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.

A one-dimensional dynamic partial differential equation (PDE) agglomeration model is derived based on the continuous Safronov agglomeration equation. A regularized PDE agglomeration model, represented by a set of convectionreaction-diffusion PDEs, can be solved within a standard adaptive-mesh implicit numerical framework that does not require additional quadrature assumptions to evaluate the aggregation integral. The PDE agglomeration model is solved numerically using a general Newton's-method-based implicit Galerkin finite-element algorithm. The applied algorithm uses an automatic Gear-type time step and nonuniform adaptive-mesh strategies, which aids solution convergence. A numerical solution of the model for an agglomeration degree of 99.9% closely matches an asymptotic analytic solution of the original Safronov equation, which confirms the accuracy of the numerical procedure used. It is also shown that the number density function predicted by the new PDE agglomeration model satisfactorily agrees with the analytic solution of the Smoluchowski agglomeration equation. For small particle sizes and first and zeroth full moments, the two models give similar solutions. However, for larger particle sizes and the second full moment, the difference between the two models increases with increasing degree of agglomeration. Industrially important gibbsite agglomeration is used as a case study to demonstrate the application of the new numerical approach for agglomeration modeling.