Computers and Chemical Engineering 32 (2008) 3224–3237 Contents lists available at ScienceDirect Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng Three-dimensional mathematical modeling of dispersed two-phase flow using class method of population balance in bubble columns Rachid Bannari a , Fouzi Kerdouss b , Brahim Selma a , Abdelfettah Bannari a , Pierre Proulx a,∗ a Department of Chemical Engineering, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada b Environnement Canada, Place Vincent Massey, 351, Boulevard St-Joseph, Gatineau, QC K1A 0H3, Canada article info Article history: Received 26 October 2007 Received in revised form 6 May 2008 Accepted 28 May 2008 Available online 8 June 2008 Keywords: Population balance equation Coalescence and break-up Interfacial area Methods of classes Bubble size Computational fluid dynamics (CFD) abstract Computational fluid dynamics (CFD) simulations of bubble columns have received recently much attention and several multiphase models have been developed, tested, and validated through comparison with experimental data. In this work, we propose a model for two-phase flows at high phase fractions. The inter-phase forces (drag, lift and virtual mass) with different closure terms are used and coupled with a classes method (CM) for population balance. This in order to predict bubble’s size distribution in the column which results of break-up and coalescence of bubbles. Since these mechanisms result greatly of turbulence, a dispersed k–  turbulent model is used. The results are compared to experimental data available from the literature using a mean bubble diame- ter approach and CM approach and the appropriate formulations for inter-phase forces in order to predict the flow are highlighted. The above models are implemented using the open source package OpenFoam. Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved. 1. Introduction Bubble columns are today widely used in many industrial pro- cesses like chemical, pharmaceutical and petrochemical because of their capability of achieving high heat and mass transfer rates with low energy input. The computational fluid dynamics (CFD) methods using modern high speed computing capabilities is a useful tool that is now often used in the industry for scaling-up or design of many types of reactors, but a lot of effort is to be made in order to develop models that will take into account all the complexity of bubble columns’ type reactors. Sound engineering judgement and detailed understanding of the underlying physics of the phenomena must be used in order to develop models that can adequately describe the observed behavior of these reactors. In bubble columns, gas phase exists as a dispersed bubble phase in a continuous liquid phase. Up to the recent years, the use of CFD modeling for bubble columns has been very limited compared to that in single-phase or dilute multiphase systems. Most work published had focused on developing closure models for interfacial bubble–liquid forces and on estimating bubble-induced turbulence. Again, until recently, relatively little attention has been given to the bubble size distri- bution problem although it is an important design parameter and ∗ Corresponding author. E-mail address: pierre.proulx@usherbrooke.ca (P. Proulx). can influence significantly the result of gas–liquid mass transfer equipment. Bubble size distribution depends extensively on col- umn geometry, operating conditions, physical properties of phases and sparger type, and the last few years have seen more and more researchers addressing this complex problem. There are essentially two approaches for the numerical calcula- tion of multiphase flow, namely the Euler–Lagrange method which considers the bubbles as individual entities tracked using trajec- tory equations (Webb, Que, & Senior, 1992; Lapin & Lübbert, 1994), and the Euler–Euler method described below. The Euler–Euler approach (E–E) has been used by several authors (Sokolichin & Eigenberger, 1997; Pfleger, Gomes, Gilbert, & Wagner, 1999; Buwa & Ranade, 2002). The basic equations have been first formulated by Ishii (1975). In order to take into account turbulence of the under- lying flow, the turbulent stress term in the mixture equation is closed by solving a standard k–  model for the mixture phase. In the present work the two-fluid methodology of Weller (2002), and Rusche (2002), is used. It is based on the standard E–E approach for prediction of dispersed phase flow at high volume fractions of the dispersed phase. This methodology involves some modifications in the closure models for inter-phase forces, volume fraction equa- tion and reformulation of turbulence model in order to compute at all phase fraction values and to reduces the complex model to the equivalent single-phase when only one of the phases is present. To incorporate the effects of the dispersed phase on the turbulence, additional source terms are used (Mudde & Simonin, 1999). Initial 0098-1354/$ – see front matter. Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2008.05.016