Particle-size control in crystallization is important for product quality control, as well as plant productivity. Since rapid crystallization traps impurities within the crystal, it is desirable to manipulate solution desupersaturation rates so as to maintain quality. However, the crystal-size distribution (CSD) is influenced by supersaturation levels through complex nucleation and growth kinetics as well as through particulate phenomena such as agglomeration and attrition. Plant productivity is severely hindered when excessive fines are produced since the solid-liquid separation system must be slowed to compensate.Significant advances have been made toward the development of dynamic CSD measurements, however, these sensors are not as yet widely accepted by industry. As a result, the conventional practice involving the control of seeding or desupersaturation profiles rather than CSD continues. The preferred CSD measurement is obtained from laser light scattering instruments, since these avoid the plugging problems associated with Coulter counters, require little calibration and are easily automated (Rawlings et al., 1993). However, combining the process and light scattering measurement model into a form suitable for model-based control is difficult.Consider a crystallization and measurement process:where for the process model The full identification of a crystallization process according to model Eqs. 1 and 2 is restricted by structural and parametric plant/model mismatch, mathematical conditioning problems and mixing pattern uncertainties. Structural mismatch is introduced in Eq. 1 by the size-axis discretation scheme used to reduce the population balance to an ODE and into Eq. 2 since the measurement itself is a subset of the process states. Parametric and structural mismatch occurs in Eq. 1 due to the difficulty in modeling, as well as identifying time-varying rate mechanisms representing nucleation, growth, agglomeration, breakage, and so on (Farrell and Tsai, 1994). Finally, the computations associated with reducing the Laser measurement to CSD are not only mathematically ill-conditioned, but also dependent on hydrodynamics, particle shape, spatial distribution of the suspended particles around the sensor and the focal plane of the laser beam (Rawlings et al., 1993;Boxman, 1992).To avoid some of the conditioning problems associated with developing a model identification and control scheme based upon Eqs. 1 and 2, Witkowski et al. (1990) proposed using the bulk crystallizer transmittance measurement in place of Eq. 2 as a CSD related secondary measurement. Transmittance can be shown to have a one-to-one relationship to the projected crystal area (second moment). Using dynamic transmittance data together with concentration measurements, Rawlings et al. (1993) discuss the successful kinetic model identification of a batch KN0,-H,O crystallizer. Using the potassium nitrate model, Miller and Rawlings (1994) formulated a model-based approach to the open-loop computation of the cooling profiles that maximizes final...

IntroductionThe onset of instability in industrial continuous crystallization processes in response to small disturbances has previously been shown to be determined primarily by the strong nonlinear dependence of nucleation rates on the relative supersaturation driving force (Buyevich et al., 1991;Jager et al., 1991). Consequently, accurate kinetic modeling of crystallization kinetics, especially nucleation, is a prerequisite to the optimal design and control of crystallization processes. Seeded batch experiments provide a convenient means of obtaining kinetic data, since batch runs permit investigations over wider ranges of supersaturations and are less time-consuming than MSMPR studies (Tavare and Garside, 1986). Using the batch data, nonlinear optimization can be used to fit kinetic parameters to the data (Eaton and Rawlings, 1990; Witkowsky et al., 1989;Stewart et al., 1992). As formulated using SimuSolv (Steiner et al., 1990), the procedure involves: (1) constructing a differentiaValgebraic process model for batch crystallization;(2) sequentially solving the model for parameter sets until a logarithmic likelihood objective function constructed from the measured response function and model-generated response and covariance is maximized. In practice, however, the identification of reaction rate models will present problems whenever limitations to the operating range of experimental apparatus d o not permit the identification of a full mechanistic model (Box and Hill, 1967).For limited sample sizes, reparameterizing the model in a close-to-linear form can improve results as well as speed convergence Watts, 1981, 1988). Ratkowsky (1985) has Correspondence concerning this article should be addressed 10 R. J . Farrell discussed applications of reparameterization to solid-phase catalytic reactions. Recently, Edgar and Wright (1992) used reparameterization to obtain preexponential and activation energy parameters for the water gas shift reaction in a fixed-bed catalytic reactor using a data set involving a narrow temperature range. Chen and Aris (1992) also used reparameterization and rescaling to obtain Arrhenius parameters. For complex models, there is little guidance regarding the proper choice for the reparameterization function. Usually, it is necessary to experiment with several transformations (Bates and Watts, 1988). In fact, suitable reparameterizations for a particular process model may change for different data sets (Ratkowsky, 1983).For batch crystallization, difficulties with CSD measurements below 4 micron can limit the information content of the data. The relative success of previous experimental studies may be explained by the relative importance of small particle measurements to the identification of a process model. Qui and Rasmunson (1 991) have successfully identified nucleation and growth kinetics for succinic acid in a batch cooling crystallizer. This system is easily identifiable since it is characterized by low nucleation rates and high growth. On the other hand, Rawlings and coworkers hav...