Ind. Eng. Chem. Process Des. Dev. 1986, 25, 66-70
66
Initial Breeding in Seeded Batch Crystallizers Martha W. Girolami and Ronald W. Rousseau' Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905
This experimental study examined factors affecting the number and characteristics of crystals produced by initial breeding in seeded batch crystallizers. The number of crystals formed was found to depend on the surface area of the seed crystals but not the supersaturation prevailing in the solution. This mechanism of secondary nucleation produced crystals in the order of 5 pm in size. The original crystal size distribution, which was determined to be very narrow, broadened significantly with the extent of growth of the crystals. Broadening of the distribution is due to growth rate dispersion. The implications of these results are significant, as they shed light on the role played by this nucleation mechanism in determining the product crystal size distribution in seeded batch crystallizers.
Batch crystallizers have the potential to produce a product with a narrow crystal size distribution, provided nucleation can be satisfactorily controlled and crystal growth is uniform. For example, if nucleation of the entire population occurs early in the batch cycle, either by primary or secondary mechanisms, subsequent steady and uniform growth of the nuclei will produce product crystals with a narrow size distribution. On the other hand, the occurrence of nucleation throughout the batch cycle or variations in crystal growth rates, due to growth rate dispersion or size-dependent growth, will produce a broad distribution. These qualitative observations hold regardless of the mode by which supersaturation is generated in the system. It is common practice to seed batch crystallizers, thereby avoiding the generation of high supersaturations which lead to excessive nucleation. Indeed, crystallization of some materials (e.g., sucrose) cannot occur under reasonable conditions without the use of seeding. Seed crystals present surface area for growth and foster secondary nucleation. This paper is concerned with initial breeding, one of many mechanisms by which secondary nucleation can occur and a potential major contributor to nucleation in seeded batch crystallizers. Initial breeding was defined by Strickland-Constable (1972) as the formation of crystals resulting from immersion of seeds into a supersaturated solution. Although no formal study of initial breeding has been reported, it is thought that crystals formed by this mechanism are the products of microscopic entities formed on the surface of seed crystals as they are dried. These entities dislodge as the seed crystals are immersed in liquor and may be thought of as nuclei from initial breeding. If the solution into which the seed crystals are immersed is supersaturated, the nuclei from initial breeding grow to form macroscopic crystals. This view of initial breeding would indicate that it plays no role in operations where seeding was by a slurry of crystals, as is the case in many sugar crystallizers. Despite their size and the fact that they were actually formed prior to seeding, microscopic crystals produced by initial breeding will be referred to in this paper as nuclei. When initial breeding dominates other nucleation mechanisms in batch crystallization, it determines the size distribution of new crystals and, ultimately, the size distribution of the product. This study was undertaken to evaluate the characteristics of nuclei produced by initial breeding and to identify those variables that may be used
* To whom correspondence should be
addressed.
0196-4305/86/1125-0066$01.50/0
to improve the control of seeded batch crystallizers. Aqueous solutions of potassium alum (KA1(S04)2-12H20) were used as a model system. The effects of seed crystal surface area, supersaturation of the solution, presence of an impurity, and seed crystal pretreatment were examined.
Experimental Section These experiments employed a glass, 200-mL batch crystallizer. The crystallizer temperature was maintained at 29.4 i 0.1 "C in all experiments. All solutions were obtained by dilution of aliquots taken from one master solution. The master solution was maintained a t a fixed temperature and with gentle agitation and large crystals of potassium alum in the bottom of the container. Before use in an experiment, a sample of the master solution was analyzed for potassium alum concentration, diluted with an amount of dionized water that would give the desired concentration, and reanalyzed prior to use. Deionized water and ACS Certified reagent-grade potassium alum were used in all preparations. Once the preheated and filtered solution was poured into the crystallizer, stirring was begun, and the system was cooled to the desired supersaturation and temperature. Seed crystal samples were taken from two carefully sieved populations and were either 250-300 or 300-355 ym in size. One population consisted of well-formed octahedral crystals that had been partially dissolved and regrown at 36 "C. The second population was made up of spherical crystals that were obtained by stirring vigorously an equilibrated, 40% solids slurry of crystals for 19 h at 29 "C. Both populations were rinsed quickly with acetone upon filtration from their respective solutions and allowed to air dry before sieving into the desired fractions. A charge of between 0.27 and 4.35 g of these large crystals was used to seed 150 mL of the supersaturated solution held in the batch crystallizer. A calibrated refractometer was used to measure the concentration of potassium alum in aqueous solutions. A precision of i0.05 g of alum/100 g of solution could be obtained with this instrument. A TA I1 Coulter Counter, fitted with a 200-pm aperature tube, was employed to monitor crystal size distribution. This aperature allowed detection of crystals having equivalent diameters between 4 and 80 ym. Particle counts in the lowest channel, however, were confounded by electrical noise and were unreliable. Because the Coulter Counter relates crystal size to the diameter of a sphere with an equivalent volume, a correction must be made in the data to account for the octahedral symmetry of potassium alum. The edge length L between (111)faces of the octahedron was determined by equating volumes of a sphere 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 67
%n
0
40
20
60
m 0
L
Figure 1. Typical number density transients (experiment 19).
having a diameter L, to the volume of an octahedron to obtain L = 1.036L,. After the solution had been cooled to the desired temperature, a charge of seed crystals from one of the populations described earlier was added. Although the seed crystals were at room temperature, adding them to the solution did not cause a measurable change in solution temperature. Two estimates of the number of nuclei generated by initial breeding were obtained from the measurements in this study: the total number of crystals counted NT and the number of crystals, NL,having size greater than the mean. The number of crystals having a size greater than the mean was evaluated only when initial breeding produced a maximum in the nuclei size distribution of an experiment, while NT was determined in all experiments. All crystal counts were corrected by subtracting the average of three background readings made prior to seeding. The surface area of the seed crystals A. may be calculated from the relationship A0 = kamo/kvPcL~c
(1)
where k, and k, are dimensionless area and volume shape factors, respectively. The density of potassium alum crystals is 1.76 g/cm3. For the octahedral crystals, k, is 2(3)1/2and k, is 2II2/3, while the values are T and u/6 for the spherical crystals.
Nuclei Formed by Initial Breeding Seeding an aqueous potassium alum solution induces instantaneous formation of microscopic crystals for initial breeding. Measurements of this crystal population 1min after seeding showed that these crystals populated a distribution in the range from 0 to 30 gm. Some of the measured crystal size distributions exhibited a mode while others did not. The appearance of a mode depended upon the conditions of the system a t seeding. Figure 1illustrates the form of population density data obtained during an experiment that gave a mode in the distribution. Analysis of these size distributions showed that the total number of crystals was nearly constant during the experiment. This is a clear indication that nuclei resulting from other secondary mechanisms, such as contact nucleation, were unimportant at these experimental conditions. It may have been that the agitation was insufficient or the magma density too low to cause
10
2 0 3 0 L b d
40
50
Figure 2. Typical transient distributions without modes (experiment 46).
significant nucleation by crystal contacts with one another or the crystallizer internals. An alternative explanation is that contact nucleation occurred, but the number of nuclei resulting from crystal contacts was much smaller than the total number of nuclei or that the nuclei from crystal contacts did not enter the monitored size interval during the experiment. Figure 2 presents a typical set of data for a distribution without a mode. Variables Affecting Numbers of Nuclei If the relative supersaturation in the solution prior to the addition of seed crystals was greater than 0.022, no impurity (defined later) was added to the solution, and seed crystals with sufficient surface area were added to the solution, then the population distribution exhibited the mode as illustrated by Figure 1. The form of the seed crystals affected the conditions a t which a mode in the nuclei population was observed; the well-formed octahedral seed crystals exhibited a population with a maximum when the surface area of the crystals was greater than 40 cm2, but the addition of 135 cm2of rounded seed crystals to a solution with a relative supersaturation of 0.046 produced only a small peak in the population. In the following discussion, these qualitative results are supported by more detailed analysis of the variables affecting initial breeding. Figure 3 presents data that show seed crystal surface area affected the number of nuclei formed by initial breeding. These data were obtained at different supersaturations, with and without the impurity Bismarck Brown R in the solution. Only octahedral seed crystals were used in these experiments. The number of nuclei obtained from initial breeding was a linear function of seed crystal surface area, and it is clear that Bismarck Brown R had an effect on the number of nuclei. The data in Figure 3 indicate that relative supersaturation also affected the number of nuclei. However, results from a subsequent series of experiments showed this effect to result from the influence of supersaturation on growth rates of these nuclei. In this later series of experiments, a charge of octahedral seed crystals having a surface area of 165 cm2 was added to supersaturated solutions. Both NT and NLwere determined for these experiments and are shown in Figure 4. Although the total number of crystals produced by initial breeding appears to be a function of supersaturation, the number greater than the mean remained constant. It is thought that these results show the
68
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 500
24m 1
L= Oppm:
2
C =Oppm: s,=O.o2
1
So = 0.06
C = BB
Oppm:
s =-0.09
i
400 16,000. 300
2
NT
(No./ml)
(No./ml)
3 4
200
100
0
A,
0
(CI&
Figure 3. Effect of seed crystal surface area on the number of crystals produced by initial breeding.
6,000 L
4,000.
2,000 .
-0
400 A,
600
800
2
(Cm
Figure 5. Effect of surface area on the number of crystals generated in undersaturated solutions.
greater than the number formed when seeding an identically supersaturated solution with the same surface area of rounded crystals. This difference may have resulted from the unequal conditions a t which the two seed types were prepared; the octahedral crystals were grown a t 36 "C while the spherical crystals were formed at 29 "C.
NT or +.,N (No./ml)
200
0.02
0.04
0.06
0.06
0.10
SO
Figure 4. Effect of supersaturation on the number of crystals produced by initial breeding.
effect of extent of growth on the number of crystals counted by the particle size analyzer. Lower supersaturation or the presence of the growth inhibitor Bismarck Brown R (Girolami and Rousseau, 1985a) means that fewer small crystals grow to a size that can be counted during the experiment. This reasoning leads to the conclusion that supersaturation does not affect initial breeding directly. Figure 5, which shows crystal counts obtained in undersaturated solutions, further supports this point; the small numbers of nuclei observed in the experiments on undersaturated solutions are the result of dissolution of crystah during the interval between seeding and counting. These observations are consistent with the initial breeding mechanism. Recall that initial breeding does not require formation of a solid phase, a process that would be affected by both supersaturation and impurities; it is the growth of microscopic crystalline matter formed prior to immersion of seed crystals in supersaturated solutions. It is reasonable, then, for neither supersaturation nor impurities to affect the number of new crystals generated but to influence the rate a t which they develop. The trends observed when seeding with octahedral crystals were similar to those resulting from seeding with rounded crystals. However, the actual number of nuclei obtained from the octahedral crystals was a factor of 3
Characteristics of Nuclei from Initial Breeding The nature of the experiments performed in this study made it impossible to measure immediately and directly the size distribution of nuclei produced from initial breeding. As can be observed in all data presented to this point, there was a 1-min interval between seeding and the first set of measurements. As a result, a method was developed to allow estimation of the characteristics of the original size distribution from those that were measured. One possibility for determination of both the number and the characteristics of nuclei produced by initial breeding was to evaluate both of these after immersion of seed crystals in a nonsolvent. It should be remembered, however, that initial breeding involves dislodging microscopic crystals from the surface of larger seed crystals. As the properties of the solution in which the seed crystals are immersed could affect the number of potential nuclei that are actually dislodged, it was considered important that the experiments be performed in aqueous solutions of the solute. Moreover, the use of a nonsolvent would present difficulties with the use of the particle size analyzer as it requires the use of electrically conductive solutions. The effects of growth on the form of successivemeasured transient distributions were characterized by determinations of mean size L, size variance uL2, and the moment coefficient of kurtosis a4. The latter two quantities are defined as
where Niis the number of crystals in the channel having size Li. The variance is a measure of the spread or width of a distribution, while the moment coefficient of kurtosis is a measure of the height or peakedness of a distribution. Figure 1 illustrates the effect of growth rate dispersion on a typical population of crystals generated by initial
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 69
I
o
I
/
I
/ " II
l5
1 -O.Ol&,
a4f = 7.35 10
lot
(rm2)la,
I 0
10
20
if 10
20
30
40
Figure 6. Effect of growth on the size variance of crystals produced by initial breeding.
breeding. As growth proceeds, the distribution height decreases and width increases. In the following analysis, the size variance U L ~and moment coefficient of kurtosis u4 of each distribution were obtained from eq 2 and 3, respectively. These parameters were correlated with the mean size of the distribution L, which is a measure of the extent of growth of the crystal population. The resulting correlations and a correlation of growth kinetics can then be used to estimate the characteristics of the distribution of initial nuclei. Using the subscript f to indicate properties of the crystal distribution measured after a growth period, values of uLf2 and u4f were determined from eq 2 and 3 using data from several batch experiments. These results are plotted as functions of Ef, the mean of the distribution a t the time of measurement, in Figures 6 and 7. The power-law expression ULf2
= pLfr
(4)
was fit to the size variance data by using linear regression of the data to the logarithmic form of eq 4 to obtain the correlation uLf2 = 0.43Lf1.75
(5)
where uLf2 is in pm2 and Lf is in pm. This function meets the basic requirements, deduced from the observed form of the distribution: uLf2 goes to 0 as goes to 0, and uLf2 increases positively as Ef increases. The 95% confidence limits for the order r are f0.12, while those for the coefficient p are from 0.31 to 0.59. The correlation coefficient is 0.968. Correlation of the values of u4f with Lf (Figure 7) gave U4f
= 7.35 x 10-0~016~f
40
Figure 7. Effect of growth on the moment coefficient of kurtosis of crystals produced by initial breeding.
1 0
30
(pm)
(6)
Because the difference term in the defining equation for u4f is raised to the fourth power, slight errors in the crystal counts as the size becomes large causes enormous variability in u4. This variability was reduced significantly by excluding crystal counts in the size ranges that made up less than 0.1-0.270 of the total number of crystals. Nevertheless, the uncertainty in udf remains great, and the correlation coefficient for eq 6 is low (0.867). The 95% confidence limits on the coefficient are 6.56-8.24, and for
the exponent, they are h0.003. Figures 6 and 7 include data from 13 experiments. The crystals examined in these experiments were generated by initial breeding resulting from seeding solutions having relative supersaturations ranging 0.015-0.078. Within the accuracy of the data, there was no dependence of the final distribution variance uLf2on supersaturation or crystal growth rate. The original size variance uh2 and moment coefficient of kurtosis can be estimated from the correlations given by eq 5 and 6, if the original mean size is known. An estimate of this value was obtained from a correlation of centroid displacement obtained in a separate study (Girolami and Rousseau, 198513). The rate of centroid displacement is defined as
G = AL/t (7) The correlation obtained in the referenced study is 108G = 4 0 6 ~ ' . ~ ~ (8) where G is m/s. When eq 7 and 8 are combined and the result is rearranged, the following equation can be obtained:
Lo = E , - 4.06~;'~~At
(9) In this expression, so is substituted for s, Ll is the mean size of the first distinct and usable transient crystal size distribution measured in an experiment, and At is the elapsed time between seeding and the initial measurements. Using the methods of the preceding paragraph, the average size of the nuclei formed by initial breeding in the 13 experiments of this study was determined to be 5 f 1 pm. Moreover, the standard deviation of the initial size distribution was evaluated as 7.3 pm2, and the moment coefficient of kurtosis of these nuclei was 6.1. The original distribution of crystals formed by initial breeding appears to be twice as peaked as a normal distribution, which has a value of u4 equal to 3 (Randolph and Larson, 1971). Therefore, 95% of the nuclei were within one standard deviation of Lo,whereas this fraction would be spread over two standard deviations of a normal distribution. These characteristics were not affected by the supersaturation in the solution to which seed crystals were added. Conclusions The number of crystals formed by initial breeding associated with seeding supersaturated potassium alum solutions was a linear function of seed crystal surface area.
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 70-75
70
The presence of a growth inhibitor affected the number of nuclei formed, but this effect may be due to the impact of a diminished nuclei growth rate on the number of crystals populating the observable size range. Similar results were obtained in examining the effect of supersaturation on the number of crystals produced by initial breeding; the number of nuclei appeared to be reduced as supersaturation was reduced but an examination of the number of crystals on the leading edge (larger than the mean) of the distribution showed that there was no change in the number of these as supersaturation was varied. The variance and moment coefficients of kurtosis of the distribution of crystals formed by initial breeding were measured as the crystals grew, and the results were correlated with extent of growth. These correlations and that for growth rate were used to calculate characteristics of the original distribution of crystals formed by initial breeding. The mean size of these crystals was estimated to be 5 pm, and 95% of the crystals was within f2.7 pm of the mean size. These results show that the distribution of crystals from initial breeding is very narrow and that the broadening of the distribution as the crystals grow is due to growth rate dispersion. Nomenclature a4 = moment coefficient of kurtosis A,, = surface area of seed crystals G = growth rate (eq 9) k , = area shape factor k , = volume shape factor
4 = crystal size Lf = mean size of crystals after growth &i = average size of channel i Lo = mean size of crystals produced by initial breeding L,, = arithmetic average size of the seed crystals used in an experiment ma = mass of seed crystals n = population density N ; = number of crystals counted in channel i N L = number of crystals having size greater than the mean NT = total number of crystals s = relative supersaturation, (z - x * ) / x * so = relative supersaturation at time zero z = solute mole fraction, g of solute/g of solution x* = solute mole fraction at equilibrium, g of solute/g of solution Greek Letters
d
= change in mean size due to growth
p, =
crystal density variance of the population density about the mean size
uL2 =
Literature Cited Girolami, M.; Rousseau, R. W. J . Cryst. Growth 1085, 7 7 , 220. Girolami, M.; Rousseau, R. W. AIChE J . , in press. Girolami, M. W. M. S. Thesis, North Carolina State University, Raleigh, NC, 1984.
Randolph, A. D.; Larson, M. A. "Theory of Particulate Processes": Academic Press: New York, 1971. Strickland-Constable, R. F. AIChESymp. Ser. No. 727 1072, 6 8 , 1.
Received for review August 9, 1984 Revised manuscript received February 25, 1985 Accepted April 30, 1985
Reversible Cyctic Behavior of Monovariant Ion-Exchange Systems Having Equilibrium Isotherms of Complex Shape Douglas D. Frey Chemical Engineering Lbpartment, Yale University, New Haven, Connecticut 06520
An analysis is given for an ion-exchange process operating in a cyclic steady state for which there is no net difference in available energy between the influent and effluent streams. The conditions apply to fixed-bed ion-exchange columns operating with reverse-flow regeneration and 100% regenerant efficiency. General equations are developed from which composition profiles and the exchanger utilization can be determined. The equations are used to investigate the performance of an ion-exchange column containing a mixture of exchangers.
Industrial ion-exchange processes generally involve saturation and regeneration cycles in which a cyclic steady state (i.e., perfectly periodic behavior) is achieved. In a typical process, the saturating and regenerating streams are fed alternately to a fixed bed of ion-exchange particles, and two product streams are produced. Most ion-exchange processes also involve additional steps which might include backwashing and/or rinsing. In this study, these additional steps are assumed not to affect the performance of the bed. Accurate calculations of the cyclic performance of ionexchange columns should take into account the variance of the exchange process, the influence of mass-transfer inefficiencies, the existence of nonuniform concentration profiles at the beginning of each half-cycle, and the detailed shape of the equilibrium isotherm. Variance is defined as the minimum number of concentration variables which can 0196-4305/86/1125-0070$01.50/0
be used to functionally express the concentrations of all other species (Golden, 1973;Vermeulen et al., 1984). Ion exchange involving two exchangeable ions is monovariant, and calculations for beds having uniform presaturation, local equilibrium between phases, no axial dispersion, and any equilibrium relation are relatively easy to perform (Vermeulen et ai., 1984). More difficult are calculations of monovariant cyclic behavior involving mass-transfer inefficiencies even when simple equilibrium relations are assumed (Pancharatnam, 1967). The objective of this paper is to develop analytical expressions that describe the cyclic performance of monovariant ion-exchange processes. These calculations assume local-equilibrium behavior, 100% regenerant efficiency, and reverse-flow regeneration and apply to equilibrium isotherms of any shape. Previous investigations of this case have apparently involved numerical calculations (Klein 0 1985 American Chemical Society
