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.     

Mathematical modeling of gas-liquid mass transfer rate in bubble columns operated in the heterogeneous regime

In this paper, a multi-scale approach is followed to study gas-liquid mass transfer in bubble columns. First, a single bubble of equivalent diameter d is considered. Its morphology and its gas to liquid relative velocity are related to the bubble diameter through the use of known correlations. Then, the gas-liquid mass transfer between the bubble and the surrounding liquid is studied theoretically. An equation describing the transport of the transferred species in the viscous boundary layer around the bubble is solved. In a second step, a bubble column of 6-10 m height is studied experimentally. The gas phase in the column is characterized experimentally by means of a gammametric technique. Finally, the two studies are linked, yielding a 1D mathematical model able to predict the gas-liquid mass transfer rate in a bubble column operated in the heterogeneous regime.     

Validation of bubble breakage, coalescence and mass transfer models for gas-liquid dispersion in agitated vessel

Bubble breakage and coalescence phenomena and multicomponent gas-liquid mass transfer were studied in a Rushton turbine agitated vessel. Local bubble size distributions (BSD) were measured from air-tap water system at several agitation conditions with capillary suction probe (CSP) technique. The CSP was compared to the digital imaging (DI) and phase Doppler anemometry (PDA) techniques in a stirred vessel. The volumetric BSDs between the CSP and DI were in agreement, but number BSDs showed notable deviation. The limitations of measurement techniques seem to be the main reason.A multiblock stirred tank model with discretized population balances for bubbles and two-film Maxwell-Stefan multicomponent mass transfer between gas and liquid was created for the agitated vessel. The model considers local mass transfer conditions in the vessel and is simple enough for the mathematical optimization of unknown model parameters. Unknown parameters in the mechanistic bubble breakage and coalescence models were fitted against measured local BSDs. After this, a parameter in the liquid film mass transfer correlation was adjusted against absorption and desorption experiments of oxygen. Local gas-liquid mass transfer areas were calculated from the population balance model. The simulations with the validated models show good agreement against experiments. On the other hand, the fitted parameters deviate from the theoretical values, which emphasizes the need of model validation against accurate experiments. Due to their fundamental character and the validation process, the fitted models seem to be useful tools for the design and scale-up of agitated gas-liquid reactors.     

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.     

Interfacial area and volumetric mass transfer coefficient in a bubble reactor at elevated pressures

The present study deals with the pressure effects on mass transfer parameters within a bubble reactor operating at pressures up to . The gas-liquid systems are N₂/CO₂-aqueous solution of Na₂CO₃-NaHCO₃ and N₂/CO₂-aqueous solution of NaOH. A sintered powder plate is used as a gas distributor. Three parameters characterizing the mass transfer are identified and investigated with respect to pressure: the gas-liquid interfacial area a, the volumetric liquid side mass transfer coefficient k_La and the volumetric gas side mass transfer coefficient k_Ga. The gas-liquid absorption with chemical reaction is used and the mass transfer parameters are determined by using the model reaction between CO₂ and the aqueous solutions of Na₂CO₃-NaHCO₃ and NaOH. For a given gas mass flow rate, the interfacial area as well as the volumetric liquid mass transfer coefficient decrease with increasing operating pressure. However, for a given pressure, a and k_La increase with increasing gas mass flow rates. The mass transfer coefficient k_L is independent of pressure. Furthermore, the pressure increase results in a decrease of k_G and k_Ga for a given gas mass flow rate. The values of the interfacial area, which are obtained from both chemical systems are found to be different. These discrepancies are attributed to the choice of the liquid system in the absorption reaction model.     

Population balance modelling for bubbly flows with heat and mass transfer

Population balance equations combined with a three-dimensional two-fluid model are employed to predict bubbly flows with the presence of heat and mass transfer processes. Subcooled boiling flow belongs to this specific category of bubbly flows is considered. The MUSIG (MUltiple-SIze-Group) model implemented in CFX4.4 is further developed to account for the wall nucleation and condensation in the subcooled boiling regime. Comparison of model predictions against local measurements near the test channel exit is made for the radial distribution of the bubble Sauter diameter, void fraction, interfacial area concentration and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Additional comparison was also performed against existing boiling model in CFX4.4 and the modified model developed in our previous work (Int. J. Heat Mass Transfer 45 (2002) 1197). Good agreement is better achieved with the local radial bubble Sauter diameter, void fraction, interfacial area concentration and liquid velocity profiles against measurements using the newly formulated MUSIG boiling model over the simpler boiling models. However, significant weakness of the model is still evidenced in the prediction of the vapour velocity. Work is in progress to circumvent the deficiency of the model by the consideration of additional momentum equations or an algebraic slip model to account for bubble separation.     

偏氟乙烯乳液聚合的气液传质效果影响因素研究

推导了表征气液传质效果的K值计算方法,并研究了聚合釜装填量、搅拌桨型和转速等因素对气液传质效果的影响。结果表明,随着聚合釜装填量和搅拌转速的增大,K值增大,气液传质效果提高;在搅拌转速大于500r/min的情况下,桨型采用上层二叶平桨、下层二叶斜桨的方式明显比单层的二叶斜桨或二叶平桨的K值大,气液传质效果好。确定了5L釜偏氟乙烯(VDF)乳液聚合必需的良好搅拌效果的条件:装填量3L,搅拌转速700r/min,桨型采用上层二叶平桨和下层二叶斜桨;VDF聚合速率达到115g/(L·h),合成出固体质量分数20%、粒径分布窄、稳定性好的聚偏氟乙烯(PVDF)乳液。     

Linear stability analysis of a two-fluid model for air-water bubble columns

Linear stability analysis is performed for the two-dimensional, two-fluid model for gas-liquid flow applied in our previous computational study of bubble columns [Monahan, S.M., Vitankar, V.S., Fox, R.O., 2005. CFD predictions for flow-regime transitions in bubble columns. A.I.Ch.E. Journal 51, 1897-1923]. The growth rate and the velocity of propagation for a small-amplitude disturbance wave are shown to be highly dependent on the wave number, the direction of propagation, and the two-fluid model parameters. Two types of vertical instabilities are identified: one corresponding to the classical analysis of Jackson [1963. The mechanics of fluidized beds. I: the stability of the state of uniform fluidization. Transactions of the Institution of Chemical Engineers 41, 13-21] for the one-dimensional model, and the other due to a second pair of roots to the characteristic equation of the linearized two-dimensional model. Numerical simulations keeping one type or the other of the roots stable (or unstable) show distinctly different dynamics and suggest that large-scale instabilities seen experimentally may be associated with the second type of instability. The latter leads to instability in the horizontal velocities and is associated with a positive lift coefficient in flows without mean shear in the presence of isotropic bubble-bubble interactions (i.e., “bubble pressure”). This instability is thus different than previously reported instabilities due to negative lift or cooperative/hindered rise.     

An improved dynamic combined model for evaluating the mass transfer performances in extraction columns

Dispersed phase droplet behavior research is very important for the design and scaling up of extraction columns. Recently, the droplet velocities at high holdup were found to be uniform, which means the conventional concept of forward mixing needs correction. The drop size distribution only influences the mass transfer coefficients and not the residence time distribution of droplets. In this work, an improved dynamic combined model considering the influence of drop size distribution has been developed, by which the axial mixing can be easily evaluated using a one-dimension search. A typical experimental system of 30% tributyl phosphate (TBP) (in kerosene)-nitric acid-water with interfacial tension of 0.00995 N/m was used to investigate the mass transfer performances in a coalescence-dispersion pulsed-sieve-plate extraction column (CDPSEC) with 150 mm in diameter. The two-point dynamic method was used to obtain the stimulus-response curves. With these results, the axial mixing in the CDPSEC was evaluated. The calculated results showed that the response curves could be predicted by the dynamic combined model with a deviation less than 0.001. This model has marked advantages over previous models in literature because of its accuracy, simple boundary conditions, and single parameter optimization.     

Theoretical modelling of the effect of surface active species on the mass transfer rates in bubble column reactors

The complex composition of the liquid media in bubble column reactors makes their understanding and theoretical modelling challenging. In this work we have studied the effect of surface tension and contaminants, salts, on the mass transfer rates from a theoretical point of view, looking for a deeper understanding on the effect of surface active species which usually reduce surface tension and modify bubble surface behaviour. The specific contact area is obtained using a population balance where the effect of the presence of contaminants is addressed by the proper theoretical closures for bubble coalescence efficiency, for partially and fully immobile surfaces, and bubble break-up. Meanwhile, the contribution of contaminants to the liquid-film resistance is implemented as function of the coverage of the surface of the bubbles. It was found that the degree of bubble surface coverage not only affects bubble coalescence but also their break-up. The ion strength defines bubbles stability and the critical Weber number can be predicted as function of ion strength. Furthermore, the mass transfer rates are function of the surface coverage by the electrolytes. The model was able to predict k_La taking into account the fact that the concentration profiles surrounding individual bubbles are not completely developed due to the presence of other bubbles, in agreement with previous results from the literature.     

The measurement of bubble diameter distributions and liquid side mass transfer coefficients in a gas-liquid agitated vessel using a real-time,high-speed image processing system

In this study, we report the measurement results of various spatial distributions, such as Sauter diameter, gas holdup ratio, and interface area per unit liquid volume, in a vessel using a real-time, high-speed image processing system developed by ourselves. We attempted to separate liquid side mass transfer coefficients, k L , from overall volumetric mass transfer coefficients, k L a, based on the results mentioned above. And we examined the relations between power consumption per unit volume, P v , and both k L and k L a in order to establish correlation equations of k L and k L a with P v , gas holdup ratio, gas superficial velocity, v s , and surface tension.     

Simultaneous measurement of flow pattern and mass transfer coefficient in bubble columns

Mass transfer studies were carried out in a bubble column using the chemical method. Catalytic oxidation of sodium sulfite was chosen for the studies and the corresponding specific rates of oxidation were obtained using a stirred cell. Laser Doppler anemometer (LDA) was used to measure the instantaneous velocities in the same stirred cell as well as in bubble columns (100 and i.d.). An efficient algorithm based on the multiresolution analysis of the velocity-time data using wavelets was used for the isolation of data belonging to the gas and liquid phases. Eddy isolation model was used for the characterization of the eddy motion including the estimation of the energy dissipation rate. Using the knowledge of eddy motion, a methodology was developed for the prediction of true mass transfer coefficient (k_L) in a stirred cell as well as in bubble columns. The predicted values of k_L have been compared with the experimental values obtained by the chemical method.     

The role of small bubbles in gas-liquid mass transfer in stirred vessels and assessment of a two-fraction model for noncoalescent or moderately viscous liquids

A two-gas-fraction model previously tested with gas-phase residence time distribution tests was used to interpret experimental transient curves during K_La determinations made with a conventional dynamic method. Large bubbles are modelled by an axial dispersion model, whereas small bubbles are considered as perfectly mixed. The use of this two-gas-fraction model leads to a significant improvement in the fitting of the experimental curves. Furthermore, the analysis carried out suggested that what is actually observed during an experiment carried out with the dynamic method is the mass transfer of large bubbles only—even in the presence of a significant fraction of small bubbles.     

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.     

Numerical study of condensing bubble in subcooled boiling flow using volume of fluid model

In this numerical study, the behavior of condensing bubble was investigated using the volume of fluid (VOF) model in the FLUENT code. In order to simulate the condensing bubble with the FLUENT code, the bubble condensation was modeled using the user-defined function (UDF). For the validation of the UDF of bubble condensation, the results of CFD simulation were compared with the results of a bubble condensation experiment performed in Seoul National University (SNU). Simulation results showed good agreements with the experimental data. Moreover, the fundamental behavior of the condensing bubble was investigated in various conditions. The effects of condensation on bubble behavior were analyzed by comparing the behavior of condensing bubbles with that of adiabatic bubbles. It was found that the behavior of the condensing bubble was different from that of the adiabatic bubble in many respects including the bubble shape, velocity, rise distance and moving trajectory.     

Four stages of the simultaneous mass and heat transfer during bubble formation and rise in a bubbly absorber

We studied nonisothermal absorption of a solvable gas from growing at an orifice and rising bubble when the concentration level of the absorbate in the absorbent is finite (finite dilution of absorbate approximation). It is shown that simultaneous heat and mass transfer at all stages of bubble growth and rise in a bubbly absorber can be described by a system of generalized equations of nonstationary convective diffusion and energy balance. Solutions of diffusion and energy balance equations are obtained in the exact analytical form. Coupled thermal effects during absorption and absorbate concentration level effect on the rate of mass transfer are investigated. It is found that the rate of mass transfer between a bubble and a fluid increases with the increase of the absorbate concentration level. The suggested approach is valid for high Peclet, Prandtl and Schmidt numbers. It is shown that for the positive dimensionless heat of absorption K thermal effects cause the increase of the mass transfer rate in comparison with the isothermal case. On the contrary, for negative K thermal effects cause the decrease of the mass transfer rate in comparison with the isothermal case. The latter effect becomes more pronounced with the increase of the concentration level of the absorbate in the absorbent. Theoretical results are consistent with the experiments of Kang et al. (Int. J. Refrigeration 25 (2002) 127) for absorption from ammonia gas bubbles rising in water and aqueous ammonia solutions.     

气液液三相体系中的气液传质特征

选取CO2-K2CO3/KHCO3为吸收体系,次氯酸钠为催化剂,甲苯、异戊醇为第2液相,应用Danckwerts图来同时确定液侧传质系数KL和界面面积a,通过实验研究了分散第2液相的加入对气液传质的影响。实验结果表明,随着分散相体积分数φ(1%—10%)的增大,或分散相形成的液滴直径的减小,以及传质组分在分散相和连续相中溶解度的比值m(即传质组分在实验体系的分配系数)增加,或在二相间的相对扩散系数增加时,可显著增强气液传质,为气液液三相体系的系统化研究提供了实验依据。     

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid-liquid extraction column.