Crystal growth and nucleation kinetics

The population balance analysis of the idealized MSMPR crystallizer is a particularly elegant method for analysing crystal size distributions at steady state in order to determine crystal growth and nucleation kinetics. Unfortunately, the latter cannot currently be predicted a priori and must be measured, as considered in Chapter 5. Anomalies can occur in the data and their subsequent analysis, however, if the assumptions of the MSMPR crystallizer are not strictly met. 

Several authors have presented methods for the simultaneous estimation of crystal growth and nucleation kinetics from batch crystallizations. In an early study, Bransom and Dunning (1949) derived a crystal population balance to analyse batch CSD for growth and nucleation kinetics. Misra and White (1971), Ness and White (1976) and McNeil etal. (1978) applied the population balance to obtain both nucleation and crystal growth rates from the measurement of crystal size distributions during a batch experiment. In a refinement, Tavare and... 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

Crystal Growth and Nucleation Kinetics from Batch Experiments... 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

Several investigators have offered various techniques for estimating crystallization growth and nucleation parameters. Parameters such as kg, 6, and ki are the ones usually estimated. Often different results are presented for identical systems. These discrepancies are discussed by several authors (13,14). One weakness of most of these schemes is that the validity of the parameter estimates, i.e., the confidence in the estimates, is not assessed. This section discusses two of the more popular routines to evaluate kinetic parameters and introduces a method that attempts to improve the parameter inference and provide a measure of the reliability of the estimates. 

Etherton studied the growth and nucleation kinetics of gypsum crystallization from simulated stack gas liquor using a one-liter seeded mininucleator with a Mixed Suspension Mixed Product Removal (MSMPR) configuration for the fines created by the retained parent seed. The effect of pH and chemical additives on crystallization kinetics of gypsum was measured. This early fundamental study has been the basis for later CSD studies. 

Crystallization kinetics. Expressions that describe crystal growth and nucleation rates from solution. CSD. Ciystal size distribution. 

The Avrami exponent ( ) depends on nucleation type, the geometry of crystal growth and the kinetics of crystal growth (see Chapter 8). The kinetics at low degrees of conversion usually follows the Avrami equation but deviates from the linear trend in the plot... 

Similarly, several authors have presented MSMPR methods for kinetics determination from continuous crystallizer operation (Chapter 3), which have become widely adopted. In an early study, Bransom etal. (1949) anticipated Randolph and Larson (1962) and derived a crystal population balance to analyse the CSD from the steady state continuous MSMPR crystallizer for growth and nucleation kinetics. Han (1968) proposed a method of kinetics determination from the moments of the CSD from a cascade of continuous crystallizers and assessed the effect of sample position. Timm and Larson (1968) suggested the use of the extra information present in transient response data to determine kinetics, followed by Sowul and Epstein (1981), Daudey and de Jong (1984) and Jager etal. (1991). Tavare (1986) applied the j-plane analysis to the precipitation of calcium oxalate, again assuming nucleation and growth only. 

The population balance concept enables the calculation of CSD to be made from basic kinetic data of crystal growth and nucleation and the development of this has been expounded by Randolph and Larson (1988), as summarized in Chapters 2 and 3. Batch operation is, of course, inherently in the unsteady-state so the dynamic form of the equations must be used. For a well-mixed batch crystallizer in which crystal breakage and agglomeration may be neglected, application of the population balance leads to the partial differential equation (Bransom and Dunning, 1949)... 

For derivation of the relationship between crosslinking density and structure fractal dimension d for the studied epoxy polymers the model proposed in paper [90] was used, the essence of which consists in the following. Competitive processes of crystal growth and nucleation were described within the frameworks of two kinetic equations [90] ... 

In addition to induction time measurements, several other methods have been proposed for determination of bulk crystallization kinetics since they are often considered appropriate for design purposes, either growth and nucleation separately or simultaneously, from both batch and continuous crystallization. Additionally, Mullin (2001) also describes methods for single crystal growth rate determination. 

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