Iterative parameter estimation for extraction of crystallization kinetics of potassium chloride from batch experiments (original) (raw)
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A simultaneous determination of nucleation and growth rates from batch spontaneous precipitation
Chemical Engineering Science, 1999
A review of the literature shows that many methods for determining the nucleation and growth rates from batch precipitation experiments are inaccurate. Therefore, a new method has been developed to derive these two rates based on the simultaneous measurement of the supersaturation and the crystal size distribution during a batch spontaneous precipitation. All experimental data are used without any previous numerical treatment and the differential kinetic equations are replaced by integral ones to avoid numeric derivation. The kinetic parameters of the two postulated laws of nucleation and crystal growth rates in kinetic equations are separately determined by non-linear optimization. This method was applied to barium sulphate precipitation. Experiments were performed in a 20 l stirred reactor. Effects of reagent adding order and stirring speed were noticed. The influence of initial supersaturation was determined. The nucleation rate was found to obey an exponential law with a primary heterogeneous dominating process. The linear rate of crystal growth follows a 2.1 power law and the process seems to be limited by a surface integration step. The kinetic results were compared to those previously reported.
Determination of the kinetics of secondary nucleation in batch crystallizers
Aiche Journal, 1974
used in this study, but it is obvious that other procedures could be employed. The convergence criterion used was that With this criterion, convergence occurred, at best, with 50 iterations. At worst more than 1000 iterations may be required. Computations were performed on a CDC 6400. On this machine the computing time was roughly 0.003 sediteration per component. As indicated, computing time increases linearly with the number of components. The advantages of this procedure lie in its ease of formulation and programming, its tie to thermodynamic principles, and its reliability. The formulation presented here can easily be extended to moR than three phases which makes it a candidate for use in computations where solid phases need be considered, as in cryogenic applications. Since the procedure minimizes free energy, it is possible to avoid spurious solutions to the equilibrium problem such as those pointed out in the discussion of Figure 5e. In the liquidliquid-vapor computations the starting point was that almost all the water was in a water rich liquid with a few percent of the total water being present in the two hydrocarbon rich phases. The hydrocarbons were distributed so that the vapor phase was rich in the light components and the liquid phase was rich in the heavy components. This assured convergence to the correct equilibrium conditions.
A new policy for the estimation of the course of supersaturation in batch crystallization
The Canadian Journal of Chemical Engineering, 1996
A continuing effort is being made in the development of on-line sensors for the monitoring and control of crystallization processes. Applied on-line calorimetry provides an estimate of the instantaneous heat release due to crystallization advancement. Two approaches to the implementation of such "soft sensors" are described and difficulties arising in the case of on-line multipurpose non-isothermal applications are outlined. Off-line estimates of concentration and supersaturation profiles during typical batch evaporative crystallizations are reported. Particular attention is paid to the detection of second, late nucleation bursts occurring in cases of poor crystal growth and rapid temperature decay. Des recherches sont continuellement menees pour perfectionner les capteurs en ligne pour la surveillance et la regulation des procedes de cristallisation. La calorirnetrie en ligne appliquee foumit une estimation du degagement de chaleur instantane dii a la progression de la cristallisation. Nous decrivons deux methodes d'implantation de ces (capteurs doux)) et soulignons les diffcultes qui surviennent dans les applications non isothermes a objectifs multiples en ligne. Nous donnons des estimations hors-ligne des profils de concentration et de sursaturation durant les cristallisations avec evaporation discontinues typiques. Une attention particuliere est accordee a la detection des eclatements de nucleation tardives et secondaires qui surviennent en cas d'une faible croissance des cristaux et d'une decroissance rapide des temperatures.
Kinetics of two-step nucleation of crystals
The Journal of Chemical Physics, 2005
When the nucleation of a stable crystalline phase directly in a supersaturated old phase is greatly retarded, the crystal nuclei might nucleate within faster-forming particles of an intermediate phase.
The Chemical Engineering Journal and the Biochemical Engineering Journal, 1994
A comparison of methods proposed for the determination of crystallization kinetics from the crystal size distribution in continuous mixed-suspension, mixed-product-removal (MSMPR) crystallizers is presented for several systems exhibiting size-dependent crystal growth rates reported in the literature. Wide variations in inferred kinetic parameters are obtained depending on the analytical method adopted, leading to uncertainty in their utility. Direct fitting of differential population density data using exponential sizedependent growth models, however, gives an improved estimation of growth rates over the whole size range and leads to higher zero-size crystal growth and nucleation rates in comparison with other models tested.
Journal of Alloys and Compounds, 1997
A procedure has been developed for analyzing the evolution with time of the volume fraction crystallized and for calculating the kinetic parameters at non-isothermal reactions in materials involving formation and growth of nuclei. By means of this method, and considering the assumptions of extended volume and random nucleation, a general expression of the fraction crystallized has been obtained, as a function of time. In the quoted expression one considers that the crystal growth rate is anisotropic. In addition, the particular case of isotropic growth rate has been studied, and the obtained equation has been integrated for the important case of nucleation frequency and growth rate independent of time, resulting an expression that may be taken as a detailed specific case of the Johnson-Mehl-Avrami relation. The kinetic parameters have been deduced, fitting a theoretical function, obtained from the JMA model to the experimental data, temperature and volume fraction crystallized. A least-squares method has been used, bearing in mind the fact that, in most non-isothermal processes, the reaction rate constant exhibits an Arrhenian temperature dependence. Finally, the theoretical derivations of the kinetic parameters have been applied to the experimental data corresponding to a set of glassy alloys, quoted in the literature, thus obtaining mean values that agree very satisfactorily with the bibliographical data. This fact shows the reliability of the developed theoretical method.
Crystal growth in carbon sequestration
Acta Crystallographica Section A Foundations of Crystallography, 2011
Known kinetic models are relatively successful in predicting steady state nucleation. Transient nucleation has been also modeled in restricted situations (constant concentration and temperature) showing the origin of some time-dependent effects. Nevertheless, current setups for crystallization from solution, especially those based in counterdiffusion methods, display a higher level of time-dependent behavior: concentration of two or more reactants do change continuously along the crystallization volume and over time and, in some setups, it is also possible to have continuous, long term changes in other parameters affecting nucleation like pH or temperature. This transient behavior can modify the output of the experiment in terms of the supersaturation at which nucleation happen, the nucleation flux, and can even have an active role in the selection of polymorph at nucleation time. Experimental evidence of these effects has been collected in our laboratory, mainly from protein crystal growth in counter-diffusion experiments. In these experiments, nucleation at variable supersaturation and crystallization of different polymorphs is often observed but have never been explained in terms of transient nucleation, that is the objective of this work. Equations describing the transient development of the precritical clusters population, coupled with the mass transport processes characteristic of counter-diffusion methods have been prepared and implemented in the form of a numerical simulation of counterdiffusion, with an special emphasis on nucleation. The output of these simulations shows the relative role of the time required to setup an steady precritical distribution and the characteristic rate of macroscopic mass transport, that result in a definite rate of change of the critical size during the experiment. The supersaturation at which nucleation happen is mainly controlled by this rate of change in the critical size, that is dependent on the position at the experimental volume. Obviously, this transient, automatic selection of the nucleation conditions lead to situations out equilibrium following a path in the phase space that is time-and position-dependent. Under these circumstances, kinetic selection of different polymorphs is possible.
A simplified approach to the operation of a batch crystallizer
Canadian Journal of Chemical Engineering, 1990
A general model for a seeded cooling batch crystallizer based on first principles is developed, incorporating either size-dependent or size-independent growth rates. A simple approach is proposed to obtain temperature-time trajectories at constant supersaturation or nucleation rate, without resorting to optimization techniques. Cooling curves at constant supersaturation, which lead to a substantial improvement (a smaller coefficient of variation and a larger mean size) of the terminal crystal size distribution, can be determined even in the absence of precise nucleation and growth kinetics, whereas properties related to the crystal size distribution are sensitive to such kinetics. Experimental results for the potassium sulfate-water system, potash alum-water system, and hexamethylenetetramine in ethanol, methanol, and 2-propanol/water are predicted reasonably well by the model. Extension to any type of batch crystallization with super-saturation induced by means other than cooling, such as reactive precipitation and salting out, is briefly discussed.On a mis au point un modèle général pour un cristalliseur discontinu inoculé à refroidissement basé sur les premiers principes, qui utilise des vitesses de croissance dépendant ou non de la taille. On propose une méthode simple pour obtenir des trajectoires temps-température à sursaturation ou à vitesse de nucléation constantes, sans faire appel à des techniques d'optimisation. Des courbes de refroidissement à sursaturation constante, qui entra îne une amélioration substantielle (coefficient de variation plus petit et taille moyenne plus grande) de la distribution terminale de taille des cristaux, peuvent être déterminées même en l'absence de nucléation précise et de cinétique de croissance, alors que les propriétés liées à la distribution de taille des cristaux sont sensibles à ces cinétiques. Les résultats expérimentaux pour le système sulfate de potassium-eau, alun de potasse-eau et hexaméthylènetétramine dans l'éthanol, le méthanol et le 2-propanol-eau sont prédits raisonnablement bien par le modèle. On analyse brièvement la possibilité d'étendre le modèle à n'importe quelle méthode de cristallisation discontinue oú la sursaturation est provoquée autrement que par refroidissement, comme la précipitation réactive et le désalaee.
AIChE Journal, 2012
The batch cooling solution crystallization of Ammonium Oxalate (AO) was performed in water at various constant cooling rates. Measurements of the solute concentration were obtained using in situ ATR FTIR spectroscopy, and discrete-time estimates of the Crystal Size Distribution (CSD) were computed thanks to in situ image acquisition and off-line image analysis. The crystallization process was then simulated using Population Balance Equations (PBE). Estimates of the nucleation and growth parameters were computed through model/experiments fitting. According to the cooling rate, the PBE model allowed distinguishing between two distinct crystallization regimes, separated by an "intermediate regime". The results allow assessing the respective contributions and shortcomings of solute concentration measurements and granulometric data to the identification of nucleation and growth kinetic parameters. It is shown in particular that no real separate estimation of nucleation and growth parameters can be obtained in the absence of CSD data.