A Combined Multifluid-Population Balance Model for Vertical Gas−Liquid Bubble-Driven Flows Considering Bubble Column Operating Conditions

Nayak, Ameeya Kumar; Borka, Zsolt; Patruno, L.E.; Sporleder, Federico; Dorao, Carlos Alberto; Jakobsen, Hugo A.

doi:10.1021/ie101664w

Cited by 38 publications

(38 citation statements)

References 22 publications

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“…The least-squares method has been applied to more comprehensive PB problems, e.g. the work on bubbly flows by Zhu et al [35], Nayak et al [36], Solsvik and Jakobsen [37], Borka and Jakobsen [38][39][40], and Sporleder et al [41]. According to Dorao and Jakobsen [42] the least-squares method is a promising numerical method.…”

Section: A N U S C R I P Tmentioning

confidence: 99%

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

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The behavior of dispersed systems such as gas-liquids or liquid-liquid systems depends on the characteristics of the dispersed phase. The population balance (PB) equation is encountered in numerous engineering disciplines in order to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behavior of the system. In the present study, the orthogonal collocation, Galerkin and least-squares methods have been adopted to solve a non-linear PB equation which consists of both breakage and coalescence terms. The performance of the methods is demonstrated by comparing the numerical solution results with the (manufactured) analytical solution of the problem. For the least-squares method, the choice of linearization technique influences the numerical performance, whereas the Galerkin and orthogonal collocation methods obtain the same numerical accuracy for both the Picard and Newton iteration techniques. The least-squares method suffers from lack of convergency using the Picard method. On the other hand, if the Newton method is employed, the least-squares method obtains the same accuracy as the Galerkin and orthogonal collocation methods.

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Section: A N U S C R I P Tmentioning

confidence: 99%

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

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“…In the third modeling approach the generic population balance equation is derived by Maxwellian averaging a statistical Boltzmann-type of equation in terms of number probability densities, in line with the work of Reyes [107], Lafi and Reyes [59], Carrica et al [11], Lasheras et al [60], and Jakobsen et al [19,91,96,[123][124][125]137]. In this framework, a Boltzmann type equation is Maxwellian averaged over the entire particle velocity space only thus the resulting dispersed phase governing equations are expressed explicitly in terms of the inner coordinates, the external physical space coordinates and time so that further details of the dispersed flow and chemical process behavior can be resolved and described directly and not only the moments of the dispersed phase properties.…”

Section: Four Alternative Population Balance Frameworkmentioning

confidence: 99%

“…A statistical description of multiphase flow might be developed based on an analogy to the Boltzmann theory of gases [11,39,60,63,66,91,125,135]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question.…”

Section: Statistical Microscopic Population Balance Formulationmentioning

confidence: 99%

“…They were able to derive Tambour's sectional equations from the kinetic level. This type of modeling framework has also been investigated by Jakobsen et al [19,91,96,[123][124][125]137] in order to derive a general set of balance equations for the dispersed phase populations. Starting out from a Boltzmann-type equation, a set of consistent PBE, mass, momentum, species mass and enthalpy (temperature) equations were derived for the dispersed phase and expressed in terms of a probability density function.…”

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confidence: 99%

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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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“…Bakker and Akker (1994) introduced a first-order model to describe the process of bubble formation in the bulk of a tank, and considered the bubble Reynolds number to calculate contours of constant gas holdup. For modeling the drop populations in a stirred tank (Shah and Ramkrishna, 1973;Nayak et al, 2011;Buffo et al, 2012), the Population Balance Model (PBM) divides the drop size distribution into a finite number of size intervals, in the meanwhile, PBM contains the bubble breakage and coalescence rate functions in turbulent flow (see Eq. (17)), and thus PBM can statistical formulate to describe phenomenon in which the bubble is changed rapidly due to growth and expansion in the agitated vessel.…”

Section: Introductionmentioning

confidence: 99%

Simulation of the Bubble Behaviors for Gas–Liquid Dispersion in Agitated Vessel

Liu

Sheng

Han

et al. 2017

J. Chem. Eng. Japan / JCEJ

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The present study mainly investigates bubble behaviors in an agitated vessel. Computational uid dynamics provides a method for exploring the complex uid ow in an agitated vessel. A stirred tank model with transport equation for the interfacial area concentration and force equilibrium equation for bubbles under multi-forces is created for a 43-dm 3 agitated vessel. The model considers the breakage and coalescence mechanism of the bubbles. The local gas holdup is measured by ber-optical probe. After this, the bubble size distributions are calculated by the interfacial area concentration model. The simulations with validated models show good agreement with the experiments. From the simulation results, the area in which the gas holdup value is greater than 0.01 extends 37% in the upper circulation region when the rotation speed is increased to 40 rad/s from 20 rad/s. However, in the lower circulation, the gas holdup maintains a low level due to the existence of a "death zone". The simulations indicate that the bubble breakage mainly occurs in the impeller out ow region and the coalescence occurs in the upper circulation region. For bubble behaviors being in the lower circulation region, bubble coalescence or breakage is determined by the ow eld pattern in the vessel. Moreover, it is di cult for the simulation process to converge when the virtual mass force is considered and there is a lack of effective technology to measure it, which has led researchers to generally ignore this force. In order to analyze the virtual mass force, the present study utilizes computer programming which is based on the virtual mass force formula to obtain the force. The results show that at positions close to the impeller tip and the center of the upper region, the virtual mass force value is 0.015 and 0.017, respectively, but the value is approximately zero in the bulk of the tank. Hence, the bubble has two signi cant stages of acceleration process in the vessel.

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A Combined Multifluid-Population Balance Model for Vertical Gas−Liquid Bubble-Driven Flows Considering Bubble Column Operating Conditions

Cited by 38 publications

References 22 publications

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

The Population Balance Equation

Simulation of the Bubble Behaviors for Gas–Liquid Dispersion in Agitated Vessel

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