Models, crystallization process kinetics

A mechanistic model for the kinetics of gas hydrate formation was proposed by Englezos et al. (1987). The model contains one adjustable parameter for each gas hydrate forming substance. The parameters for methane and ethane were determined from experimental data in a semi-batch agitated gas-liquid vessel. During a typical experiment in such a vessel one monitors the rate of methane or ethane gas consumption, the temperature and the pressure. Gas hydrate formation is a crystallization process but the fact that it occurs from a gas-liquid system under pressure makes it difficult to measure and monitor in situ the particle size and particle size distribution as well as the concentration of the methane or ethane in the water phase. 

Clearly this is a very interesting problem and of great practical relevance, very well suited to Monte Carlo simulation. At the same time, simulations of such problems have just only begun. In the context of crystal growth kinetics, models where evaporation-condensation processes compete with surface diffusion processes have occasionally been considered before . But many related processes can be envisaged which have not yet been studied at all. 

The observed transients of the crystal size distribution (CSD) of industrial crystallizers are either caused by process disturbances or by instabilities in the crystallization process itself (1 ). Due to the introduction of an on-line CSD measurement technique (2), the control of CSD s in crystallization processes comes into sight. Another requirement to reach this goal is a dynamic model for the CSD in Industrial crystallizers. The dynamic model for a continuous crystallization process consists of a nonlinear partial difference equation coupled to one or two ordinary differential equations (2..iU and is completed by a set of algebraic relations for the growth and nucleatlon kinetics. The kinetic relations are empirical and contain a number of parameters which have to be estimated from the experimental data. Simulation of the experimental data in combination with a nonlinear parameter estimation is a powerful 1 technique to determine the kinetic parameters from the experimental... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

A complete solution to the problem of modelling the crystallization process requires the determination of the space-time distribution of temperature and crystallinity. These distributions can be predicted using the thermal kinetic approach formulated above with the following assumptions ... 

Modelling of crystallization was discussed in Section 2.8. Now, we shall develop a model for superimposed polymerization and crystallization processes. This model is important for calculating temperature evolution during reactive processing, because an increase in temperature, regardless of its cause, influences the kinetics of both polymerization and crystallization. This concept is expressed by the following equation for the rate of heat output from the superimposed proceses 102,103... 

Batch crystallization process control using the first-principles and direct design approaches were discussed. The first-principles approach utilizes crystallization process models, which require the associated determination of crystallization kinetics. The optimal seed characteristics and/or supersaturation profile to obtain the desired product characteristics are then computed. The direct design approach involves feedback control of a state measurement, in this case the solution... 

In the absence of a full kinetic description of the crystallization process, the assumption that the solidification of chocolate can be modeled using effective heat capacity data as a function of temperature and cooling rate is an acceptable engi-... 

Most of the kinetics equations established at the early stage of the studies were based on elementary functions. Therefore, the form of these equations varied from one study to another. Because the establishment of these equations was not based on the theoretical model, and most of them were represented by elementary functions, the theoretical and experimental crystallization curves just partially matched each other. Later, some crystallization kinetics equations based on designed theoretical models were established as well. Recently, the crystallization process was further simulated by the computational modeling approach. 

With the advancement of online measurement techniques such as focused beam reflectance measurement (FBRM) and Fourier transform infrared (FTIR), it is now possible to obtain particle size distribution and solution concentration information rapidly through these in-situ probes. In one experiment, hundreds of data points can be generated. With proper experiment design, the model-based experimental design for crystallization is capable of obtaining high-quality crystallization kinetic data with a small number of experiments. This approach can thus save significant experimental effort and time in the development of crystallization processes. 

The first two terms on the right-hand side refer to the underlying signal and the last two to the response to the modulation. The inclusion of the expression Hfla)Ae EIRT simply refers to the kinetic model used to describe the crystallization process. 

The development and operation of industrial crystallization processes can be made significantly easier if some data on the kinetics of crystal growth are available. This information can be incorporated in process models, can be used in process and crystallizer design, and can shed light on the observed behavior of the system. 

Parameter Estimation. The kinetic parameters of the model given above that must be estimated for model identification include kg, g, Eg, kb, b, Eb,j. Parameter estimation for this type of model is quite difficult because the parameters appear nonlinearly, the nucleation rate parameters enter only in the boundary condition, and availability of accurate data is limited. Certainly a model that describes the behavior of a nonisothermally operated crystallizer is needed if the temperature is to be manipulated, but there have been only a few studies of the effect of temperature on crystallization processes (Kelt and Larson 1977 Randolph and Cise 1972 Rousseau and Woo 1980). For isothermal crystallization, the terms involving Eg and Eh are absorbed into kg and kb, and only kg, g, kb, b, and i need to be estimated. 

Theoretical and experimental studies of the role of solvent on polymorphic crystallization and phase transformations abound in the literature of the last few years and some pertinent examples are described here. For solvent-mediated transformations, the driving force is the difference in solubility between different polymorphs. An important earlier paper on the kinetics of such phase transformations [51 ] described a model featuring two kinetic processes in sohd to solid phase changes via a solution phase, namely dissolution of the metastable phase and growth of the stable one. 

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