Agglomeration Process Modeling Based on a PDE Approximation of the Safronov Agglomeration Equation

Abstract

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.

Comparison of FEM and DPB Numerical Methodologies for Dynamic Modeling of Isothermal Batch Gibbsite Crystallization

“…A FEM-based gibbsite crystallization model will be built here by consolidating two separate modules of a continuous PBE model of gibbsite crystallization that have been developed recently by Bekker and Livk. 1,11,14,21 Using particle volume as an internal coordinate, the FEM-based model is stated as 22 G L is the sizeindependent linear crystal growth rate, and β 0 is the sizeindependent agglomeration kernel, with the rate expressions as those presented in eq 2.…”

“…The formulation of the agglomeration rate is based on the Safronov agglomeration equation, which was adapted for use within the FEM-based modeling framework by Bekker and Livk. 1 To make it fit into the implicit FEM framework, the original Safronov agglomeration equation was transformed into a PDE agglomeration The mass balance equation and the equation for the total surface area of crystals, eq 2, were transformed to a form amenable to the implicit numerical scheme as…”

“…The focus of this work is on applying a novel FEM framework for modeling gibbsite crystallization 1,11 and comparing the new solution to a well-established DPB model solution. 5,12 The novel FEM numerical approach consists of two modules: (i) nucleation and growth module 11 and (ii) agglomeration module.…”

“…5,12 The novel FEM numerical approach consists of two modules: (i) nucleation and growth module 11 and (ii) agglomeration module. 1 The two modules are consolidated in this work to provide a unified framework for modeling gibbsite crystallization, with all three crystallization mechanisms taking place simultaneously. Unlike Industrial & Engineering Chemistry Research ARTICLE previous crystallization models, the current FEM-based model has been formulated using the Safronov agglomeration equation 13 and its new partial differential equation (PDE) approximation, as derived for the gibbsite crystallization by Bekker and Livk.…”

Crystallization Kinetics of Monosodium Aluminate Hydrate in Concentrated Sodium Aluminate Solutions

“…MAH is usually produced by the cooling crystallization of concentrated sodium aluminate solutions such as a postleaching solutions of bauxite . This process is quite different from the extensively investigated gibbsite precipitation, − which is usually performed from dilute sodium aluminate solutions through carbonation decomposition in the sintering process for alumina production or seeded hydrolysis in the Bayer process. Little systematic investigation of the crystallization of MAH has been carried out.…”