Desupersaturation curve

Garside, J., Gibilaro, L.G. and Tavare, N.S., 1982. Evaluation of crystal growth kinetics from a desupersaturation curve using initial derivatives. Chemical Engineering Science, 37, 1625-1628. 

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). 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

Figure 4(a). Effect of seed size on nucleation at 70°C, 200 g/1 seed density. Desupersaturation curve. 

Figure 6. Desupersaturation curve at different temperatures for seed density of200g/l.

All of the direct measurement techniques are time consuming and require a significant number of experiments to obtain sufficient data to obtain kinetic parameters. This has led a number of investigators (Garside et al. 1982 Tavare and Garside 1986 Qiu and Rasmussen 1990 Witkowski et al. 1990) to look at indirect methods for the estimation of both growth and nucleation kinetics. Most of the indirect methods are based on the measurement of the solution concentration versus time in a seeded isothermal batch experiment. This is often called the desupersaturation curve since the concentration and the solubility can be used to calculate the supersaturation of the system versus time. 

A number of methods can be used to estimate growth kinetics from the desupersaturation curve obtained during batch-seeded isothermal experiments. The simplest of these methods was developed by Garside et al. (1982) and involves using the derivatives of the desupersaturation curve at time zero. It is assumed in this method that the concentration change is due only to crystal growth... 

More sophisticated techniques for the estimation of growth kinetics involve the use of the entire desupersaturation curve with parameter estimation techniques (Qiu and Rasmussen 1990). The combination of the desupersaturation curve and the crystal size distribution can be used to estimate both growth and nucleation... 

Parameter Estimation. As discussed in Section 9.4.1., model identification requires the determination of the parameters kg, g. Eg, kj, b, Ej, and j for a particular chemical system and batch crystallizer configuration using the available measurements. Gar-side et al. (1982) proposed a method using the initial first and second derivatives of the transient desupersaturation curve to calculate the growth rate parameters kg and g. The disadvantages of this method are that it is limited to cases in which the nucleation rate can be assumed to be negligible, and since numerical differentiation of experimental data is required, it is sensitive to measurement noise. 

If the initial portion of a measured desupersaturation curve is approximated by a second-order polynomial... 

The analysis of batch crystallizers normally requires the consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances), together with appropriate nucleation and growth kinetic equations. The solution of these nonlinear partial differential equations is relatively difficult. Under certain conditions, these batch conservation equations can be solved numerically by a moment technique. Several simple and useful techniques to study crystallization kinetics and CSDs are discussed. These include the thermal response technique, the desupersaturation curve technique, the cumulative CSD method, and the characterization of CSD maximum. 

Figure 5.12. A desupersaturation curve (diagramatic) c = equilibrium saturation, tn = nucleation time, ti d = induction time, t = latent period...

Values of dc/dt may be obtained from the measured desupersaturation curve. IF is a constant for a given run and the surface area, A, of the added seeds can be estimated from their total mass, M, and characteristic size, L ... 

A different approach was adopted by Garside, Gibilaro and Tavare (1982) who suggested that crystal growth rates could be evaluated from a knowledge of the first two zero-time derivatives of a desupersaturation curve which had been approximated by an nih. order polynomial. The analytical procedures adopted are fully described in the above paper, together with an example of the application of the approach to the growth of potassium sulphate crystals in a fluidized bed crystallizer. 

Figure 5. A typical desupersaturation curve for pure gypsum crystallisation (Ca concentration decreases as crystallisation progresses, 1 residence time =15 minutes)...

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