Statistics of multiple particle breakage accounting for particle shape

Hill, Priscilla J.

doi:10.1002/aic.10091

Cited by 19 publications

(14 citation statements)

References 29 publications

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“…In the currently available literature, little is reported concerning morphological population balance equations for complex phenomena such as breakage and agglomeration. − In the framework of the development and characterization of the 3-stage process, a mathematical model for the breakage, that could consider also the morphology of the particles, had to be created. The population balance equation for a continuous rotor-stator wet mill, which is described as a plug-flow tubular apparatus (i.e., under conditions of perfectly segregated flow), operated at steady-state, under the assumption of constant temperature and supersaturation, reads as follows: Note that τ represents the residence time along the tubular apparatus and that no material balance for the solute is needed (see eq ) because breakage does not impact the concentration in solution.…”

Section: Mathematical Modelmentioning

confidence: 99%

Manipulation of Particle Morphology by Crystallization, Milling, and Heating Cycles—A Mathematical Modeling Approach

Salvatori

Mazzotti

2017

Ind. Eng. Chem. Res.

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A new process consisting of a combination of crystallization, milling, and dissolution stages is presented. At first, a mathematical model for the different unit operations, as well as their coupling, is developed, based on the basic phenomena characterizing the evolution of the ensemble of particles in terms of both size and shape. Because of the broad design space and the plethora of interparticle phenomena, the mathematical model stands out as a vital tool for an in silico assessment of process feasibility. Subsequently, computer simulations can be exploited for a thorough understanding of the effect of the different operating conditions. The results collected are discussed considering the particle size and shape distribution in its entirety and by means of its properties, such as the average sizes and the variance. A final assessment is performed by comparing the combined process with a single cooling crystallization stage, both with and without a milling of the final products.

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Section: Mathematical Modelmentioning

confidence: 99%

Manipulation of Particle Morphology by Crystallization, Milling, and Heating Cycles—A Mathematical Modeling Approach

Salvatori

Mazzotti

2017

Ind. Eng. Chem. Res.

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“…Parameter k max χ τ {r l } Value 400 2 −52 3/5 10 −9 {0, 10 −10 , 10 −8 , 10 −6 , 10 −4 , 10 −2 , 0.1, 0.5, 1} For all examples we use the a daughter droplet distribution which is based on the purely statistical daughter droplet distribution function of Hill and Ng [26]. Recall that N (v ) is the average number a particle of size v splits.…”

Section: Numerical Resultsmentioning

confidence: 99%

A numerical comparison of the method of moments for the population balance equation

Müller

Klar

Schneider

2019

Mathematics and Computers in Simulation

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We investigate the application of the method of moments approach for the one-dimensional population balance equation. We consider different types of moment closures, namely polynomial (P N ) closures, maximum entropy (M N ) closures and the quadrature method of moments QM OM N . Realizability issues and implementation details are discussed. The numerical examples range from spatially homogeneous cases to a population balance equation coupled with fluid dynamic equations for a lid-driven cavity test case. A detailed numerical discussion of accuracy, order of the moment method and computational time is given.Keywords: population balance, maximum entropy, quadrature method of moments 2010 MSC: 35L40, 45K05, 35R09 IntroductionPopulation balance equations are widely used in engineering applications, including aerosol physics, high shear granulation, pharmaceutical industries, polymerization and emulsion processes, evaporation and condensation processes in bubble column reactors, bioreactors, turbulent flame reactors and many others, see [13,19,24,24,[43][44][45] and references therein. These polydisperse processes are characterized by two phases: one of them is the continuous and the second is a dispersed phase consisting of particles. The particles can take different forms like crystals, drops or bubbles with several possible properties such as volume, chemical composition, porosity and enthalpy. In this work only the volume is considered. The dynamic evolution of the particle number distribution which is described by the population balance equation (PBE) of the dispersed phase depends not only on the particle-particle interactions, but also on the continuous phase, due to interaction of these particles with the continuous flow field in which they are dispersed. These interactions usually result in the common mechanisms aggregation, breakage, condensation, growth and nucleation. In our work we concentrate on binary aggregation, namely the Smoluchowski aggregation [42] and multiple breakages, since binary breakage is not sufficient for some of these applications. Binary aggregation is the process of merging two particles to a larger particle, whereas in a breakage process, a particle breaks into several smaller fragments. Often included in a typical PBE are spatial transport terms, i.e. advection and diffusion terms. We use the simplest case and assume that the mean particle velocity is the same as those of the fluid. The resulting population balance equations range from integro-differential to partial integro-differential equations of hyperbolic or parabolic type in phase space, in addition to the differential terms for advection and diffusion in the physical space.

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“…For instance, one could consider the proportion of round pebbles versus angular fragments, or a continuous shape distribution, by characterizing the distribution of curvatures along the pebbles contour as proposed by Durian et al [2007]. Following Hill [2004], the shape distribution could then be integrated into the Population Balance Equation framework by developing joint probability distribution accounting for both the shape and the size.…”

Section: Discussionmentioning

confidence: 99%

A new framework for modeling sediment fining during transport with fragmentation and abrasion

Bouteiller

Naaim-Bouvet²,

Mathys³

et al. 2011

J. Geophys. Res.

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[1] A new framework for modeling sediment fining during transport is proposed. The model is based on a physical description of both abrasion and fragmentation processes. It attempts to describe and explain the changes in the grain-size distribution of sediments during their transport in a water flow. Abrasion and fragmentation are modeled through the use of breakage frequency functions and daughter size distributions. These functions are determined experimentally for the specific case study which is presented. Comparison between the predictions from the model and experimental data from the case study show that (1) both abrasion and fragmentation are involved in the degradation, (2) fragmentation efficiency decreases during the experiment, and (3) the shape of the pebbles provides feedback between abrasion and fragmentation. This type of model is therefore appropriate for explaining degradation mechanisms, which is promising for a future application to rivers.Citation: Le Bouteiller, C., F. Naaim-Bouvet, N. Mathys, and J. Lavé (2011), A new framework for modeling sediment fining during transport with fragmentation and abrasion,

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Statistics of multiple particle breakage accounting for particle shape

Abstract: A method is presented for developing multiple particle breakage distribution functions that include the shape factor, as well as the particle size.

Cited by 19 publications

References 29 publications

Manipulation of Particle Morphology by Crystallization, Milling, and Heating Cycles—A Mathematical Modeling Approach

Manipulation of Particle Morphology by Crystallization, Milling, and Heating Cycles—A Mathematical Modeling Approach

A numerical comparison of the method of moments for the population balance equation

A new framework for modeling sediment fining during transport with fragmentation and abrasion

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