Models, crystallization process crystallizer volume

Both growth rate dispersion and size-dependent growth affect the crystal size distribution obtained from laboratory and industrial crystallizers. They must, therefore, be taken into account when analyzing the modeling crystallization processes. More information on this topic can be found in Chapter 4 of this volume. 

Jager etal. (1992) used a dilution unit in conjunction with laser diffraction measurement equipment. The combination could only determine, however, CSD by volume while the controller required absolute values of population density. For this purpose the CSD measurements were used along with mass flow meter. They were found to be very accurate when used to calculate higher moments of CSD. For the zeroth moment, however, the calculations resulted in standard deviations of up to 20 per cent. This was anticipated because small particles amounted for less then 1 per cent of volume distribution. Physical models for process dynamics were simplified by assuming isothermal operation and class II crystallizer behaviour. The latter implies a fast growing system in which solute concentration remains constant with time and approaches saturation concentration. An isothermal operation constraint enabled the simplification of mass and energy balances into a single constraint on product flowrate. 

Overall crystallization kinetics can be described experimentally by the ratio Q i)IQ of the heat Q t) liberated between the onset of crystallization and time t to the total heat of crystallization <2- (see Fig. 15.4). To a first approximation, this ratio can be assimilated to the volume fraction a transformed into spherulites, and described by the laws for overall crystallization kinetics, for example, Ozawa s equation (see Section 15.3.2). Logically, the a T) curves are shifted to lower temperatures when the cooling rate increases. Until recently, classical calorimetry allowed us to determine the kinetic law only at low or moderate cooling rates, typically from 0.125 to 40°C/min. This raised questions when these data were used to model crystallization in processes where the cooling rates were much higher. As a result... 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

The dynamic stability of the quasi steady-state process suggests that active control of the CZ system has to account only for random disturbances to the system about its set points and for the batchwise transient caused by the decreasing melt volume. Derby and Brown (150) implemented a simple proportional-integral (PI) controller that coupled the crystal radius to a set point temperature for the heater in an effort to control the dynamic CZ model with idealized radiation. Figure 20 shows the shapes of the crystal and melt predicted without control, with purely integral control, and with... 

Modelling non-isothermal crystallization is the next important step in a quantitative description of reactive processing. This is particularly important, because crystallization determines the properties of the end product. Therefore, the development the spatial distribution of crystallinity, a, and temperature, T, with time throughout the volume of the reactive medium must be calculated. It is also noteworthy that crystallization and polymerization processes may occur simultaneously. This happens when polymerization proceeds at temperatures below the melting point of the newly formed polymer. A typical example of this phenomenon is anionic-activated polymerization of e-caprolactam, which takes place below the melting temperature of polycaproamide. 

There is another important fact about the melting process. When many ion lattices are melted, there is a 10 to 25% increase in the volume of the system (Table 5.10). This volume increase is of fundamental importance to someone who wishes to conceptualize models for ionic liquids because one is faced with an apparent contradiction. From the increase in volume, one would think that the mean distance apart of the ions in a liquid electrolyte would be greater than in its parent crystal. On the other hand, from the fact that the ions in a fused salt are slightly closer together than in the solid lattice. 

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