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.     

Effect of solids on homogeneous-heterogeneous flow regime transition in bubble columns

Experiments were conducted to study the effect of the presence of the solid phase on the homogeneous-heterogeneous flow regime transition in a bubble column 0.14 m diameter. Air, distilled water and calcium alginate beads (2.1 mm, ) at concentrations c=0-30% (vol.) were the phases. The basic data were the voidage-gas flow rate dependences. The critical point, where the homogeneous regime loses stability and the transition begins, was evaluated by the drift flux model. The critical values of voidage and gas flow rate were the quantitative measures of the homogeneous regime stability. These were plotted against the solid phase concentration. It was found, that both the voidage and the critical values increased with the solid content at low solid loading, approx. c=0-3%, and decreased at higher loading, c>3%. The homogeneous regime was thus first stabilized and then destabilized. To explain this dual effect, possible physical mechanisms of the solid phase influence on the uniform bubble bed were discussed.     

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 estimation of the regime transition point in bubble columns

Using literature data of gas hold-up as a function of the superficial velocity from 16 different sources, a data bank of regime transition points was elaborated. It comprises 83 data related to a total of 20 systems, covering a wide range of physical properties and operating parameters, for both perforated and porous plate spargers. This data bank was employed to critically assess the quality of the predictions of the regime transition point given by available literature correlations. All correlations tested failed to provide a proper representation of the data bank, with rather high mean absolute deviations (always greater than 37%) and, in some cases, even physically inconsistent values were obtained. Thus, new empirical formulas were proposed for estimating the gas superficial velocity at the point of regime transition in bubble columns and the corresponding gas hold-up, whose mean absolute deviations were respectively equal to 17.7 and 21.1%.     

Eulerian–Lagrangian simulation of bubble coalescence in bubbly flow using the spring-dashpot model

The Eulerian–Lagrangian simulation of bubbly flow has the advantage of tracking the motion of bubbles in continuous fluid, and hence the position and velocity of each bubble could be accurately acquired. Previous simulation usually used the hard-sphere model for bubble–bubble interactions, assuming that bubbles are rigid spheres and the collisions between bubbles are instantaneous. The bubble contact time during collision processes is not directly taken into account in the collision model. However, the contact time is physically a prerequisite for bubbles to coalesce, and should be long enough for liquid film drainage. In this work we applied the spring-dashpot model to model the bubble collisions and the bubble contact time, and then integrated the spring-dashpot model with the film drainage model for coalescence and a bubble breakage model. The bubble contact time is therefore accurately recorded during the collisions. We investigated the performance of the spring-dashpot model and the effect of the normal stiffness coefficient on bubble coalescence in the simulation.The results indicate that the spring-dashpot model together with the bubble coalescence and breakage model could reasonably reproduce the two-phase flow field, bubble coalescence and bubble size distribution. The influence of normal stiffness coefficient on simulation is also discussed.     

Three-dimensional mathematical modeling of dispersed two-phase flow using class method of population balance in bubble columns

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 diameter 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.     

CFD simulation of gas–liquid flow in a high-pressure bubble column with a modified population balance model

In this study, based on the Luo bubble coalescence model, a model correction factor C_e for pressures according to the literature experimental results was introduced in the bubble coalescence efficiency term. Then, a coupled modified population balance model (PBM) with computational fluid dynamics (CFD) was used to simulate a high-pressure bubble column. The simulation results with and without C_e were compared with the experimental data. The modified CFD-PBM coupled model was used to investigate its applicability to broader experimental conditions. These results showed that the modified CFD-PBM coupled model can predict the hydrodynamic behaviors under various operating conditions.     

The influence of electrolytes on gas hold-up and regime transition in bubble columns

Experiments were conducted in a 0.12-m-in-diameter bubble column to investigate the effect of electrolytes on gas hold-up (ε) and on the regime transition point in bubble columns. Air was used as the dispersed phase and aqueous solutions of three different salts (NaCl, Na₂SO₄ and NaI), as well as double-distilled water, were utilised as the continuous phase, varying the gas superficial velocity (u_G) in the range 0-0.26 m/s. The ε×u_G curves were a function of both the chemical nature and the concentration of the electrolytes. However, similar ε×u_G profiles were obtained regardless of the electrolyte for a given ratio between the concentration in the solution and the critical concentration of the electrolyte for bubble coalescence. This ratio therefore presents itself as a promising modelling parameter to account for the chemical nature of electrolytes. The gas hold-up data were employed to compute the regime transition point according to two different methods, evidencing its non-linear dependence on the concentration of electrolytes in the liquid.     

Two regime transitions to pseudo-homogeneous and heterogeneous bubble flow for various liquid viscosities

The gas hold-up variation and regime transition were investigated with different liquid viscosities ranging from 1.0 mPa s to 31.5 mPa s using a 0.15-m-in-diameter bubble column. In contrast to common observations, the gas hold-up graph with the superficial gas velocity could be categorized into three flow regimes: homogeneous, pseudo-homogeneous and heterogeneous flow regimes. The formation of large bubbles caused a transition from the homogeneous to the pseudo-homogenous flow regime, in which large bubbles rose vertically without oscillatory turbulence. According to the results from the dynamic gas disengagement (DGD) technique, large bubbles began to form at the transition superficial gas velocity to the pseudo-homogeneous flow regime. The transition to a heterogeneous flow regime was initiated by the turbulent movement of large bubbles. The transition superficial velocities to pseudo-homogeneous and heterogeneous flow regimes, u_t1 and u_t2, decreased with increasing liquid viscosity below a critical viscosity and converged to a certain value above that viscosity. However, the correlations from the literatures could not make a reasonable estimation of the transition superficial velocities because they did not consider the possible transition to a pseudo-homogeneous flow regime. Therefore, the two transition points should be predicted separately.     

A novel algorithm for solving population balance equations: the parallel parent and daughter classes. Derivation, analysis and testing

A novel numerical method, the parallel parent and daughter classes (PPDC) technique, for solving population balance equations (PBEs) is presented in this paper. In many practical applications, the PBE of particles under investigation is coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, the PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to an additional computational load. The computational cost becomes intractable when techniques such as methods of classes (CM) or Monte Carlo method are used. Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) are accurate and require a relatively low additional computational cost when applied to CFD. The PPDC is shown to be as accurate as QMOM and DQMOM, and even more accurate in some cases, when the same number of classes is used. In the present work, the PPDC technique has been derived and tested. This technique can be used for solving a wide class of problems involving PBE such as polymerization, aerosol dynamics, bubble columns, etc. Numerical simulations have been carried out on aggregation processes with different kernels and on simultaneous aggregation and breakage processes. The numerical predictions are compared either with analytical solutions, when available, or with the numerical solutions obtained by methods of classes.     

Effect of spent grains on flow regime transition in bubble column

It is well known that two main flow regimes are present in bubble columns, being the evaluation of transition between homogeneous and heterogeneous regimes of crucial importance for reactor design. For air–water systems, several models have been satisfactorily proposed to explain this phenomenon. However when gas–liquid–solids systems are considered, solid particles influence on regime transition is not yet clear, in spite of the amount of research developed over the past years.The objective of this work is to evaluate the effect of a specific solid phase – spent grains – on homogeneous regime stability and regime transition. Spent grains are cellulose-based particles that have been used to immobilize cells on biotechnology process. These particles are wettable and have a density close to water and its influence on bubble column reactors is particularly important in order to establish the limits were both regimes prevail.A cylindrical Plexiglax BC of 18 L volume was used with air, water and spent grains at different concentrations (0–20% (wt._{WET BASIS}/vol.)) as gas, liquid and solid phases. Regime transition was determined according to the drift-flux and slip speed concept.It was found that at studied concentrations of spent grains, critical gas hold-up decreases as solids concentration increases. At the highest solids concentration and lowest gas flow rates no fluidization of the solid phase was observed. It is believed that the critical hold-up decrease was mainly due to bubble coalescence, as larger bubbles were observed when heterogeneous regime was present. This coalescence may be caused by the non-uniform distribution of solid phase on the column and the interaction of spent grains with bubbles in the liquid–gas interface     

Explorations on the multi-scale flow structure and stability condition in bubble columns

Physical understanding of heterogeneous flow structure is of crucial importance for modelling and simulation of gas-liquid systems. This article presents a review and report of recent progress in our group on exploratory application of the variational (analytical) multi-scale approach to gas-liquid systems. The work features the closure of a hydrodynamic model with the incorporation of a stability condition reflecting the compromise between the dominant mechanisms in the system. A dual-bubble-size (DBS) model is proposed to approximate the heterogeneous structure of gas-liquid systems based on a single-bubble-size (SBS) model previously established. Reasonable variation of the gas holdup and the composition of the two bubble species with operating conditions have been calculated and the regime transition can therefore be reasonably predicted for air-water system, suggesting that stability condition may provide an insightful concept to explain the general tendencies in gas-liquid systems out of their hydrodynamic complexity, and to give simple models of their overall behaviors. Of course, the diversity of the correlations for drag force and minimum bubble size and the sensitivity of the model predictions to these correlations may suggest the necessity to clarify further the essential and robust results in the current model and to reduce the uncertainties involved.     

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.     

Stability analysis of bubble columns: Predictions for regime transition

A criterion for the transition from the homogeneous to the heterogeneous regime in a bubble column is developed based on the theory of linear stability. Hydrodynamics of bubble column is described by two-fluid model incorporating the interphase forces like drag force and added mass force. Added mass force affects the hydrodynamics of gas-liquid flows significantly and is formulated by taking into account the bubble deformation. A proper understanding of the nature of gas-liquid interface (clean or contaminated) is desired for the reliable predictions of the added mass coefficient. Data from the literature on the transition in bubble columns is critically analyzed. A good agreement has been obtained between the experimental transition gas hold-up and the predictions of the same obtained by the theory developed in this work.     

Effect of surfactant on homogeneous regime stability in bubble column

Experiments were carried out to investigate the effect a surface active agent on homogeneous-heterogeneous flow regime transition in a laboratory scale bubble column. Air and water with various amount of CaCl₂ were the phases. The (voidage e) - (gas flow rate q) dependence was measured. The critical point where the homogeneous regime loses stability and the transition begins was evaluated by several methods. These methods are based on the slip speed concept and the drift flux model. The critical values of voidage and gas flow rate were taken as the quantitative measures of the homogeneous regime stability. They were plotted against the surfactant concentration. It was found that the surfactant has a dual effect on both the voidage and the regime transition: low concentration stabilizes and larger concentration destabilizes the homogeneous bubble bed. At present, we do not have an explanation to these observations. Possible physical mechanisms of the surfactant effect are expected to be revealed by further experiments, which are currently under way.     

On the measurements of regime transition in high-pressure bubble columns

Two approaches are adopted in this study to identify the flow transition in a bubble column from the homogeneous regime to the heterogeneous regime at pressures up to 15.2 MPa and temperatures up to 78°C. These approaches, which yield essentially identical results, include those based on the standard deviation of the pressure fluctuation and the drift flux model. The experimental results obtained indicate that the regime transition velocity is delayed when the system pressure and temperature increase. The correlation proposed for the transition velocity by Wilkinson et al. (1992) predicts the present results to a reasonable extent, provided that the experimentally measured values for the physical properties of the fluids are used in the correlation. It is clear, however, that an improved correlation or model is needed for a quantitative account of the transition velocity for high-pressure bubble column systems with different gas distributing capability.     

Simulation of polydisperse multiphase systems using population balances and example application to bubbly flows

In this work the relationship between multiphase computational fluid dynamics models and population balance models is illustrated by deriving the main governing equations from the generalized population balance equation. The resulting set of equations, consisting of the well known two-fluid model coupled with a bivariate population balance model, is then implemented in the CFD code OpenFOAM. The implementation is used to simulate a particular multiphase problem: bubbly flow in a rectangular column. Results show that, although the different mesoscale models for drag force, coalescence, breakup and mass transfer, can be improved, the agreement with experiments is nevertheless good. Moreover, although the problem investigated is quite complex, as the evolution of bubbles is solved in real-space, time and phase-space (i.e. bubble size and composition) the resulting computational costs are reasonable. This is due to the fact that the bivariate population balance model is solved here with the so-called conditional quadrature method of moments, that very efficiently deals with these problems. The overall approach is demonstrated to be efficient and robust and is therefore suitable for the simulation of many polydisperse multiphase flows.     

Numerical simulation of two-phase flow with bubble break-up and coalescence coupled with population balance modeling

A computational fluid dynamic (CFD) model was developed with an improved source term based on previous work by Hagesaether et al. [1] for bubble break up and bubble coalescence to carry out numerical prediction of number density of different bubble class in turbulent dispersed flow. The numerical prediction was based on two fluid models, using the Eulerian–Eulerian approach where the liquid phase was treated as a continuum and the gas phase (bubbles) was considered as a dispersed phase. Bubble–bubble interactions, such as breakage due to turbulence and coalescence due to the combined effect of turbulence and laminar shear were considered. The result shows that the radial distributions of number densities of lower bubble classes are more than its higher counterpart. The result also shows that the Sauter mean diameter increases with the increase of height up to 1 m and then become steady. Simulated results are found to be in good agreement with the experimental data.     

The effect of sparger geometry on gas holdup and regime transition points in a bubble column equipped with perforated plate spargers

This paper investigates the effect of sparger geometry on flow regime of a bubble column. The experiments presented in this study were performed under atmospheric pressure with water/air in a cylindrical Plexiglas^® column of 33.0 cm i.d. and 3.0 m height. Three different perforated plate spargers were employed. Hole diameter was varied in the range of 1–3 mm, while the free area was 1.0%.The theory of linear stability is used for the prediction of regime transitions in the bubble column and a comparison has been presented between the predictions and the experimental observations. A good agreement between the predictions and the experimental values of transition gas holdup has been obtained.In addition, the data from the literature has been analyzed. Experimental values of transition gas holdups and predictions by the theory of linear stability have been compared with those of literature.A correlation based on dimensionless numbers (Archimedes, Froude, Eötvös and Weber) and the group (d_o/D_C) for the prediction of gas holdup in homogeneous regime is proposed. The average error between the correlation predictions and experimental values remains under ±10%.The proposed correlation is compared with the published data and found to be in fairly good agreement.