Extended method of moment for general population balance models including size dependent growth rate, aggregation and breakage kernels

Falola, Akinola; Borissova, Antonia; Wang, Xue Zhong

doi:10.1016/j.compchemeng.2013.04.017

Cited by 34 publications

(20 citation statements)

References 23 publications

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“…Logashenko et al [24] employed CFD and PBM but they utilized 1D CFD model. Falola et al [25] and Rane et al [26] have simulated crystallization by coupling CFD and PBM. However, part (b) has been sparsely done.…”

Section: Previous Workmentioning

confidence: 99%

CFD simulation and comparison of industrial crystallizers

et al. 2014

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Simulation of flow patterns in eleven industrial crystallizers (standard Messo, Cerny direct contact, Swenson draft tube baffled, Swenson walker, Swenson evaporative, Oslo cooling, Oslo Krystal, APV Kestner, Batch Vacuum, stirred tank with disc turbine, and stirred tank with pitched blade impeller) have been carried out using computational fluid dynamics (CFD). Population balance modelling (PBM) is coupled with CFD to obtain better results. In all the cases, the crystallizer volume was 100 liter and the power consumption per unit volume was 1 kW/m 3 . The simulation results have been presented in terms of mean velocity components, turbulent kinetic energy and dissipation rate. On the basis of the flow patterns, these eleven crystallizers have been compared for crystal size distribution (CSD). It was found that Swenson Evaporative, Krystal and Batch Vacuum provide relatively low CSD.

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Section: Previous Workmentioning

confidence: 99%

CFD simulation and comparison of industrial crystallizers

et al. 2014

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“…In general, the birth and death terms are given in terms of the distribution function, f , as well as the independent coordinates r and ξ as shown in (9.87)-(9.90). The moment form of the birth and death terms can be expressed as [14,25,75,81,82]: κ C (ξ, r; ξ , r)f (1) (ξ , r, t)dξ dξ (9.144) in which the Jacobian determinant used in (9.143) to change the integration variable is defined as d ξ = ξ 2 / ξdξ. For convenience, the B C (r, t) term can been manipulated by reversing the order of integration and introducing the size of the third particle in a coalescence event, thus ξ 3 = ξ 3 + ξ 3 .…”

Section: Moment Transformation Of the Population Balance Equationmentioning

confidence: 99%

The Population Balance Equation

Jakobsen

2014

Chemical Reactor Modeling

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The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s. The familiar application of population balance principles to the modeling of flow and mixing characteristics in vessels was formally organized by Danckwerts [18]. Certain distribution functions were then defined for the residence times of fluid elements in a process vessel. The residence time distribution function give information about the fraction of the fluid that spends a certain time in a process vessel. Himmelblau and Bishoff [37] describe how the residence time and other age distributions are defined, how they can be measured, and how they can be interpreted. This chapter focuses on the derivation and use of the general population balance equation (PBE) to describe the evolution of the fluid particle distribution due to the advection, growth, coalescence and breakage processes.The historical derivation of the general population balance equation for countable entities is outlined. Two fundamental modeling frameworks emerge formulating the early population balances, in quite the same way as the kinetic theory of dilute gases and the continuum mechanical theory were proposed deriving the governing conservation equations in fluid mechanics. A third less rigorous approach is also used formulating the population balance directly on the macroscopic averaging scales, an analogue to the multiphase mixture models. A fourth less computationally demanding approach is to formulate a moment form of the population balance equation.Considering dispersed two-phase flows, a few research groups adopted a statistical Boltzmann-type equation determining the rate of change of a suitable defined distribution function. Performing the Maxwellian averaging integrating all terms over the whole velocity space to eliminate the velocity dependence, one obtains the generic population balance equation. Rigorous closures are required for the unknown terms resulting from the averaging process. However, by employing the conventional Chapman-Enskog approximate expansion method solving the Boltzmann-type equation this theory provides rational means of understanding for the one way coupling between microscopic particle physics and the average macroscopic continuum properties. Moreover, to represent complex problem physics an optimized choice of H. A. Jakobsen, Chemical Reactor Modeling,

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“…The model presented in this work includes integro‐differential equations of moments. One of the most suitable algorithms for solving these equations is the so‐called “Extended Method of Moments” (EMOM), as it is particularly effective to resolve moments equations with size dependent growth rates . However, this method presents instability problems due to the existence of ill‐conditioned matrices, whereby Maya and Chejne recently proposed an improved version of EMOM that includes a regularization process and the usage of the Simpson's rule.…”

Section: The Modelmentioning

confidence: 99%

Novel model for the sintering of ceramics with bimodal pore size distributions: Application to the sintering of lime

Maya

Bhatia

2016

AIChE Journal

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A mathematical model for the sintering of ceramics with bimodal pore size distributions at intermediate and final stages is developed. It considers the simultaneous effects of coarsening by surface diffusion, and densification by grain boundary diffusion and lattice diffusion. This model involves population balances for the pores in different zones determined by each porosimetry peak, and is able to predict the evolution of pore size distribution function, surface area, and porosity over time. The model is experimentally validated for the sintering of lime and it is reliable in predicting the so called “initial induction period” in sintering, which is due to a decrease in intra‐aggregate porosity offset by an increase inter‐aggregate porosity. In addition, a novel methodology for determination of mechanisms based on the analysis of the pore size distribution function is proposed, and with this, it was demonstrated that lattice diffusion is the controlling mechanism in the CaO sintering. © 2016 American Institute of Chemical Engineers AIChE J, 63: 893–902, 2017

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Extended method of moment for general population balance models including size dependent growth rate, aggregation and breakage kernels

Cited by 34 publications

References 23 publications

CFD simulation and comparison of industrial crystallizers

CFD simulation and comparison of industrial crystallizers

The Population Balance Equation

Novel model for the sintering of ceramics with bimodal pore size distributions: Application to the sintering of lime

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