OPOSPM: A Reduced Population Balance Model that Works

Abstract

The presentation will focus on the fundamental mathematical framework elucidated by success stories with real applications in Chemical Engineering and Computational Fluid Dynamics. The key points include: Introduction to the Population Balance Equation: Importance and Applications, Frequent Solution Methods, SQMOM: Model Concept, PPBLab Software Implementation, From SQMOM to OPOSPM, Mathematical Foundation and Hyperbolic Analysis, OPOSPM: A Nonlinear Autocorrelation Model with real applications, OPOSPM Validation using CFD Solvers: FLUENT, OpenFoam, COMSOL and FPM (Finite Point Set Method), Implementation of OPOSPM in SIMULINK , Dynamic and Steady State Experimental Validation, and finally Conclusions.

In this presentation, I deal with a critical problem in the numerical modelling of the dispersed multiphase flow systems, which is composed of dispersed particles in a continuous phase. The numerical modelling of such systems calls for the population balance equation (PBE) which is an integro-partial differential equation with no general analytical solution. In spite of the intensive research in the last two decades, which is concerned with fast numerical solvers based on the moment methods, numerical solutions which are able to conserve particle integral properties are still expensive from CPU time point of view and suffer from losing the particle size distribution. The latter is required in industrial particulate systems where it is used to determine mechanical and physiochemical properties and for online control purposes. Therefore, I will introduce a numerical framework for solving the PBE based on the selective conservation of the total number and volume concentrations of the particulate system population. In one-dimensional particle property space, the key idea is to represent the population by a single Lagrange particle which moves along the positive real axis to conserve two low-order moments of the underlying particle distribution. The mean position of the particle is related algebraically to the total volume and number concentrations. In multi-particle property space, these particles carry information about the distribution as it evolves in space and time. This information includes averaged quantities such as total number, volume and solute concentrations, which are tracked directly through a system of coupled hyperbolic conservation laws with nonlinear source terms. In the framework of the Sectional Quadrature Method of Moments (Attarakih, M, Drumm, C., and Bart, H.-J., 2009, Solution of the population balance equation using the sectional quadrature method of moments (SQMOM). Chemical Engineering Science, 64, 742--752), this moving particle is called the “Secondary Particle” where its position coincides exactly with that of the “Primary Particle”. Therefore, the method is called the One Primary and One Secondary Particle Method (OPOSPM) where the resulting discrete hydrodynamic model for the PBE consist of two continuity equations for the total number and volume concentrations in the most general case. These equations are found exact when compared to those derived from the continuous PBE for many popular breakage, aggregation and growth functions. The accuracy of the method can be easily improved by increasing the number of primary particles in case it deviates from the exact solution. The method has been proved as an efficient engineering model and stable numerical tool for modeling physical and engineering problems that have discrete and multi-scale nature with multivariate internal particle properties.

Keywords: Population balances, OPOSPM, Mathematical modeling, Model reduction, Two- phase flow, CFD.