Multi‐Phase‐Multi‐Size‐Group Model for the Inclusion of Population Balances into the CFD Simulation of Gas‐Liquid Bubbly Flows

A CFD model for the simulation of gas‐liquid bubbly flow is developed. In the model, the multi‐phase flow is simulated by an Eulerian‐Eulerian approach using several phase definitions (from 3 to 10). The bubble size distribution is simulated by a solution of the discretized population balance equation with coalescence and break‐up of bubbles. The number of the discretized population balance equations in the model is larger than the number of the phases used in the flow field simulation. A desired accuracy in the simulation can be achieved by choosing a suitable number of phases as a compromise between accuracy and computational cost. With this model, more detailed flow hydrodynamics and bubble size distribution can be obtained. The model was tested with different operating conditions and for different numbers of dispersed phases in a bubble column, and was verified with a bubble size distribution obtained experimentally.     

Numerical simulation of bubble columns flows: effect of different breakup and coalescence closures

Two-dimensional axisymmetric Eulerian/Eulerian simulations of two-phase (gas/liquid) transient flow were performed using a multiphase flow algorithm based on the finite-volume method. These numerical simulations cover laboratory scale bubble columns of different diameters, operated over a range of superficial gas velocities ranging from the bubbly to the churn turbulent regime. The bubble population balance equation (BPBE) is implemented in the two-fluid model that accounts for the drag force and employs the modified k-ε turbulence model in the liquid phase. Several available bubble breakup and coalescence closures are tested. Quantitative agreements between the experimental data and simulations are obtained for the time-averaged axial liquid velocity profiles, as well as for the kinetic energy profiles, only when model predicted breakup rate is increased by a factor of ten to match the coalescence rate. The calculated time-averaged gas holdup profiles deviate in shape from the measured ones and suggest that full three-dimensional simulation is needed. Implementation of BPBE leads to better agreement with data, especially in the churn-turbulent flow regime, compared to the simulation based on an estimated constant mean bubble diameter. Differences in the predicted interfacial area density, with and without BPBE implementation, are significant. The choice of bubble breakup and coalescence closure does not have a significant impact on the simulated results as long as the magnitude of breakup is increased tenfold.     

Gas-liquid flow and bubble size distribution in stirred tanks

This work is aimed at investigating the turbulent two-phase flow and the bubble size distribution (BSD) in aerated stirred tanks by experiments and computational fluid dynamics (CFD) modelling. The experimental data were collected using a two-phase particle image velocimetry technique and a digital image processing method based on a threshold criterion. With the former technique, the liquid and the gas phase ensemble-averaged mean and r.m.s. velocities are measured simultaneously, while with the latter the dimensions of the bubbles dispersed inside the liquid are evaluated. On the modelling side, a CFD approach, based on the solution of Reynolds average Navier-Stokes equations in an Eulerian framework for both phases, is adopted. As for the bubble dimensions modelling, besides the mono-dispersed assumption, a population balance method, named MUSIG, with bubble break-up and coalescence models is considered. The BSD and the axial and radial velocity of the gas and the liquid phase are presented and discussed. The outcomes of the computational work are evaluated on the basis of the experimental results.     

CFD simulation of bubble columns incorporating population balance modeling

A computational fluid dynamics (CFD)-code has been developed using finite volume method in Eulerian framework for the simulation of axisymmetric steady state flows in bubble columns. The population balance equation for bubble number density has been included in the CFD code. The fixed pivot method of Kumar and Ramkrishna [1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332] has been used to discretize the population balance equation. The turbulence in the liquid phase has been modeled by a k-ε model. The novel feature of the framework is that it includes the size-specific bubble velocities obtained by assuming mechanical equilibrium for each bubble and hence it is a generalized multi-fluid model. With appropriate closures for the drag and lift forces, it allows for different velocities for bubbles of different sizes and hence the proper spatial distributions of bubbles are predicted. Accordingly the proper distributions of gas hold-up, liquid circulation velocities and turbulence intensities in the column are predicted. A survey of the literature shows that the algebraic manipulations of either bubble coalescence or break-up rate were mainly guided by the need to obtain the equilibrium bubble size distributions in the column. The model of Prince and Blanch [1990. Bubble coalescence and break-up in air-sparged bubble columns. A.I.Ch.E. Journal 36, 1485-1499] is known to overpredict the bubble collision frequencies in bubble columns. It has been modified to incorporate the effect of gas phase dispersion number. The predictions of the model are in good agreement with the experimental data of Bhole et al. [2006. Laser Doppler anemometer measurements in bubble column: effect of sparger. Industrial & Engineering Chemistry Research 45, 9201-9207] obtained using Laser Doppler anemometry. Comparison of simulation results with the experimental measurements of Sanyal et al. [1999. Numerical simulation of gas-liquid dynamics in cylindrical bubble column reactors. Chemical Engineering Science 54, 5071-5083] and Olmos et al. [2001. Numerical simulation of multiphase flow in bubble column reactors: influence of bubble coalescence and breakup. Chemical Engineering Science 56, 6359-6365] also show a good agreement for liquid velocity and gas hold-up profiles.     

Computational modeling of microbubble coalescence and breakup using large eddy simulation and Lagrangian tracking

The flow of dispersed microbubbles was studied with an Eulerian–Lagrangian technique using large eddy simulation to predict the continuous liquid flow and Lagrangian tracking to compute bubble trajectories. The model fully accounts for bubble coalescence and breakup and was applied to horizontal and vertical channel flows. With low levels of turbulence, gravity in horizontal, and lift in vertical, channel flows govern the bubble spatial and collision distribution. When turbulence is sufficiently high to, at least partially, oppose bubble preferential concentration, more uniform collision and coalescence distributions are found, although these remain peaked near the wall in both configurations. Almost 100% coalescence efficiency was always found, due to bubbles colliding along similar trajectories, with breakup only recorded in a flow of low surface tension refrigerant R134a. Models like this can provide the required quantitative understanding of the microbubbles complex behavior, as well as supporting the development of more macroscopic modeling closures.     

Effect of interfacial forces and turbulence models on predicting flow pattern inside the bubble column

The numerical approaches have been used in many studies to predict the flow pattern inside the bubble column reactors because of the difficulties that are still found in designing and scaling-up the bubble columns. This review makes an effort to show suitable interfacial forces i.e., drag force, lift force, turbulent dispersion models and virtual mass and turbulence models such as standard k–ɛ model, Reynolds Stress Model, Large Eddy Simulation to predict flow pattern inside the bubble column using Eulerian–Eulerian. The effect of various interfacial forces and turbulence models on gas–liquid velocity and gas hold-up in bubble column is critically reviewed.     

Predicting the diameters of droplets produced in turbulent liquid–liquid dispersion

The droplet size distribution in liquid–liquid dispersions is a complex convolution of impeller speed, impeller type, fluid properties, and flow conditions. In this work, we present three a priori modeling approaches for predicting the droplet diameter distributions as a function of system operating conditions. In the first approach, called the two-fluid approach, we use high-resolution solutions to the Navier–Stokes equations to directly model the flow of each phase and the corresponding droplet breakup/coalescence events. In the second approach, based on an Eulerian–Lagrangian model, we describe the dispersed fluid as individual spheres undergoing ongoing breakup and coalescence events per user-defined interaction kernels. In the third approach, called the Eulerian–Parcel model, we model a sub-set of the droplets in the Eulerian–Lagrangian model to estimate the overall behavior of the entire droplet population. We discuss output from each model within the context of predictions from first principles turbulence theory and measured data.     

Bubble size modeling approach for the simulation of bubble columns

The constant bubble size modeling approach (CBSM) and variable bubble size modeling approach (VBSM) are frequently employed in Eulerian–Eulerian simulation of bubble columns. However, the accuracy of CBSM is limited while the computational efficiency of VBSM needs to be improved. This work aims to develop method for bubble size modeling which has high computational efficiency and accuracy in the simulation of bubble columns. The distribution of bubble sizes is represented by a series of discrete points, and the percentage of bubbles with various sizes at gas inlet is determined by the results of computational fluid dynamics (CFD)–population balance model (PBM) simulations, whereas the influence of bubble coalescence and breakup is neglected. The simulated results of a 0.15 m diameter bubble column suggest that the developed method has high computational speed and can achieve similar accuracy as CFD–PBM modeling. Furthermore, the convergence issues caused by solving population balance equations are addressed.     

The important role of local dispersed phase hold-ups for the calculation of three-phase bubble columns

A computational fluid dynamics model is used to calculate a three-phase (air-water-solid particles) flow in a bubble column. The calculation of multi-phase flows is significantly influenced by the formulation of the inter-phase drag and the modelling of the turbulence. Both are influenced by the dispersed phases. The k-ε turbulence model extended with terms accounting for the bubble-induced turbulence is used to calculate the eddy viscosity of the liquid phase. Bubble-bubble and particle-particle interactions are considered as well as a direct momentum transfer between the two dispersed phases bubbles and solid particles. The local volume fractions of the dispersed phases are considered for the calculation of the drag coefficients between the dispersed phases and the continuous phase. Measured local gas and solid hold-ups as well as measured liquid velocities are compared with the corresponding calculated results. The measured and the calculated results show good agreement.     

Theoretical prediction of flow regime transition in bubble columns by the population balance model

Theoretical prediction of flow regime transition in bubble columns was studied based on the bubble size distribution by the population balance model (PBM). Models for bubble coalescence and breakup due to different mechanisms, including coalescence due to turbulent eddies, coalescence due to different bubble rise velocities, coalescence due to bubble wake entrainment, breakup due to eddy collision and breakup due to large bubble instability, were proposed. Simulation results showed that at relatively low superficial gas velocities, bubble coalescence and breakup were relatively weak and the bubble size was small and had a narrow distribution; with an increase in the superficial gas velocity, large bubbles began to form due to bubble coalescence, resulting in a much wider bubble size distribution. The regime transition was predicted to occur when the volume fraction of small bubbles sharply decreased. The predicted transition superficial gas velocity was about 4 cm/s for the air-water system, in accordance with the values obtained from experimental approaches.     

Computational study of bubble coalescence/break-up behaviors and bubble size distribution in a 3-D pressurized bubbling gas-solid fluidized bed of Geldart A particles

A computational study was carried out on bubble dynamic behaviors and bubble size distributions in a pressurized lab-scale gas-solid fluidized bed of Geldart A particles. High-resolution 3-D numerical simulations were performed using the two-fluid model based on the kinetic theory of granular flow. A fine-grid, which is in the range of 3–4 particle diameters, was utilized in order to capture bubble structures explicitly without breaking down the continuum assumption for the solid phase. A novel bubble tracking scheme was developed in combination with a 3-D detection and tracking algorithm (MS3DATA) and applied to detect the bubble statistics, such as bubble size, location in each time frame and relative position between two adjacent time frames, from numerical simulations. The spatial coordinates and corresponding void fraction data were sampled at 100 Hz for data analyzing. The bubble coalescence/break-up frequencies and the daughter bubble size distribution were evaluated by using the new bubble tracking algorithm. The results showed that the bubble size distributed non-uniformly over cross-sections in the bed. The equilibrium bubble diameter due to bubble break-up and coalescence dynamics can be obtained, and the bubble rise velocity follows Davidson’s correlation closely. Good agreements were obtained between the computed results and that predicted by using the bubble break-up model proposed in our previous work. The computational bubble tracking method showed the potential of analyzing bubble motions and the coalescence and break-up characteristics based on time series data sets of void fraction maps obtained numerically and experimentally.     

Phase distribution in dispersed liquid–liquid flow in vertical pipe: Mean and turbulent contributions of interfacial force

We applied an Eulerian–Eulerian two‐fluid model on an upward dispersed oil–water flow in vertical pipe with 80 mm diameter and 2.5 m length. The numerical profiles of the radial distribution of the oil drops at 1.5 m from the inflow are compared to the experimental data of Lucas and Panagiotopoulos (Flow Meas Instrum. 2009;20:127–135) This article analyzes the roles of turbulence and interfacial forces on the phase distribution phenomenon. In liquid–liquid flow the relative velocity is low and the distribution of the dispersed phase is mainly governed by the turbulence. This work highlights the important role of the turbulent contribution obtained by averaging the added mass force on the radial distribution profiles of the oil drops. The numerical results present improved profiles of the dispersed phase comparing to the experimental data when this turbulent contribution is taken into account in the momentum balance. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4214–4223, 2017     

Calculation of flow fields in two and three-phase bubble columns considering mass transfer

The dimension of bubble column reactors is often based on empirical correlations. Very popular is the axial dispersion model. However, the applicability of these models is limited to the experimental conditions for which the dispersion coefficients are measured, because backmixing depends strongly on the columns dimension and the flow regime. This paper presents a numerical method for the calculation of the three-dimensional flow fields in bubble columns based on a multi-fluid model. Therefore, the local bubble size distribution is considered by a transport equation for the mean bubble volume, which is obtained from the population balance equation. For comparison with experimental results, the axial dispersion coefficients in the liquid and gas phase are calculated from the instationary, three-dimensional concentration fields of a tracer. The model is then extended to include mass transfer between the gas and liquid phase. Increasing mass transfer rates significantly influence the flow pattern. For several applications, a dispersed solid phase is added. For the calculation of three-phase gas-liquid-solid flow, the solid phase is considered numerically by an additional Eulerian phase.     

Simulations of Gas Distributors in the Design of Shallow Bubble Column Reactors

Computational fluid dynamics (CFD) was used to simulate the effect of sparger construction in gas holdup and liquid axial velocity in a shallow bubble column reactor for the air‐water system. Model parameters were evaluated in 2‐ and 3‐D simulations by using a two‐fluid model and the standard k‐? turbulence model. The Eulerian‐Eulerian approach was employed to predict the height of column that is affected by the sparger. It was found that increasing the number of orifices in the sparger increases the total gas holdup. Moreover, each orifice causes an increase in the circulation and mixing of liquid in the column. The results of the simulations follow the trends observed in the findings of Dhotre and Joshi [1].     

Modelling of dispersed bubble and droplet flow at high phase fractions

The present paper describes an Eulerian two-fluid model for the prediction of dispersed two-phase (gas/liquid and liquid/liquid) flow at high volume fractions of the dispersed phase. The model is based on the standard Eulerian approaches for modelling two-phase flow that have hitherto been limited in validity to dilute systems. An extension to high phase fractions is made here and this involves two aspects. First, the closure models for inter-phase forces (namely drag and lift) are modified to account for the high concentration of the dispersed phase. Second, a turbulence model based on the k-ε equations for the mixture of the two phases is formulated. This turbulence model is suitable for computations at all phase fraction values and reduces to the equivalent single phase model in the extremes when only one or other of the phases is present. The model uses a response function to link the turbulent fluctuations of the continuous and dispersed phases. The variation of this response function with phase fraction is determined from experimental evidence made available recently. The overall model is applied to the prediction of air/water bubble flow in a pipe with a sudden enlargement where phase fractions can reach 25% and for which experimental data exist. The calculations show that marked improvement in the quality of the predictions, as compared to measurements, is obtained over the available model for dilute systems.     

Numerical simulations of gas-liquid mass transfer in bubble columns with a CFD-PBM coupled model

Gas-liquid mass transfer in a bubble column in both the homogeneous and heterogeneous flow regimes was studied by numerical simulations with a CFD-PBM (computation fluid dynamics-population balance model) coupled model and a gas-liquid mass transfer model. In the CFD-PBM coupled model, the gas-liquid interfacial area a is calculated from the gas holdup and bubble size distribution. In this work, multiple mechanisms for bubble coalescence, including coalescence due to turbulent eddies, different bubble rise velocities and bubble wake entrainment, and for bubble breakup due to eddy collision and instability of large bubbles were considered. Previous studies show that these considerations are crucial for proper predictions of both the homogenous and the heterogeneous flow regimes. Many parameters may affect the mass transfer coefficient, including the bubble size distribution, bubble slip velocity, turbulent energy dissipation rate and bubble coalescence and breakup. These complex factors were quantitatively counted in the CFD-PBM coupled model. For the mass transfer coefficient k_l, two typical models were compared, namely the eddy cell model in which k_l depends on the turbulent energy dissipation rate, and the slip penetration model in which k_l depends on the bubble size and bubble slip velocity. Reasonable predictions of k_la were obtained with both models in a wide range of superficial gas velocity, with only a slight modification of the model constants. The simulation results show that CFD-PBM coupled model is an efficient method for predicting the hydrodynamics, bubble size distribution, interfacial area and gas-liquid mass transfer rate in a bubble column.     

On the effect of liquid temperature upon bubble coalescence

An experimental study concerning the influence of liquid temperature on bubble coalescence is presented. Bubble collisions in two different liquids (water and ethanol) were analysed at four distinct temperatures , using air as the dispersed phase in all cases. A quantitative criterion was developed to compute the critical velocity for bubble coalescence based on the relative velocities and the outcome (coalescence or bouncing) of the recorded collisions. The critical velocity, and consequently bubble coalescence, was shown to increase as the liquid temperature was raised, an effect whose intensity depended on the kind of liquid employed. Using some simplifying assumptions, a comparison between the experimentally observed trends and the predictions of literature models for the film drainage was drawn. The experimental data were used together with available literature data in the development of a dimensionless correlation for predicting the critical velocity as a function of the liquid properties.     

Bubble size distribution modeling in stirred gas–liquid reactors with QMOM augmented by a new correction algorithm

Local gas hold‐up and bubbles size distributions have been modeled and validated against experimental data in a stirred gas–liquid reactor, considering two different spargers. An Eulerian multifluid approach coupled with a population balance model (PBM) has been employed to describe the evolution of the bubble size distribution due to break‐up and coalescence. The PBM has been solved by resorting to the quadrature method of moments, implemented through user defined functions in the commercial computational fluid dynamics code Fluent v. 6.2. To overcome divergence issues caused by moments corruption, due to numerical problems, a correction scheme for the moments has been implemented; simulation results prove that it plays a crucial role for the stability and the accuracy of the overall approach. Very good agreements between experimental data and simulations predictions are obtained, for a unique set of break‐up and coalescence kinetic constants, in a wide range of operating conditions. © 2009 American Institute of Chemical Engineers AIChE J, 2010