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  • chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

    Contents lists available at ScienceDirect

    Chemical Engineering Research and Design

    journa l h om epage: www.elsev ier .com/ locate /cherd

    CFD-PBM approach with modified drag model for the gas–liquid flow in a bubble column

    Xiao-Fei Liang, Hui Pan, Yuan-Hai Su, Zheng-Hong Luo ∗

    Department of Chemical Engineering, School of Chemistry and Chemical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

    a r t i c l e i n f o

    Article history:

    Received 29 March 2016

    Received in revised form 25 May


    Accepted 8 June 2016

    Available online 18 June 2016


    Bubble column

    Euler–Euler approach


    Drag model

    Correction factor

    a b s t r a c t

    In this work, numerical simulations of cylindrical bubble column are performed using the

    Euler–Euler approach incorporated with a population balance model (PBM). First, three drag

    models and their corresponding modified models with the wake acceleration are incorpo-

    rated into the coupled approach in order to evaluate the effectiveness of these drag models.

    The simulated time-averaged local gas holdups and normalized axial liquid velocities using

    different drag equations are compared with the experimental data, showing that only the

    PBM-customized drag model with the wake acceleration (cf., the application of a correction

    factor) can reproduce the measured flow field data. Subsequently, the applicability of the

    coupled approach with the effective drag model is further evaluated at various superficial gas

    velocities and gas distributors. The simulated results accord well with the experimental data

    at high gas velocities. However, the model greatly underestimates the radial local gas holdup

    and the total gas holdup at low gas flow rates. Additionally, the simulated results demon-

    strate that the opening area and orifice geometry play a significant role in total aeration and

    the triple-ring gas distributor produces more uniform radial profiles of local gas holdup and

    normalized liquid velocity than the multi-orifice one, thus leading to poor mixing efficiency

    in the bubble column. © 2016 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

    imental results in the heterogeneous regime (Ishii, 1975). A

    1. Introduction

    Bubble columns serve as multiphase contactors and reactors in the chemical, petrochemical, biochemical, and metallurgi- cal industries, primarily because of their simple construct with low price, simple operation, ability to handle solids, excellent heat and mass transfer characteristics, etc. In spite of their simplicity in mechanical design, fundamental properties of the two-phase hydrodynamics associated with the operation of bubble column reactors essential for scale-up and process optimization, are still not fully understood because of the complex nature of multiphase flow (Chen et al., 2005; Tabib et al., 2008).

    A great deal of research on gas–liquid flows has been carried out with computational fluid dynamics (CFD) simula-

    tions for engineering purposes. Two approaches including the

    ∗ Corresponding author. Tel.: +86 21 54745602; fax: +86 21 54745602. E-mail address: (Z.-H. Luo). 0263-8762/© 2016 Institution of Chemical Engineers. Published by Elsev

    Euler–Lagrange (E-L) approach (Delnoij et al., 1997; Sokolichin et al., 1997) and the Euler–Euler (E-E) approach are primar- ily used (Drew, 1983; Ishii, 1975; Krishna et al., 1999; Lehr et al., 2002). The E-L approach shows its advantages on a clear physical description but its disadvantages on high computa- tional cost and the difficulties to involve the forces on the deformable bubbles and bubble breakup and coalescence. For the E-E approach, the model equations for each phase have the same form, showing obvious advantages in view of the com- puter memory and simulation time limitations. However, the E-E two-fluid model with the assumption of a constant bubble diameter gives reasonable predictions only for the homoge- nous regime in which the bubble size distribution is narrow. There are obvious deviations between simulated and exper-

    profound progress in the CFD simulation of bubble columns

    ier B.V. All rights reserved.

  • chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 89


    ai,j bubble aggregation rate, s−1

    b(v) bubble breakup rate, s−1

    CD drag coefficient d bubble diameter, m dc critical size of bubbles having wake effect for

    bubble coalescence, m Eo Eötvös number, g(�l − �g)d2b/� flarge fraction of large bubbles g gravity, m2 s−1

    G the production rate of turbulent energy, W m−3

    k turbulent kinetic energy, m2 s−2

    kb,large correction factor accounting for bubble wake effect

    ki,j the covariance of the velocities of phase i and phase j

    Mo Morton number, dimensionless n number of particles per unit volume, m−3

    p pressure, Pa Q gas flow rate, m3 s−1

    Re bubble Reynolds number, dimensionless t time, s tdrainage film drainage time, s tcontact bubble contact time, s ui,j the relative velocity between the liquid and dis-

    persed phases U velocity vector, m2 s−2

    Ug superficial gas velocity, m2 s−2

    v bubble volume, m3

    xi pivot bubble size of the bubble interval (vi, vi+1), m3

    Greek Letters ˛ phase holdup ˇ(v, xk) daughter bubble size distribution function ε turbulent dissipation rate, m2 s−3

    � viscosity, kg m−1 s−1

    �1(v, xi), �2(v, xi) bubble redistribution coefficient ωi,jk transfer coefficient between bubble groups aris-

    ing from bubble coalescence i,k transfer coefficient between bubble groups aris-

    ing from bubble breakup ˘k,i bubble-induced turbulence term, W m−3

    ˘ε,i bubble-induced turbulence term, W s−1 m−3

    � density, kg m−3

    � surface tension, N m−1

    Subscripts b bubble index g gas index l liquid index i,j phase index large large bubbles index

    i C 2 a a s a

    s the coupling of the population balance model (PBM) into FD models, namely the CFD-PBM coupled model (Duan et al., 011; Wang and Wang, 2007; Wang et al., 2006). The prominent dvantages of the coupled model could be expressed briefly s follows: (1) it combines the PBM into the CFD framework

    o that bubble breakup and coalescence can be taken into ccount, beneficial for describing the bubble size distribution

    and gas holdup in different flow regimes, and (2) it consid- ers the influence of bubble size on the interphase interaction, which allows it to well predict the local gas–liquid interfacial area and the flow behavior in diverse flow regimes (Xing et al., 2013).

    While the closure models are widely applied in quantita- tive and qualitative description of the heat and mass transfer in reactors. In these models, bubble induce turbulence of liq- uid phase, interaction forces between gas and liquid phase, and local bubble size distribution are emphasized and highly coupled with each other (Laborde-Boutet et al., 2009). It is generally believed that the drag is the predominant force in modeling the gas–liquid flows of bubble columns (Chen, 2004) and the magnitude of the drag force was found to be over 100 times than that of the other forces such as lift force, added mass force, and turbulent dispersion force (Laborde-Boutet et al., 2009). Although there have been many studies on the lift force (Díaz et al., 2009; Kulkarni, 2008; Lucas and Tomiyama, 2011; Van Nierop et al., 2007), an appropriate model for the lift coefficient is not available and the role of lift force or turbulent dispersion force in two-fluid modeling is not clear. In some of these studies, the lift force was only considered in order to adjust CFD simulations, and the turbulent dis- persion force was involved to compensate the destabilizing action induced by a negative lift coefficient (Lucas et al., 2005). Many drag models such as Schiller and Naumann (1935), Ishii and Zuber (1979), Tomiyama (1998), Zhang and VanderHeyden (2002) and Grace et al. (1976) models have been used to calcu- late the drag force in bubble columns (Gupta and Roy, 2013; Laborde-Boutet et al., 2009; Olmos et al., 2001; Silva et al., 2012; Zhang et al., 2006). For small spherical bubbles, the Schiller–Naumann model is frequently applied (Schiller and Naumann, 1935). While for larger bubble size, a model was proposed by Ishii and Zuber (1979) which is suitable for vari- ous shapes of bubbles such as sphere, ellipse and cap shape (Ishii and Zuber, 1979; Xu et al., 2013). Recently, Yang et al. (2011) suggested a dual-bubble size (DBS) model (Chen et al., 2009; Yang et al., 2007, 2010) where the force balance equa- tions of two bubble classes are closed via a stability condition. Afterwards, they developed a stability-constrained multi-fluid (SCMF) model for the CFD simulation of bubble columns. In this SCMF model, the structure parameters are calculated via the DBS model, and then the ratio of drag coefficient to bub- ble diameter can be obtained and fed into the two-fluid or multi-fluid model frameworks. The simulation with a constant bubble


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