Numerical Benchmarking of Stabilized Finite Element Methods Applied to Population Balance Equations for Modeling Crystal Size Distributions

Resumo

Crystallization fouling is a prevalent phenomenon observed in heat exchangers, involving the accumulation of crystals on the heat transfer surfaces. It degrades equipment performance and increases operational costs in industry. Crystals are transported by diffusion from higher to lower concentration regions, with a reaction rate governing their incorporation into the crystal lattice formed on the heat exchanger walls. Its removal is promoted by the action of hydrodynamic forces and the effect of shear stresses. Mathematical models are developed to reproduce both the deposition and removal processes. This paper presents a preliminary study intended to support future research in which a new mathematical formulation for the removal term will be proposed. Removal is directly affected by crystal size, as its variation influences mechanical properties such as material strength, Young’s modulus, and hardness, as well as thermophysical properties, including thermal conductivity and specific mass. Thus, characterizing crystal size distribution within the fouling layer is crucial. Population balance equations (PBEs) are typically used to model size distribution dynamics, where different mathematical methods, such as the Method of Characteristics or Finite Volume Schemes, have been proposed for solving these equations. In this paper, we investigate the performance of several Stabilized Finite Element Methods (SFEMs) applied to PBEs for modeling the growth, aggregation, and nucleation processes involved in the crystallization fouling process. The stabilized formulations examined in this work include the Streamline Upwind Petrov-Galerkin (SUPG) method with various stabilization parameters, the Unusual Stabilized Finite Element Method (USFEM), and the Rothe’s method, employed here as a less conventional discretization technique for transient transport equations. The performance of these SFEMs is assessed based on their ability to reproduce expected physical behavior, providing insights for future model development.