EXPERIMENTAL TECHNIQUES
Direct Measurement of Crystal Nucleation and Growth Rate Kinetics in a Backmixed Crystal Slurry Study of the K2S04 System A. D. Randolph’ and Krishnaswami Rajagopal University of Florida,Gainesville, Flu.
A novel method of generating a fine crystal-size distribution in the 0- to 50-micron size range was developed. This distribution was generated in a mixed suspension of larger-sized crystals, obeyed the exponential population decay law, and was measured by in situ particle counting in the crystallizer discharge using a Model B Coulter particle counter. The specific system studied was K2S04 but the method is applicable to other crystal systems. Quantitative nucleation and growth rate kinetics for the KzS04 system were obtained by fitting the experimental fine crystal-size distribution to the theoretical exponential distribution. Nucleation rate s~ I , is the nucleation rate, was found to correlate with power-law kinetics of the form I , = k & 4 ~ ~ where k ~ a,, and b are constants, MT the solids concentration, and s the supersaturation. Linear crystal growth rate, r, was found to b e proportional to supersaturation. Thus r = k,s where k, i s a constant.
CRYSTAL
SUCLCATION and growth rate kinetics are of paramount importance in determining the absolute size and, to some extent, the size distribution (CSD) of crystals produced in various types of crystallizers. I n turn, the size and size distribution of product crystals are among the most important factors in the design and economics of large scale crystallizers. Thus, there is a great need for a laboratory apparatus which can study quantitative nucleation-growth rate kinetics over a wide range of conditions and in an environment a t least resembling that of a continuous backmixed crystal suspension. With such empirical kinetics in hand, recent theoretical studies of CSD (.4begg et al., 1969; Bransom, 1960; Canning and Randolph, 1967; Randolph, 1965; Randolph and Larson, 1962; Robinson and Roberts, 1957; Saeman, 1956) provide a sound basis for using these data to “design” CSD, in a process sense, in large scale crystallizers. Considerable work has been done in the measurement of crystal growth and nucleation rates. However, measurements of growth rates are usually divorced from associated nucleation measurements (measurements are usually made in regions where nucleation is essentially absent), while nucleation rates are studied in unrealistic unstirred, unseeded systems which in no way resemble a backmixed crystal slurry. Classic examples of crystal growth rate studies are those of Cartier et al. (1959) on the itaconic and citric acid systems and Ishii (1965) on the K2S04 system. These studies are good examples of two different ways of measuring crystal growth rates, by observing the change in length of a single supported crystal, or obtaining the average growth rate calculated by the mass change in a swarm of crystals suspended in an upward-flowing liquor or equivalently the concentration change of the liquor. 1 Present address, Department of Chemical Engineering, University of Arizona, Tucson, Ariz. 85721.
These growth rate studies give insight into the molecular driving forces (supersaturation) necessary to produce crystals of a given size, but in themselves are not sufficient to predict CSD because they give no information concerning nucleation rates. Similarly, nucleation studies are normally divorced from growth rate measurements, and further, are often of a semiquantitative nature-e.g., measurements of the degree of subcooling before observing a “shower” of nuclei with a Tyndal beam (Ting and McCabe, 1934). Much theoretical work has been done in the area of homogeneous nucleation, dating back to the pioneering work of Willard Gibbs (1928), but such theories of homogeneous nucleation have little predictive value and in fact probably have little to say concerning the secondary nucleation mechanisms actually operating in a backmixed suspension. Several reviews of the theory of homogeneous nucleation, as developed and refined by Gibbs, Doring, Becker, and others, have been presented by LaMer (1952), Hill et al. (1963), Nancollas and Purdie (1964), Jackson (1964), and Hirth and Pound (1963). Basically, this theory develops the idea that the formation of a new second phase in a homogeneous original phase is accompanied by an increase in free energy due to the formation of new interfacial surface between the phases together with a decrease in free energy due to the over-all difference in free energy between the bulk phases. Thus, the free energy change is given as Differentiation of Equation 1 with respect to nucleus size, d , indicates a critical nucleus size given as
Ki K2
d, = 2/3-
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165
while the free energy change at this size can be shown t o be (3)
where CY is the supersaturation ratio. Finally, the rate at which the embryos reach the critical size-i.e., the nucleation rateis given by reaction rate theory as the product of the concentration of critical-sized embryos times the frequency with which they grow to stable size by the addition of another molecule. The former term is assumed to be a Uoltzman distribution in terms of the free energy of formation of the nuclei and thus
filially obtain quantitative nucleatioii and growth rates by fitting the CSD data to the form of CSD predicted by theory. Uransom et aZ. (1949) were the first to utilize this method in their classic work with cyclonite. TF7ithoutdefining it as such, they introduced the concept of the NSMPR c r y ~at11’izer (Randolph, 1965) as a kinetic tool. Thus, the entire crystallizer operated with mixed suspension and a mixed product removal. Under such conditions the form of CSD can be predicted from a population balance (Bransoin et al., 1949; Randolph a i d Larson, 1962; Raridolph, 1964, 1965). This population balance statement can be written for a mixed suspension (Randolph, 1964) as
AF,
Jo = -4exp - --kT
(4)
where A is t.he pre-exponential or frequeucy fact’or. The net effect of t,he supersaturation term in the deiiorninat,or of the exponent is to make the nucleation rat’e essentially a critical phenomenon in terms of supersaturation ratio--Le., there is a crit,ical supersaturation level below which essentially no nucleation occurs and above which enormous nucleation rates occur. LaMer (1952) and Kill et al. (1963) both give exaniples of Equation 4 applied to condensatmionnucleation. Even in such experiments, ext’reme care must be taken to rcinove all particles from the vapor and nucleation often occurs much lower than t,he crit,ical supersaturation ratio. Truly homogeneous nucleation phenomena appear to be crit’ical i n nature with respect to supersatmurationand perhaps this qualitative agreement with t’heory is the most that ran be said about homogeneous nucleation theory. ‘rliere is ainple experiment’al evidence t’hat the theory of homogeneous nucleation, as briefly outlined above, has little or no relationship to actual nucleation rat’ei n a continuous, backmixed crystal suspension. This fact has been brought out both by Nancollas and I’urdie (1964) in a review of crystallization kinetics and Jackson (1963) in a review of hornogeneous nucleation theory. This writer (ADR) has had direct expericnce with nucleation measurement in several tliffcrent inorganic cryst,al systems operatcd as continuous hackmixed crystdlizers, and can state that in a typical backmixed suspension t,here is no resemblance to the behavior expected on the basis of homogeneous theory, with the possible exception of syst,eins t,hat are forced into a region of “mass” nucle a t‘ion (a nuclei “shower”) by conditions which force the supersaturation to higher than normal levels-e.g., with excessive fines dissolving. Thus, bcrause of inailq~.aciesof the theory of liomogeneous nucleation, when applied to real backniixed systems, the qualitative nature of most nucleation studies, and tmheabsence of nucleation measurements in quantitat.ive crystal growth studies, t,hcrc is a great need to develop techniques for quantitative simultaneous nucleation-growth rate ineasurcinents in enviroiiments resembling R continuous backmixed crystallizer. This paper describcs ways i n d i i c h t’he siniultaneous processes of nucleation and growth can be quantit’ativelynieasurcd in backmixed crystal suspensions. Nucleation-growth rate kinetics for the KzS04 system, measured over a liinit’ed range of conditions, are presented. Theory of Method
The essence of the proposed method is to generate a seed crystal di.;triGution under controlled coiiditions, carefully measure the distribution over a suitahle finite size range, and 166
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
IJnder the following conditions, which can norinally be realized in a well-mixed laboratory crystallizer, Equation 5 can be reduced to
which has t,hc solution n(L) = no exp[-LQ,/rV]
(7)
Thc conditions are steady state, no crystals in feed liquor, negligible breakage, mixed suspension, mixed product removal, and linear growth rate independent of size. The last,, referred to as McCabc’s AL law, is a property of t’hc crystd system, but more often than not holds true i n a well-mixed suspcnsion. Larson and coworkers ( h i i n and Larson, 1968; hfurray and Larson, 1965) have utilized Equation 7 to obtain nucleation-growth rate kinetics from a variety of inorganic systems from analysis of the entire CSD produced in a AIShIPR crystallizer. Thus, a semilog plot of In n 2’s. L is made and n o is obtained from the intercept while t’he growth rate, r, is obtained froin the slope of the plot and a knowledgc of the retention time, V/Qo. The inethod described in this 1)aper utilizes the population balance theory einbodicd in Equation 7, except that such mixed product removal is allowed to occur only over a suitable sniall size range of fine crystals, and thus Equation 7 holds over only a finite size range, say 1, i n (0, I,,). This is accomplished by allowing the liquor to discharge from the backmixed crystallizer t,hrough a fine retaining screen. Thus, crystals smaller than size L,, say 50 microns, are unimpeded by tlic screen arid satisfy the assun~ptionof mixed product removal. Crystals between L , and L,, the screen size, arc somewhat impeded with respect to niixcd discharge, while crystals >I,, i n size are positivcly ret,ained in the niixed environment. Thiis, in thcory, it is possible to study iiurleation by sampling the distribution of sizes
