from eq 3; the slope should equal 0 . 5 ~ .Half the average value of p is 13.9 wm2/pm, which is approximately the value of the slope in Figure 2, 13.1 pm2/pm. This indicates that the RF model is internally consistent but does not really answer the question of whether it is applicable. Figure 2 also exhibits a large amount of scatter. To analyze the data in terms of the CCG model, the variance of the growth rate distribution uG2 was calculated by plotting the variance of the size distribution vs. time squared and finding the slope of the linear regression line. Figure 3 shows a plot of uG2 vs. mean growth rate. A power law relation given as uG2
= 0.828G''14
(11)
was fit to the data. A correlation coefficient of 0.9505 was found for the fit of the constant to the expression linearized by logarithmic transformation.
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
176
24000
20000
16000
.
N
N
&5.
12000
-
0 7OoC
0 60°C
-
A
50°C
o 4OoC
I
“0
50
100
150
200
250
300
G ,um/hr
Figure 3. rate
Variance of the growth rate distribution vs. mean growth for sucrose crystals [data from Berglund (1980)].
In principle, the form of the variance of the CSD vs. time relationship should allow discrimination between the two models presented. In practice, however, the CCG model fits only slightly better. It is clear from previous experimental data on single crystals (Valcic, (1975; Shanks and Berglund, 1985) that the CCG model should be appropriate. This determination would be difficult to make on the basis of these data. In all of these data there is a considerable amount of scatter. This is due to the difficulty in the accurate measurement of the variance of the CSD in a nearly monosized population. The experiments were performed such that no supersaturation change was observed; i.e., these are essentially initial rate experiments. For purposes of testing between the two models it is necessary to perform the experiments to long times. This would make it clear whether the variance of the CSD was linearly (RF model) or parabolically (CCG model) related to time. Initial rate experiments as performed in the present work are useful in estimating growth rate dispersion if the model is known. However, testing for the specific model should probably not be attempted using such experiments. Further work should be undertaken to determine the criteria necessary for accurate discrimination between models. Finally, it should be understood that the two models proposed are probably just limiting cases of the true
phenomenon (Garside, 1984). That is, crystals grow at some inherent rate with fluctuations superimposed on this rate. This possibility should lead to further growth rate studies to determine the presence of such phenomena. Conclusions 1. Growth rate dispersion is an important factor in determining the crystal size distribution in sucrose crystallization. 2. The “random fluctuation” model and the “constant crystal growth” model fit the crystal size distribution data equally well. However, since the latter model is based on a physical mechanism that has been experimentally verified for the sucrose-water system, it is to be preferred in this case. 3. Precise modeling of growth rate dispersion requires quite accurate measurements of the crystal size distribution to long times. 4. The possibility of superimposing the two models into a more general model should be explored. Nomenclature DG = growth rate diffusivity, L2/t G = growth rate, L/t L = size, L Lo = initial size, L L = mean size, L Lo = mean initial size, L p = dispersion coefficient,L2/L t = time, t uG2 = variance of the growth rate distribution, L2/t2 uL2 = variance of the crystal size distribution, L2 uL: = variance of the initial crystal size distribution, L2 Registry No. Sucrose, 57-50-1.
Literature Cited Berglund, K. A. M.S. Thesis, Colorado State Unlversity, Ft. Collins, CO, 1960. Berglund, K. A. Ph.D. Dissertation, Iowa State University of Science and Technology, Ames, IA, 1961. Berglund, K. A.; Larson, M. A. AIChE Symp. Ser. 1982, 78, 9. Garside, J. UMIST, Manchester, England, UK, private communication, 1984. Ramaharayanan, K. A.; Berglund, K. A,; Larson, M. A,, paper presented at the 75th Annual Meeting of AIChE, Los Angeles, CA, Nov 14-18, 1982. Randolph, A. D.; White, E. T. Chem. Eng. Sci. 1977, 32, 1067. Shanks, B. H.; Berglund, K. A. AIChE J . 1985, 3 1 , 152. Valcic. A. V. J . Crust. Growth 1975, 30, 129. White, E. T.; Wright, P. G. Chem. Eng. f r o g . , Symp. Ser. 1971, 67, 81.
Department of Agricultural and Chemical Engineering Colorado State University Ft. Collins, Colorado 80523
Kris A. Berglund’’ Vincent G . Murphy
Received f o r review November 8, 1984 Accepted October 7, 1985 Current address: Departments of Agricultural and Chemical Engineering, Michigan State Unlversity, East Lansing, M I 48824.
