Challenges and opportunities concerning numerical solutions for population balances: a critical review

Abstract

Population balance models are tools for the study of dispersed systems, such as granular materials, polymers, colloids and aerosols. They are applied with increasing frequency across a wide range of disciplines, including chemical engineering, aerosol physics, astrophysics, polymer science, pharmaceutical sciences, and mathematical biology. Population balance models are used to track particle properties and their changes due to aggregation, fragmentation, nucleation and growth, processes that directly affect the distribution of particle sizes. The population balance equation is an integro-partial differential equation whose domain is the line of positive real numbers. This poses challenges for the stability and accuracy of the numerical methods used to solve for size distribution function and in response to these challenges several different methodologies have been developed in the literature. This review provides a critical presentation of the state of the art in numerical approaches for solving these complex models with emphasis in the algorithmic details that distinguish each methodology. The review covers finite volume methods, Monte Carlo method and sectional methods; the method of moments, another important numerical methodology, is not covered in this review.

Structured population balances to support microalgae-based processes: Review of the state-of-art and perspectives analysis

“…These limitations can be attributed to the increasing computational difficulties that are introduced with increasing the number of internal variables and to the lack of quantitative experimental data about the biochemical composition of single cells. However, the increasing availability of computational resources combined with the development of efficient computational solution schemes [112] , [113] and the rapid advancement in the development of analytical methods for single-cell analysis recorded over the past few years could help overcome these limitations [16] , [114] . Analytical techniques have been continuously improved, allowing the collection of quantitative data on intracellular lipids, starch, and protein as well as the growth rate.…”

Rate of convergence and stability analysis of a modified fixed pivot technique for a fragmentation equation

“…Due to the complex nature of the fragmentation equation, few analytical solutions are available even for simple fragmentation kernels [2,[28][29][30]. For complex kernels, various numerical methods have been proposed during the last few decades including the fixed pivot technique [8], the cell average technique [9] and finite volume schemes [7,17,21,23,25]. Some other numerical methods related to multidimensional fragmentation equations can be found in [16,18,20,22].…”

Investigating Chemical Heterogeneity in Inclusion Populations: A Multivariate Population Balance Model Study in Gas-Stirred Ladles

“…Several techniques are available in the literature to solve the PBEs numerically. Some popular methods are sectional methods, − method of moments, − semianalytical homotopy methods, and the Monte Carlo method. , A detailed review of these solution methods is available in ref . The Monte Carlo method is a probabilistic method that generates random numbers to obtain statistical approximations of the entities under consideration.…”