CFD simulation of bubble column hydrodynamics with a novel drag model based on EMMS approach

https://doi.org/10.1016/j.ces.2021.116758Get rights and content

Highlights

  • A novel drag model termed DBS-Local was proposed based on EMMS approach.

  • The new drag model was validated by experiments in all flow regimes.

  • The DBS-local and DBS-Global models predict the total and local gas holdup well.

  • The DBS-Local model shows better agreement with experiments in terms of centerline liquid velocity.

Abstract

Drag model is of critical importance for CFD simulation of flow characteristics in bubble columns. In this study, a novel drag model termed DBS-Local for the ratio of effective drag coefficient to bubble diameter is derived from the Dual-Bubble-Size (DBS) model based on Energy-Minimization Multi-Scale (EMMS). A thorough comparison demonstrates that without adjustable parameters, both the DBS-Local model and the DBS-Global model are capable of improving the prediction of total and local gas holdup as well as the flow characteristics in different flow regimes. By contrast, other drag models either overestimate or underestimate the total or local gas holdup in certain flow regimes. Moreover, the centerline liquid velocity predicted by DBS-Local model shows best agreement with experiments, whereas the other models, especially the Ishii-Zuber model, over-estimate liquid velocity. This study demonstrates the significance of modeling the bubble-swarm effect based on EMMS approach.

Introduction

Bubble columns have been widely applied as efficient gas-liquid contactors in chemical and process industries owing to their simple structure as well as good heat and mass transfer. However, the hydrodynamics in such non-equilibrium systems is complex, particularly when the column is operated in the heterogeneous regime at high superficial gas velocity. CFD simulation based on Euler-Euler approach has been commonly applied to enhance the understanding of complex flow characteristics in bubble columns in view of acceptable computational burden in comparison with the enormous cost of Euler-Lagrange approach or direct numerical simulation (DNS) (Gupta and Roy, 2013, Mowla et al., 2020, Zhu et al., 2020). Closure laws or sub-models, including interfacial forces, turbulence models as well as bubble coalescence and breakage kernel functions, are needed to reflect the impacts of micro-scale interphase and intra-phase interactions on macro-scale flow behavior. Among the various interfacial forces, it is generally acknowledged that drag is most dominant in simulation of bubble columns (Jakobsen et al., 2005, Sokolichin et al., 2004, Yan et al., 2019).

The standard drag coefficient of a single bubble in an infinite media (CD0) depends on the dimensionless number Eo or ReB (Clift et al., 1978). The presence of surface active substances in liquid phase may harden bubble surface and then increase the drag coefficient (Kulkarni et al., 1987). When a bubble is surrounded by other bubbles, the effective drag coefficient (CD) would be completely different from the standard drag coefficient, and a correction factor (usually liquid volume fraction with a power n) is introduced to take account of the swarm effect. Two opposing effects (hindrance effect and accelerating effect) have been reported in the literature, and different factors were correspondingly proposed to either increase or decrease the standard drag coefficient (Garnier et al., 2002, Krishna et al., 1999, Loisy et al., 2017, Zhang and Fan, 2003).

Three approaches have been commonly utilized to determine the drag coefficient in bubble swarms: theoretical analysis, experiments or CFD simulation (Garnier et al., 2002, Ishii and Zuber, 1979, McClure et al., 2017, Simonnet et al., 2007). Ishii and Zuber (1979) analytically derived the drag coefficient for bubbles in different shape regimes based on a simple similarity criterion and a mixture viscosity model. The model suggests that the surrounding bubbles tend to increase the drag coefficient in the viscous or distorted bubble regime and to decrease the coefficient in the capped bubble regime.

In experiments, the effective drag coefficient can be indirectly measured from the force balance between drag and effective gravity, and is expressed as functions of bubble slip velocity, bubble size and gas holdup. Therefore, these parameters should be measured locally (Garnier et al., 2002, McClure et al., 2017, Simonnet et al., 2007). Assuming that the bubble column of interest was operated in the homogeneous regime, some researchers (Colombet et al., 2011, Colombet et al., 2015, Lane et al., 2016, Mach et al., 2020) estimated gas (liquid) velocity through the ratio of gas (liquid) flux to the global gas (liquid) holdup. Some experiments (Colombet et al., 2011, Colombet et al., 2015, Garnier et al., 2002, Mach et al., 2020) showed an increase of drag coefficient with gas holdup, whereas McClure et al. (2017) reported a decrease for the polydisperse bubbles. In contrast, Simonnet et al. (2007) found a hindrance effect at the gas holdup lower than 15% and an acceleration effect at the gas holdup higher than 15% for bubbles with diameter larger than 7 mm. These diversified experimental observations demonstrate the complexity and difficulty of correlating the effective drag coefficient for bubble warms as a function of gas holdup.

It is possible to resolve all flow details around each bubble through DNS, including bubble shape oscillation and wake vortex shedding, and then to estimate the effective drag coefficient. Esmaeeli and Tryggvason, 1999, Bunner and Tryggvason, 2002 pioneered such an approach and developed the front-tracking model to capture the decrease of bubble rising velocity in bubble swarms. Besides, the results revealed that the freely-evolving bubble array rises slower than the regular one due to different microstructures. Sankaranarayanan et al. (2002) derived a drag correlation in regular arrays of bubbles via lattice Boltzmann simulations. The correlation contains the entire range from the hindered to the cooperative motion of bubble swarms. Roghair et al., 2011, Roghair et al., 2013a, Roghair et al., 2013b) simulated freely rising monodisperse and bi-disperse bubble swarms based on a front-tracking model, and the proposed drag correction factor increases linearly with gas holdup. These micro-scale simulations of bubble dynamics enhance the understanding of bubble-fluid and bubble-bubble interactions in bubble swarms, but the drawbacks are the limited number of bubbles in simulation and the finite size of computational domain which may not represent the real chaotic flow characteristics in bubble columns.

Drag models have also been developed by tuning model parameters in the simulations of Two-Fluid Model (TFM) to fit with experimental data. For a multiple-bubble-class model, Olmos et al. (2003) adjusted the correction factor for each bubble class under various superficial gas velocities. The model simulated regime transition based on the variation of total gas holdup with superficial gas velocity. It should be noticed that the factor decreases drag coefficient and reflects the accelerating effect of surrounding bubbles. Behzadi et al. (2004) proposed an empirical corrector as a combination of an exponential function and a power law. The corrector is larger than unity and reflects the hindrance effect. Buffo et al. (2016) proposed a drag model taking account of the effects of bubble swarm and micro-scale turbulence in bubbly flow regime. Both the effects decrease bubble rising velocity and increase drag coefficient. Yang et al., 2017, Yang et al., 2018) proposed a drag corrector involving both the wake-accelerating effect of large bubbles and the hindering effect of small bubbles in CFD-PBM simulation of bubble size distribution. To fit the experiments, the hindrance effect at low gas holdup in the correlation of Simonnet et al. (2007) was removed by McClure et al. (2014). Fletcher et al. (2017) further decreased the corrector of Simonnet et al. (2007) by 20% in the accelerating regime at high gas holdup. Gemello et al. (2018) modified the correction factor of Simonnet et al. (2007) by adding a minimum constant 0.15 to fit the measured total gas holdup. Through parameter-fitting, CFD simulation indeed improves greatly in some cases, but the critical issue lies in whether such fitted correlations could be extended to a broader range of operating conditions or physical properties.

Fig. 1 compares some of the developed drag correction to take account of bubble swarm effect. The deviation among various models is evident. Either the hindrance effect or the accelerating effect was taken into account in some models, whereas others involve the hindrance effect at low gas holdup and the accelerating effect at high gas holdup. It is therefore of significance to discribe the swarm effect based on more fundamentals and develop a drag closure without parameter fitting.

Yang et al., 2007, Yang et al., 2010) proposed a dual-bubble-size (DBS) model based on Energy-Minimization Multi-Scale (EMMS) concept. The model is zero-dimensional, involving three simplified conservation equations for two bubble classes and a stability condition reflecting the compromise of two dominant mechanisms, namely, a liquid-dominated mechanism favoring gas holdup in the form of small bubbles, and a gas-dominated mechanism favoring the formation of larger bubbles. Each one can be expressed as an extremum tendency of some energy consumption, and the stability condition was then formulated as the minimization of the sum of the two counterparts. The model is conceptual, and supplies another physical constraint to the complex systems obeying mass and momentum conservations. More details have been reported in Yang et al., 2007, Yang et al., 2010). When this conceptual model was applied to the overall system of a bubble column, it can qualitatively describe the evolution of the so-called structure parameters of the two bubble classes with superficial gas velocity, and interpret the macro-scale regime transition in terms of an abrupt change of structure parameters. Afterwards, Yang et al., 2011, Xiao et al., 2013, Xiao et al., 2017) derived a drag closure for the lumped parameter (the ratio of effective drag coefficient to bubble diameter Cd/db) from the structure parameters in the DBS model. The CFD simulation indicated that the new drag model without adjustable parameters improved the predictions of global and local flow characteristics in bubble columns. For simplicity, the ratio Cd/db was only correlated with the overall superficial gas velocity, and the model was therefore termed DBS-Global drag. This approach has also been extended to develop drag closure for bubble columns with different operation modes (Guan and Yang, 2019) and solid volume fraction (Zhou et al., 2020). Jiang et al. (2016) attempted to correlate the ratio Cd/db as functions of superficial gas and liquid velocities in local computational units, and the model predicted the gas holdup well in an external-loop airlift reactor.

It should be noticed that when the zero-dimensional DBS model is applied to a local computational unit rather than the whole bubble column, a gas volume fraction can also be obtained from the structure parameters within this local domain. The gas volume fraction in this case embodies the structure information of this unit which follows both the conservation equations and the stability condition. On the other hand, the local average Cd/db of this unit has an one-to-one correspondence to the gas volume fraction at steady state under the given superficial flow rates of gas and liquid entering into this domain. It is therefore feasible to correlate the ratio as a function of gas volume fraction rather than superficial flow rates of local units, so that a more convenient way to apply the DBS model to local computational domain could be derived. Once the correlation of this ratio as a function of gas volume fraction is obtained, it could then be integrated into the complete conservation equations of CFD simulation.

Thereby, this work aims to develop a novel drag model correlating the ratio Cd/db with gas volume fraction from the DBS model calculation of structure parameters under various superficial gas velocities. To validate this new model, experiments were performed in a 0.15 m ID bubble column under different superficial gas velocities (1–23 cm/s), covering homogeneous to heterogeneous regimes. For comparison, CFD simulations were also carried out with other six drag models in literature to evaluate the performance of different models in prediction of global and local flow characteristics.

Section snippets

Experimental

Experiments were performed in a semi-batch Plexiglas column with ID 0.15 m and height 1.6 m. Air was introduced through a perforated plate with 206 holes of 0.5 mm in diameter, evenly arranged on 7 circles. Superficial gas velocity was manipulated within the range of 0.008–0.23 m/s to cover different flow regimes. Total gas holdup was obtained from the static pressure difference between two taps within an axial distance 0.64 m. The pressure difference was detected by a differential pressure

The EMMS drag model

Yang et al., 2007, Yang et al., 2010) developed the DBS model based on the EMMS approach. In the conceptual model, the dispersed phase of a gas-liquid system was resolved into a small-bubble phase and a large-bubble phase based on the experimental observations in literature (Camarasa et al., 1999, Krishna and Ellenberger, 1996). Three structure parameters for each bubble phase, i.e., bubble size di, holdup αg,i and superficial gas velocity Ug,i (i = L for large-bubble phase and i = S for

Governing equations

In the TFM model, both phases are treated as interpenetrating continua and the mass and momentum conservation equations are solved for each phase, as summarized in Table 1. The interfacial forces Fi,k consist of drag, lift, turbulent dispersion and wall forces. More information about these interfacial forces models can be referred to Guan and Yang (2017b). Other six drag models (DBS-Global, Ishii-Zuber, Schiller-Naumann, Tomiyama, McClure and Gemello) are also listed in Table 2 for evaluation.

Total gas holdup

Fig. 4 compares the total gas holdup predicted by different drag models. The measured total gas holdup increases monotonously with superficial gas velocity at the beginning, and then the increasing trend slows down after the onset of regime transition. To identify the regime transition point, the drift-flux was illustrated in Fig. 5. The first transition from homogeneous to transition regime is marked as the point where the measured drift flux deviates from the theoretical drift flux (Guan and

Conclusion

A novel drag model termed DBS-Local, derived from the zero-dimensional conceptual DBS model based on EMMS approach, was proposed to correlate the ratio of effective drag coefficient to bubble diameter as a function of gas holdup. To validate the model, the total and local gas holdup as well as bubble size were measured in a bubble column of 0.15 m in diameter, and the operating flow rates cover from the homogeneous to heterogeneous regimes. Seven drag models, i.e. DBS-Local, DBS-Global,

CRediT authorship contribution statement

Xiaoping Guan: Methodology, Investigation, Software, Data curation, Formal analysis, Visualization, Writing - original draft, Funding acquisition. Ning Yang: Conceptualization, Methodology, Formal analysis, Resources, Supervision, Writing - review & editing, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors acknowledge the long-term support from the National Natural Science Foundation of China (21925805, 91834303, 21808222, 22061130204), and Innovation Academy for Green Manufacture, Chinese Academy of Sciences (IAGM-2019-A13) and Newton Advanced Fellowship, Royal Society (NAF\R1\201305).

References (56)

  • X. Guan et al.

    Modeling of co-current and counter-current bubble columns with an extended EMMS approach

    Particuology

    (2019)
  • A. Gupta et al.

    Euler-Euler simulation of bubbly flow in a rectangular bubble column: Experimental validation with Radioactive Particle Tracking

    Chem. Eng. J.

    (2013)
  • X. Jiang et al.

    Computational fluid dynamics simulation of hydrodynamics in the riser of an external loop airlift reactor

    Particuology

    (2016)
  • R. Krishna et al.

    Rise velocity of a swarm of large gas bubbles in liquids

    Chem. Eng. Sci.

    (1999)
  • C. Lane et al.

    Investigation of bubble swarm drag at elevated pressure in a contaminated system

    Chem. Eng. Sci.

    (2016)
  • D. Lucas et al.

    Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows

    Chem. Eng. Sci.

    (2007)
  • J. Mach et al.

    Effect of pressure on the drag coefficient of individual bubbles in a contaminated polydisperse swarm

    Chem. Eng. Sci.

    (2020)
  • P. Maximiano Raimundo et al.

    Hydrodynamics and scale-up of bubble columns in the heterogeneous regime: Comparison of bubble size, gas holdup and liquid velocity measured in 4 bubble columns from 0.15 m to 3 m in diameter

    Chem. Eng. Sci.

    (2019)
  • D.D. McClure et al.

    Experimental investigation into the drag volume fraction correction term for gas-liquid bubbly flows

    Chem. Eng. Sci.

    (2017)
  • E. Olmos et al.

    Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model

    Chem. Eng. Sci.

    (2003)
  • I. Roghair et al.

    Direct numerical simulations of the drag force of bi-disperse bubble swarms

    Chem. Eng. Sci.

    (2013)
  • I. Roghair et al.

    On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers

    Chem. Eng. Sci.

    (2011)
  • M. Simonnet et al.

    Experimental determination of the drag coefficient in a swarm of bubbles

    Chem. Eng. Sci.

    (2007)
  • M. Simonnet et al.

    CFD simulation of the flow field in a bubble column reactor: Importance of the drag force formulation to describe regime transitions

    Chem. Eng. Process.

    (2008)
  • Q. Xiao et al.

    Simulation of the multiphase flow in bubble columns with stability-constrained multi-fluid CFD models

    Chem. Eng. J.

    (2017)
  • Q. Xiao et al.

    Stability-constrained multi-fluid CFD models for gas–liquid flow in bubble columns

    Chem. Eng. Sci.

    (2013)
  • P. Yan et al.

    CFD simulation of hydrodynamics in a high-pressure bubble column using three optimized drag models of bubble swarm

    Chem. Eng. Sci.

    (2019)
  • G. Yang et al.

    Numerical simulation of the bubble column at elevated pressure with a CFD-PBM coupled model

    Chem. Eng. Sci.

    (2017)
  • Cited by (18)

    • Exploration on the stability conditions in bubble columns by noncooperative game theory

      2022, Chinese Journal of Chemical Engineering
      Citation Excerpt :

      Based on this zero-dimensional model, Yang et al. [26] proposed a new drag model and gave a correlation for Cd/db as a function of superficial gas velocity Ug. The drag model was further improved [27,28] and recently correlated with the local hydrodynamics of CFD cells [29]. Submodels to calculate the drag coefficients in conservation equations and definitions of different energy terms are given in Table 2.

    • Using mesoscale drag model-augmented coarse-grid simulation to design fluidized bed reactor: Effect of bed internals and sizes

      2022, Chemical Engineering Science
      Citation Excerpt :

      This complexity further gives rise to difficulty in reliable use of coarse models for design of fluidized bed units. Over the past several decades, many research groups have contributed to mesoscale modeling of nonuniform gas-particle or gas–liquid flows based on different methodologies including the energy minimization multi-scale (EMMS) theory (Li and Kwauk, 1994; Qi et al., 2007; Wang et al., 2011; Ullah et al., 2013; Li et al., 2018; Adnan et al., 2021; Guan and Yang, 2021; Hu and Liu, 2021; Musango et al., 2021) and filtered subgrid theory (Milioli et al., 2013; Ozel et al., 2013; Schneiderbauer and Pirker, 2014; Gao et al., 2018; Cloete et al., 2019; Zhu et al., 2020b; Niaki et al., 2021). The main purpose of these mesoscale modeling contributions is to establish effective subgrid closure relations that can be used in later efficient coarse-grid or coarse-grained simulations while the effect of inhomogeneous structures is accounted for (Rauchenzauner and Schneiderbauer, 2022).

    • Pore-scale mechanisms and simulations for gas–water two-phase transport processes in natural gas reservoirs

      2021, Journal of Natural Gas Science and Engineering
      Citation Excerpt :

      To better understand these processes, the gas–liquid two-phase flow problem should be modeled at the pore scale, especially for the development of natural gas reservoirs. Bubble flow processes in porous media are characterized by mass, momentum, and energy transfer (Guan and Yang, 2021). Traditionally, the evaluation of bubble flows primarily relies on engineering experience and empirical correlations, which can be expressed as macroscopic parameters based on the studies by Ergun and Jo et al. (Ergun, 1952; Jo and Revankar, 2010).

    View all citing articles on Scopus
    View full text