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Differential and integral parameter estimation techniques were used in this work to estimate aluminum trihydroxide crystallization kinetics (i.e., growth, agglomeration, and source term rates) in batch and semibatch, constant-composition crystallization systems. Of the four differential techniques tested, the methods of Bramley et al. (J.

Numerical solution for a 1-D dynamic population balance crystallization model that includes gibbsite secondary nucleation and crystal growth kinetics was developed. The implemented numerical algorithm combines an implicit Galerkin formulation of the finite element method (FEM) with Newton iterations, variable Gear-type time-step and adaptive nonuniform mesh strategies. The numerical solution of the crystallization model is compared to the analytical solution derived for the case of constant gibbsite crystallization kinetics. For this case, it is shown that the numerical solution was considerably stabilized with the introduction of the artificial diffusion term and reduction of the relative error of Newton’s iterative step. Furthermore, the numerical algorithm is tested for the case of nonlinear gibbsite crystallization kinetics demonstrating its ability to deliver consistent solutions for both nucleation and crystal growth dominant cases. In each of the cases considered, the model solution, valid for an isothermal batch homogenously mixed crystallizer, predicted evolutions of the crystal size distribution (CSD) and relative supersaturation. Using the developed modeling technique, it is also shown that the initial seed loading strongly influences the shape of the product CSD, leading in some cases to multimodal CSDs.

A variation on the unreacted shrinking core model has been developed for calcination and similar non-catalytic solid-to-gas decomposition reactions in which no gaseous reactant is involved and the reaction rate decreases with increasing product gas concentration. The numerical solution of the model has been validated against an analytical solution for the isothermal case. The model parameters have been tuned using literature data for the thermal dehydration (calcination) of gibbsite to alumina over a wide range of temperatures, from 490 K to 923 K. The model results for gibbsite conversion are agreed well with the published experimental data. A reaction order with respect to water vapour concentration of n = -1 was found to give a good fit to the data and yield activation energies consistent with literature values. Predictions of the non-isothermal unreacted shrinking core model compare well with a more complex distributed model developed previously by the authors.

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Crystallization from solution is a unit operation of great economic importance in the chemical industry. The uses of this separation and purification technique range from the small-scale manufacture of fine chemicals to bulk production, such as the precipitation of aluminum trihydroxide in the Bayer process. In addition to the purity and shape of crystals, Ž . the crystal-size distribution CSD is usually a key quality consideration for a crystallization product. The product CSD is determined by the relative rates of crystal growth, agglomeration and nucleation, which are controlled by the process conditions and the chemistry of the system. A population-bal-Ž . ance PB -based model, the theory for which can be found in Ž . Randolph and Larson 1988 , provides a mathematical tool to describe the relationship between the CSD and process conditions.This research note describes a method based on a discretized PB to estimate the rates of growth, agglomeration, and nucleation from experimental batch CSD data. This socalled ''inverse'' problem is posed as a nonlinear optimization, where an objective function based on the difference between the experimental and model-predicted values is minimized with respect to the kinetic parameters. It is shown that in the case when the objective function is formulated in terms of the time derivatives of the measured variables, the original problem can be reduced into three smaller subproblems that can be rapidly solved.The method is illustrated using data from a batch aluminum trihydroxide precipitation experiment. The validity of the estimates is evaluated by comparing the experimental CSD with that obtained from the simulation using the estimated kinetics. Estimates from our method are also compared to the estimates obtained using the differential tech-Ž . nique of Bramley et al. 1996 .