Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

Bubble column reactors are used for several processes in the chemical industry, e.g. hydrogenation or oxidation reactions. At the bottom of the reactor a gaseous phase is dispersed into a continuous liquid phase with suspended particles. The resulting bubble swarm induces three-dimensional, time-dependent velocity and concentration fields, which are predicted numerically. All phases are described by an Eulerian approach. The numerical calculations of the local interfacial area density and the interphase transfer terms for mass and momentum are based on a population balance equation approach which enables an effective way to couple population balance and computational fluid dynamics. In three-phase gas-liquid-solid flow particles with diameters of 100 μm are considered as catalyst for a heterogeneous chemical reaction. The influence of particles on bubble coalescence has been investigated in order to extend an existing model for the kernel functions in the population balance equation describing bubble coalescence and dispersion. The resulting three-dimensional, time-dependent velocity and concentration fields are described and graphically presented for the hydrogenation of anthra-chinone.

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An analytical solution to the fixed pivot fragmentation population balance equation

Chemical Engineering Science ◽

10.1016/j.ces.2019.08.008 ◽

2019 ◽

Vol 208 ◽

pp. 115150

Author(s):

Andreas Håkansson

Keyword(s):

Analytical Solution ◽

Balance Equation ◽

Population Balance ◽

Population Balance Equation

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An Improved Analytical Solution of Population Balance Equation Involving Aggregation and Breakage via Fibonacci and Lucas Approximation Method

International Journal of Chemical Reactor Engineering ◽

10.1515/ijcre-2018-0096 ◽

2018 ◽

Vol 17 (5) ◽

Cited By ~ 1

Author(s):

Zehra Pınar ◽

Abhishek Dutta ◽

Mohammed Kassemi ◽

Turgut Öziş

Keyword(s):

Analytical Solution ◽

Approximation Method ◽

Balance Equation ◽

Population Balance ◽

Travelling Wave ◽

Population Balance Equation ◽

Auxiliary Equation ◽

Auxiliary Equation Method ◽

Analytical Approaches ◽

Complementary Equation

AbstractThis study presents a novel analytical solution for the Population Balance Equation (PBE) involving particulate aggregation and breakage by making use of the appropriate solution(s) of the associated complementary equation of a nonlinear PBE via Fibonacci and Lucas Approximation Method (FLAM). In a previously related study, travelling wave solutions of the complementary equation of the PBE using Auxiliary Equation Method (AEM) with sixth order nonlinearity was taken to be analogous to the description of the dynamic behavior of the particulate processes. However, in this study, the class of auxiliary equations is extended to Fibonacci and Lucas type equations with given transformations to solve the PBE. As a proof-of-concept for the novel approach, the general case when the number of particles varies with respect to time is chosen. Three cases i. e. balanced aggregation and breakage and when either aggregation or breakage can dominate are selected and solved for their corresponding analytical solution and compared with the available analytical approaches. The solution obtained using FLAM is found to be closer to the exact solution and requiring lesser parameters compared to the AEM and thereby being a more robust and reliable framework.

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An analytical solution of the population balance equation for simultaneous Brownian and shear coagulation in the continuum regime

Advanced Powder Technology ◽

10.1016/j.apt.2020.03.008 ◽

2020 ◽

Vol 31 (5) ◽

pp. 2128-2135

Author(s):

Kaiyuan Wang ◽

Suyuan Yu ◽

Wei Peng

Keyword(s):

Analytical Solution ◽

Balance Equation ◽

Population Balance ◽

Population Balance Equation ◽

The Continuum

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