A New Crystal Nucleation Theory for Continuous Precipitation of Silver Halides (CSTR)
Ingo H. Leubner, Journal of Imaging Science and Technology 42(4):355-364 (1998)
A new theory of crystallization in the continuous stirred tank reactor (CSTR, or mixed-suspension, mixed-product-removal, MSMPR) system was developed, which is based on a dynamic balance between growth and nucleation.
The model is based on non-seeded systems with homogeneous nucleation, diffusion controlled growth, and the nucleation model previously derived for such systems in controlled double-jet batch precipitations. No assumptions of size-dependent growth are needed.
The model predicts the correlation between the average crystal size and the residence time, solubility, and temperature of the reaction system. It allowed to determine useful factors that are experimentally hard to determine like L/Lc the ratio of average to critical crystal size, the supersaturation ratio, S*, the maximum growth rate, Gm, the ratio of nucleation to growth, Rn/RI, and the size of the nascent nuclei, Ln.
Results of continuous precipitations of silver chloride were chosen to support predictions of the model. The model predicts that the average crystal size is independent of reactant addition rate and suspension density, which was experimentally confirmed. However, the experiments indicate that the width of the crystal size distribution increases with suspension density and is independent of reactant addition rate.
The model predicts that the average crystal size and residence time are linearly related when the average crystal size is significantly larger than a certain limiting size, which is given by the equation. At smaller crystal sizes, the average crystal size is larger than predicted by the linear part of the correlation. These predictions are confirmed by the experimental results. The model also predicts that when the residence time approaches zero, the average crystal size approaches a positive limiting value.
The condition where the residence time approaches zero (0.205 m m) is equal to that predicted for nucleation in plug-flow reactors, and thus predicts a lower limit of average crystal size for these systems.
The ratio or the fraction of the input stream used for nucleation, Rn, to the fraction used for crystal growth Ri (Rn/Ri) was calculated from the experimental results and varied from 4.79 at a residence time of 0.5 min to 0.12 at 5.0 min residence time. The size of the nascent (newly formed) crystals at steady state, Ln, was determined to be in the range of 0.194-0.221 m m for the residence time range from 0.5 to 5.0 min. The ratio of average to critical crystal size, L/Lc, was determined to 5.73*103 (batch precipitation: 1.02-1.09), the supersaturation ratio, S*, to 12.2 (0.54, L = 0.5 m m), the supersaturation to 8.2*10-8 (12.7*10-9, L = 0.5 m m), and the maximum growth rate, Gm, to 4.68 A/s (1.20-4.25). The data in the brackets are for equivalent batch precipitations.
The results of the present model and experiments may be used to modify the Randolph-Larson theory to model the experimental crystal size distributions more accurately. While the present model was developed for homogeneous nucleation under diffusion limited growth conditions and unseeded systems it may be modified to model seeded systems, systems containing ripening agents or growth restrainers, and systems where growth and nucleation are kinetically controlled.