Inverse Population Balance Problems: Heuristics for Aggregation/Breakage of Solid Clusters in Liquid Metals

Abstract

Populations balance models have been applied to the simultaneous aggregation and breakage of solid clusters in agitated liquid metal. Challenges for these systems lie in kinetics constants determination (kernels) from experimental clusters concentrations measurements. However, due to difficulties inherent to molten metals experimentation (severely limiting available data points and replication), and lack of literature information concerning the physical phenomena involved, more advanced inverse methods were needed. Taking into account physical characteristics, theoretical approaches using reactions networks proved the existence of at least one stable positive equilibrium state. This result allowed the construction of two distinct fitting algorithms, aimed at solving the corresponding inverse problem (kernels determination from experimental data). The first of these heuristic methods accurately identifies kernels parameters from perfect steady state data, while the second, based on transitory states, reliably leads to correct results with as few data points as 2, and measurements errors as high as 5%.

Application of Stochastic Methods to Description of the Evolution of the Dispersed Composition of Solid Phase Particles in Technological Apparatus of Periodic and Continuous Operation

“…Another example of the effectiveness of equation ( 1) in the quantitative description of real practically important processes that are accompanied by a change in the dispersed composition of the solid phase can be the use of this equation for the analysis of particle agglomeration in the processing of powdered materials. Many works are currently devoted to the study of agglomeration [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. When agglomerating particles in a flow apparatus of ideal mixing, equation ( 1) is simplified and can be formulated as follows:…”

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

“…This type of work, led by KTH in Stockholm and the University of Urbana–Champaign, was considered a benchmark in this field in the early 2000s. Later on and thanks to the increase in computational power, PBMs have been coupled with CFD to track the size distribution of NMIs. ,− In 2014, Rimbert et al developed a reduced-order multicomponent PBM to track the NMI population of titanium aluminates in liquid steel. The authors considered nucleation–growth, aggregation, and removal as the fundamental mechanisms of the process and accounted for different chemical components because of the coupling with thermodynamic databases.…”

Mathematical Modeling of Changes in the Dispersed Composition of Solid Phase Particles in Technological Apparatuses of Periodic and Continuous Action

“…Another example of the effectiveness of Equation ( 1) in the quantitative description of real practically important processes that are accompanied by a change in the dispersed composition of the solid phase is in the analysis of particle agglomeration in the processing of powdered materials. Many works are currently devoted to the study of agglomeration [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. When agglomerating particles are placed in a flow apparatus of ideal mixing, Equation ( 1) is simplified and can be formulated as follows: (9) where f 0 (m, t) is the density of the distribution function of the incoming particle flow.…”