Ind. Eng. Chem. Res. 1991,30,804
804
moAm, - m,Amo - mlAmo =G,-(3) mo2At mo2At Thus, the average growth rates determined from the batch experiments are usually higher than those determined from a flow cell or single-crystal studies. Uncertainties associated with moments of population density from a seeded batch experiment may perhaps be large because of the possible discontinuity between population densities of seed and newly generated crystals and tail effects. As only moments of growth rate activity distribution for crystals grown from both a seeded and unseeded batch crystallizer can be derived from the moments of observed transient CSDs, the efficacy of this approach appears uncertain in subsequent crystallizer performance evaluation.
Gi=
Nomenclature G, = growth rate of the jth group of crystals, pmls Gj= average growth rate of many single isolated crystals,pm/s
a, = average growth rate of an ensemble, pm/s
mk = kth moment of population density with respect to size, number (pm)k/ g of suspension
Nj = number of crystals in the jth group, number C N j = total number of crystals t = time, s Registry No. Na2S04,7757-82-6.
Literature Cited Klug, G. L.; Pigford, R. L. The Probability Distribution of Growth Rates of Anhydrous Sodium Sulfate Crystals. Znd. Eng. Chem. Res. 1989,28, 1718-1725.
N.S . Tavare Department of Chemical Engineering University of Manchester Institute of Science & Technology (UMZST) P.O. Box 88 Manchester, England M60 1 QD
Response to Comments on "The Probability of Growth Rates of Anhydrous Sodium Sulfate Crystals" Sir: Professor Tavare has correctly pointed out a matter that we have taken into consideration in our experimental and mathematical analysis of crystal growth, namely, the problem of determining growth kinetics from batch experiments in which both growth and nucleation are simultaneously occurring. This was accounted for in the following ways: (a) The batch crystallizer was seeded with a mass of large, monodisperse crystals that could be "tracked" and identified independent of secondary nuclei (as illustrated in Figure 4 of the paper). (b) In the case of unseeded crystallizer experiments, a fixed number, motof the largest particles were tracked for kinetic analysis. In both cases, the kinetic analysis is limited to a fixed population of crystals of constant number, m,,, As a result, the seond term in Tavares eq 3 becomes negligible, and the growth rates of fixed populations measured from eith-er the batch crystallizer or flow cell are equivalent, Gi = G,. The derivation of eq 19 from eq 18 in our paper is, in fact, based on this assumption. The accuracy of both approach a and approach b was evaluated in our study. The total number of particles, m,, from seeded experiments was calculated for each particle size distribution sample measured and found to remain essentially constant throughout each run. The experimental particle size distributions in Figure 4 show that the two distributions (seed and secondary nuclei) can be readily distinguished, and all subsequent kinetic analysis derived from this data showed remarkably little scatter, as shown by Figures 5-7. Hence, the assumption Amo = 0 for the seeded runs was satisfactorily met. The second approach, used for the analysis of the unseeded crystallizer, has somewhat less accuracy due to the fact that crystals are growing into and out of the tracked population at the lower size limit. However, if the number of particles in the tracked population is large, the "blurring" at the lower size limit has only a negligible effect on the overall measured behavior of the crystal population. As a result, the comparison of kinetic data derived from
the unseeded batch experiments with the independent kinetic measurements of single crystals in a flow cell is found to be in good agreement, as shown by the comparison in Figure 2. Hence, we find no fundamental reason why the kinetics of crystal growth measured from bdtch vs single crystal studies should differ as suggested by Tavare, unless hydrodynamics or other environmental factors come into play. It seems that with proper consideration of experimental design and resulting mathematics, such a conclusion is, on fundamental grounds, unfounded, and contradictory to our general findings. Finally, regarding the use of moment equations, it is well-known that particle size distributions can be approximated by the use of moment equations and that the approximation improves with the number of moments used. Equation 16 in the manuscript can be used to calculate as many moments of the growth rate distribution as desired. It is important, however, that the number of moments employed for kinetic measurements reflects the accuracy of the basic experimental data (in this case, particle size distributions). The accuracy of most particle size distribution data warrants consideration of only the first several moments. Greater mathematical resolution than this normally exceeds the precision of the experimental data several-fold and in no way improves the ability to obtain kinetic data or gauge subsequent crystallizer performance. Thus, the mathematical approach used does not present inherent limitations in this regard. As the accuracy of particle size measurement techniques improves, an appropriately larger number of moments may be calculated and employed for crystal growth determination. Registry No. Na2S04,7757-82-6. Diana L.Klug E. Z. du Pont de Nemours & Company Engineering Department, Experimental Station P.O. Box 80304 Wilmington, Delaware 19880-0304
0888-58~5/91/2630-0804$02.50/0 0 1991 American Chemical Society
