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Empirical size-dependent growth rate models are studied for their effect on the population density distributions from a continuous, mixed suspension, mixed product removal (CMSMPR) crystallizer. The growth rate models and/or their corresponding population density distributions are examined for continuity, convergence of moments, versatility, and their obility to fit experimental data.A new empirical size-dependent growth rate model is proposed which has properties superior to those of previous models. Experimental steady state data are presented to illustrate the application of the model to actual CMSMPR crystallization systems.It is now general1 held that crystal growth consists of three basic steps (6r:1. the diffusion of solute molecules from the bulk of the solution to the crystal-solution interface, followed by 2. a surface reaction as the solute molecules arrange themselves into the crystal lattice, and 3. the diffusion of the heat of crystallization from the crystal solution interface back into the bulk of the solution. The effect of the last step on the overall crystal growth rate has not been extensively studied. However, in most crystallization systems where the heat of crystallization is relatively low, the effect of step ( 3 ) on the overall growth process is probably small in comparison with the first two steps.In studying the crystallization of sodium chloride, Rumford and Bain (8) found the growth rate to be essentially diffusion controlled at temperatures above 50°C. and reaction controlled at temperatures lower than 50°C. Hence, in crystal growth both diffusion and surface reaction can be important or only one mechanism can be controlling.For most crystallization systems, the diffusion resistance is less than the resistance due to the surface reaction, and the growth rate is reaction controlled. These systems obey McCabe's AL law and have crystal growth rates which are independent of crystal size.However, a number of crystalline materials exhibit crystal growth rates which are a function of crystal size where u is the relative crystal-solution velocity, ri is the interfacial growth rate, and r', and B are constants.From this investigation it was concluded that crystal growth rate is independent of crystal size per se. The apparent effect of size [Equation ( l ) ] results from the larger crystals having a higher settling velocity and hence a greater relative crystal-solution velocity. Since the diffusion boundary layer and the diffusion resistance decrease as the velocity increases, the growth rate of crystals in a mixed suspension would be expected to increase with size, if the growth rate is not reaction controlled.Bennett (1 ) , however, presents data which indicate crystal growth rates inversely proportional to crystal size. Bennett believes that this effect is caused by classification taking place at boiling surfaces where the supersaturation may be considerably higher than in the bulk of the crystal suspension. He proposes that this surface classification dominates the opposite tendency of t...

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A mathematical model for prediction of the crystal size distribution from a continuous crystallizer is presented. The kinetic data used for the model were obtained from batch contact nucleation experiments with citric acid monohydrate. In these experiments, the distribution of growth rates as well as the initial size distribution were estimated. Results from the model indicate that the excess number of crystals usually present at small sizes in continuous crystallizers is due to growth rate dispersion (where crystals of the same size may have different growth rates) and not size dependent growth. SCOPEThe population balance technique developed by Randolph and Larson (1971) has been used extensively for both kinetic measurement and modeling of continuous mixed suspension, mixed product removal (MSMPR) crystallizers. When the assumptions of size independent crystal growth, all crystals with equal growth (i.e., no growth rate dispersion) and all nuclei formed at a near zero size are invoked, a semilogarithmic relation is predicted between crystal population density and size. Experimental evidence from continuous crystallizers, however, has shown that at lower crystal sizes (<20 pm) orders of magnitude more crystals are present than are predicted by this relation. Clearly any or all of the assumptions may be in error.The importance of the deviation from the model is that the semilogarithmic population density-crystal size plot is used to determine kinetic data. When the model holds and a straight line is produced, the growth rate is determined from the slope and the nucleation rate is determined from both the slope and intercept. When curvature occurs, the slope no longer has a single value and the intercept must be determined by some means of nonlinear extrapolation. In order to develop unambiguous kinetic models, it is necessary to understand the causes for the curvature.The present study made use of data taken previously (Berglund, and Larson, 1982) in contact nucleation experiments with the citric acid monohydrate-water system. These data suggest that size independent growth rate, growth rate dispersion, and initial size distribution are present. Use was then made of probability transform techniques to develop a model for a continuous MSMPR crystallizer that accounts for these phenomena. Studies with the model concentrated on the effects of growth rate distribution, initial size distribution, and the interaction between the two distributions on the product population density.

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