Computational Fluid Dynamics Population Balance Model Research Articles

Characteristics of gas–liquid two-phase flow in a stirred tank with double-layer punched impeller

This study investigates the gas–liquid two-phase flow characteristics in a stirred tank equipped with a double-layer punched impeller. Numerical simulations are conducted to analyze flow dynamics, gas holdup, bubble sizes, and distributions under various operational conditions. The results show a high degree of agreement between experimental and simulated power values and gas holdup distributions, validating the reliability of the computational fluid dynamics–population balance model coupling approach. The combination of the punched four-inclined-blade up-pumping turbine and the punched Rushton impeller exhibits excellent bubble dispersion characteristics, with overall small bubble sizes. Increasing the rotational speed can enhance turbulence within the flow field and accelerate the liquid phase velocity, which facilitates gas diffusion and improves gas–liquid mixing efficiency. Additionally, higher rotational speed further intensifies the shear effect of the punched impeller, resulting in a reduction in average bubble size.

Mass transfer and bubble hydrodynamics in stirred tank with multiple properties fluid via a CFD‐PBM method

AbstractThe performance of a stirred bioreactor was evaluated in this study in terms of the bubble hydrodynamics and the mass transfer efficiency, using a non‐viscous Newtonian fluid of water, a viscous non‐Newtonian fluid of xanthan, and a viscous non‐Newtonian fluids of xanthan with dispersed soybean powder, respectively. The computational fluid dynamics–population balance model (CFD‐PBM) method coupled with the viscosity model and the mass transfer model is established to simulate the gas–liquid mass transfer process and bubble size distribution in the stirred bioreactor. The results demonstrate that the rheological properties of the fluid play an important role in determining the gas holdup, the mass transfer efficiency, and the bubble size distribution. Viscosity of the fluid exhibits a negative impact on gas–liquid mass transfer rate and gas holdup. Moreover, by properly adjusting the operating conditions such as the stirrer speed, it is possible to modulate the gas dispersion and mass transfer rate in the reactor.

CFD-PBM coupled modeling of the liquid-liquid dispersion characteristics and structure optimization for Kenics static mixer

Kenics static mixers (KSM) are extensively used in industrial mixing–reaction processes by virtue of high mixing efficiency, low power homogenization and easy continuous production. Resolving liquid droplet size and its distribution and thus revealing the dispersion characteristics are of great significance for structural optimization and process intensification in the KSM. In this work, a computational fluid dynamics–population balance model (CFD–PBM) coupled method is employed to systematically investigate the effects of operating conditions and structural parameters of KSM on droplet size and its distribution, to further reveal the liquid–liquid dispersion characteristics. Results indicate that higher Reynolds numbers or higher dispersed phase volume fractions increase energy dissipation, reducing Sauter mean diameter (SMD) of dispersed phase droplets and with a shift in droplet size distribution (DSD) towards smaller size. Smaller aspect ratios, greater blade twist and assembly angles amplify shear rate, leading to smaller droplet size and a narrower DSD in the smaller range. The degree of impact exerted by the aspect ratio is notably greater. Notably, mixing elements with different spin enhance shear and stretching efficiency. Compared to the same spin, SMD becomes 3.7–5.8 times smaller in the smaller size range with a significantly narrower distribution. Taking into account the pressure drop and efficiency in a comprehensive manner, optimized structural parameters for the mixing element encompass an aspect ratio of 1–1.5, a blade twist angle of 180°, an assembly angle of 90°, and interlaced assembly of adjacent elements with different spin. This work provides vital theoretical underpinning and future reference for enhancing KSM performance.

Investigation of hydrodynamic performance in a staggered multistage internal airlift loop reactor

The multistage internal airlift loop reactor (MIALR) has shown promising application prospects in gas–liquid–solid reaction systems. However, traditional MIALRs have a global circulation with strong interstage liquid-phase exchange. This paper proposes a staggered multistage internal airlift loop reactor (SMIALR) that incorporates special guide elements to create a staggered flow. Both experiments and computational fluid dynamics-population balance model simulations were conducted to investigate the hydrodynamic performances of MIALR and SMIALR. The results demonstrate that SMIALR exhibits a local circulation at each stage. Bubbles have a longer residence time in SMIALR, resulting in a 28.35%–55.54% increase in gas holdup and a 7.27%–13.69% increase in volumetric mass transfer coefficient. The gas–liquid mass transfer coefficient of SMIALR was improved by increasing the gas–liquid interfacial area. Additionally, the radial distribution of solids was found to be more uniform. This study offers insights for optimizing MIALR and provides a theoretical foundation for the design and scale-up of SMIALR.

Study on Dynamic Characteristics of Magnetic Coagulation of Fe-Based Fine Particles in Iron and Steel Industry

Fine dust, represented by Fe-based fine particles and emitted from the production process of the iron and steel industry, is the primary factor causing many diseases represented by industrial pneumoconiosis, and ultra-low dust emission has always been a thorny problem to be solved urgently. To explore the magnetic coagulation effect of Fe-based fine particles in the magnetic field when removing them from industrial flue gas by the magnetic field effect in the iron and steel industry, using FLUENT software, magnetic dipole force was added between particles through user defined function (UDF) based on the computational fluid dynamics-discrete phase model (CFD-DPM) method so that the collision process of particles was then equivalent to their mutual trapping process. Next, the effects of particle size, particle volume fraction, external magnetic field strength, and particle magnetic susceptibility on the magnetic coagulation process were comprehensively studied. Meanwhile, the proton balance equation (PBE) was solved using the partition method on the basis of the computational fluid dynamics-population balance model (CFD-PBM) to compare the coagulation removal effect under random and aligned orientations of magnetic dipoles, respectively. The results showed that the magnetic coagulation strength under the random orientation of magnetic dipoles was greater than that under the aligned orientation. When the particle size of Fe-based fine particles increased from 0.5 μm to 1.5 μm, the magnetic coagulation coefficient decreased from 0.5414 to 0.2882, and the difference in the removal efficiency under the two different orientations of magnetic dipoles became smaller. When the particle volume fraction increased from 0.01 to 0.03, the magnetic coagulation coefficient increased from 0.2353 to 0.5061, and the difference in the removal efficiency under two orientations was enlarged. When the applied external magnetic field strength increased from 0.5 T to 1.0 T, the magnetic coagulation coefficient increased from 0.3940 to 0.5288, and the magnetic susceptibility increased from 0.0250 to 0.0500, the coagulation coefficient increased from 0.3940 to 0.5288, and the difference under two orientations basically stayed unchanged.

Study on the Numerical Model of Gas–Liquid Dispersion Driven by a Coaxial Mixer

As the flow basis for gas–liquid mixing, turbulence plays an important role in the movement of bubbles in a stirred tank. Motivated by the fact that the current models in the framework of computational fluid dynamics-population balance model lack an adequate assessment of the turbulence effects, the coaxial mixer was chosen as the subject of study, and different turbulence models, drag models, and breakup and coalescence models were compared and optimized. The results showed that the RNG k-ε model better characterized the turbulent scale and achieved a higher exactness for the gas-phase concentration in the axial direction. The Luo–Luo model modified based on the eddy capture mechanism could effectively describe the coalescence and breakup processes of bubbles. Additionally, in the presence of strong turbulence, the Ishii & Zuber model and the Brucato model outperformed other models in predicting gas dispersion and flow structure but still failed to capture the peak gas holdup. By analyzing the available data, a new correlation was proposed to elucidate the increase in drag coefficient caused by turbulence, which significantly improved the prediction accuracy of gas holdup.

Modelling Flocculation in a Thickener Feedwell Using a Coupled Computational Fluid Dynamics–Population Balance Model

The flocculation that takes place in the central feedwell of the thickener plays a crucial role in the coal-slurry thickening process, which is not only complex but also largely influenced by the flow characteristics. A coupled computational fluid dynamics–population balance model (CFD–PBM) was used to model the complex flocculation-thickening behaviour in an industrial-scale gravity thickener. The initialisation parameters of the inlet flow were obtained through self-designed image-recognition experiments, and then the effects of different types of conical deflectors on the floc distribution were simulated and analysed using them. The results showed that, under the condition that the angle of the conical deflector’s sides in the vertical plane was known, a reasonable increase in the height of the bottom surface could reduce the annular spanwise vortices at the underflow of the feedwell, thereby avoiding the erosion of the inlet flow and the annular spanwise vortex on the floc deposition layer. However, excessive height on the part of the conical deflector could affect the flocculation effect of solid particles. For the same central feedwell size of the thickener as in the simulation, the best flocculation effect was achieved at an angle of α = 24° in the vertical plane of the conical deflector. Turbulence regulation of the conical deflector promotes the aggregation of fine particles in the fluid of the feedwell, providing a new method for the intensification of the flocculation-concentration process.

CFD‐PBM simulation of the column flotation unit of FCMC: Importance of gas–liquid interphase forces models

AbstractThe cyclonic micro‐bubble flotation column (FCMC) is an efficient flotation device for the separation of fine minerals, but its mechanisms are rarely studied using computational fluid dynamics (CFD). This paper reports the air–water two‐phase computational fluid dynamics‐population balance model (CFD‐PBM) simulations for the column flotation unit of an FCMC. The shear stress transport (SST) k‐ω model with curvature correction (CC) is used to simulate turbulence effects. Then, the interphase forces models considering bubble size distribution are selected according to the experimental data in a bubble column, which is in analogy to the column flotation unit of the FCMC. Finally, the optimal combination of interphase forces models (i.e., the Ishii–Zuber drag force model, the Hosokawa–Frank wall lubrication force model, and the Lopez de Bertodano turbulent dispersion force model) is applied to simulate an FCMC with a superficial gas velocity of 0.0144 m/s. The results show that the CFD‐PBM simulation can achieve a relative error of 9.09% for gas volume fraction and −5.45% for bubble rising velocity, indicating the reliability of the selected combination of interphase forces models.

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.

Emulsions Using a Vortex-Based Cavitation Device: Influence of Number of Passes, Pressure Drop, and Device Scale on Droplet Size Distributions.

Liquid-liquid emulsions are used in a variety of industry sectors, including personal care, home care, food, and nutrition. The development of compact and modular systems and devices for creating emulsions with desired droplet size distribution (DSD) is becoming increasingly important. In this work, we have shown use of vortex-based cavitation devices for producing emulsions at nominal flow rate of 1 LPM and 20 LPM. We present new experimental results providing quantitative information on influence of multiple passes through the vortex based hydrodynamic cavitation (HC) device, type of oil and device scale on the breakage process and resulting DSDs. Multiple pass experiments were performed for generating oil-in-water emulsions containing 5 and 15% of oil. Rapeseed oil (RO) and tetrachloroethylene (TCE) were used as oil phases with densities of 915 and 1620 kg/m3, respectively. The effect of pressure drop across the HC device in the range of 50-250 kPa on DSD was examined. The HC device was shown to exhibit significant higher efficiency compared to alternative emulsion making devices (i.e., homogenizers, venturi, and orifice-based HC devices), and the Sauter mean drop size was found to reduce from 66 μm to less than 2 μm after about 50 passes in all the device scales. The DSD of the RO-water system showed a bimodal nature, whereas monomodal DSD was found for TCE-water system. Preliminary simulations using the computational fluid dynamics-population balance model (CFD-PBM) models developed in the previous work indicated the inadequacy of developed models to capture the influence of cavitation on DSDs. By carrying out Hinze scale analysis of bimodal DSD, we for the first time showed the existence of two different mechanisms (one based on conventional turbulent shear and the other based on collapsing cavities) of droplet breakage in HC devices. The order of magnitude of turbulence energy dissipation rates generated due to collapsing cavity estimated using Hinze scale analysis showed good agreement with the values reported from cavity dynamics models. The presented experimental results and analysis will be useful for researchers and engineers interested in developing computational models and compact devices for producing emulsions of the desired DSD.

Investigation of the Role of Impeller Structural Parameters on Liquid–Liquid Mixing Characteristics in Stirred Tanks

Liquid-liquid mixings in stirred tanks are commonly found in many industries. In this study, we performed computational fluid dynamics (CFD) modeling and simulation to investigate the liquid-liquid mixing behavior. Furthermore, the population balance model (PBM) was used to characterize the droplet size distribution. The PBM model parameters were calibrated using the experimental data of droplet sizes at different agitation speeds. Additionally, we employed the steady-state Sauter mean droplet size to validate the developed CFD-PBM coupled model at different dispersion phase holdups. Then, the validated CFD-PBM coupled model was employed to evaluate the role of impeller structural parameters on the liquid-liquid mixing efficiency based on a user-defined mixing index. It was found that the position of impellers significantly affects the mixing efficiency, and an increase in stirring speed and the number of impellers improved the mixing efficiency.

CFD-PBM modelling of tailings flocculation in a lab-scale gravity thickener

Flocculation occurring in a feedwell plays a critical role in tailings slurry thickening process, which is complicated and significantly influenced by flow characteristics. This work presents a numerical approach to explore the effect of flow characteristics on flocculation performance. It combines an aggregation kernel and a breakage kernel, used to describe the polymer-bridging flocculation kinetics, with a conventional Computational Fluid Dynamics-Population Balance Model (CFD-PBM) coupling to model the complex flocculation-thickening behavior in a lab-scale gravity thickener. The solid-liquid phase interaction is described by an Euler-Euler approach with a modified Schiller-Naumann drag model. The turbulence of liquid phase is resolved by the RNG k-ε turbulence model, while the solid kinematic eddy viscosity is described by a dispersed phase zero equation model. The capability of this proposed model is validated by a good agreement between experimental and predicted results in terms of single-liquid velocity and floc size distribution. The momentum and turbulence dissipation rates are investigated in and around the feedwell over a wide range of feed velocities, showing that the momentum and turbulence dissipation rates have a positive correlation with feed velocity. The momentum and turbulence dissipation rates decrease with the increase in scaled depth in the feedwell. The formation of large vortexes in the feedwell may cause a locally low turbulence dissipation rate. A reasonable increase of feed velocity favours the flocculation, however, an excessive feed velocity can cause a decrease in mean floc size. Modelling tailings flocculation is of great significance for understanding the flocculation behavior and revealing the effect of flow characteristics on flocculation performance.

Numerical and Orthogonal Study on Optimization Analysis of Structure Parameters of Bubble Breaker for Electrical Submersible Pump System

Abstract Under slug flow conditions, electrical submersible pumps (ESPs) show a low efficiency due to Taylor bubbles, which cause pressure surging and gas pockets and the further deterioration of pressure boosting ability. In this study, a novel downhole bubble breaker is designed for mitigating the impact in ESP under slug flow conditions. The computational fluid dynamics-population balance model (CFD-PBM) coupled approach was used to calculate the bubble breaker's average bubble diameter to evaluate its efficiency. Meanwhile, experimental studies were conducted and compared with numerical results. Also, matlab and DIP-image technology was used to calculate the bubble diameter. Compared with experimental results, the simulation results agree well. Furthermore, the novel bubble breaker's performance was studied by orthogonal approach. The best result of range analysis is A2B3C4D1E4 (α = 30 deg, L = 300 mm, R = 2:1, vsg = 0.2 m/s, and vsl = 0.08 m/s), and sensitively analysis results present that the range of impact intensity are A (inlet angle) > E (superficial gas velocity) > B (total length) > D (superficial liquid velocity) > C (ratio of the gas–liquid channel). The optimal structure's bubble diameters are all less than that of the original structure, with a superficial gas velocity range of 0.2–0.6 m/s. The downstream bubble diameter of the optimal bubble is about 31.6% lower than the original structure at the maximum value point.

Effect of feed solid concentration on tailings slurry flocculation in a thickener by a coupled CFD-PBM modelling approach

The flocculation technique has been proven fruitful for separating beneficiation wastewater from tailings slurry in the thickening process. The tailings flocculation is significantly affected by the feed solid concentration and flow characteristics, however, it is difficult to capture the local solid concentration and the resulting variation in flow characteristics by a conventional experimental approach. This paper thus employed a coupled Computational Fluid Dynamics-Population Balance Model (CFD-PBM) modelling approach to explore the effect of feed solid concentration on the flocculation performance in and around the thickener feedwell. The flocculation kinetics induced by polymer-bridging was incorporated into the PBM framework by aggregation and breakage kernels. The proposed model was validated against the experimental results in terms of liquid velocity and floc size distribution. The designed inner feedwell with deflectors has been demonstrated to have better flocculation performance. The effects of the feed solid concentration on the flow characteristics (e.g. effective viscosity of the slurry, and turbulence dissipation rate) and floc size distribution have been investigated. Results mainly show that the local solid concentration and effective viscosity of the slurry increase with feed solid concentration, while the mean floc size first increases and then decreases. A moderate feed solid concentration facilitates the flocculation performance. This study helps to understand the essential effect of feed solid concentration on the flow characteristics and flocculation performance, thereby guiding the flocculation-thickening process to achieve the sustainable use of beneficiation wastewater and mineral resources.

Parametric numerical study and optimization of mass transfer and bubble size distribution in a gas-liquid stirred tank bioreactor equipped with Rushton turbine using computational fluid dynamics

Abstract Designing and optimizing a bioreactor can be an especially challenging process. Computational modelling is an effective tool to investigate the effects of various operating parameters on bioreactor performance and identify the optimum ones. In this work, a computational fluid dynamics-population balance model (CFD-PBM) was developed to elucidate the effect of different geometrical and operating parameters on the hydrodynamics and mass transfer coefficient of a batch stirred tank bioreactor. The validated model was projected to predict the effect of different parameters including the gas flow rate, the impeller off-bottom clearance, the number of agitator blades, and rotational speed of the impeller on the velocity profiles, air volume fraction, bubble size distribution, and the local gas mass transfer coefficient (K l a) in the bioreactor. Air bubble breakup and coalescence phenomena were considered in all simulations. Factorial experimental design approach was employed to statistically investigate the impacts of the aforementioned operating and geometrical parameters on K l a and bubble size distribution in the bioreactor in order to determine the most significant parameters. This can give an essential insight into the most impactful factors when it comes to designing and scaling up a bioreactor.

Numerical Simulation of the Oil Droplet Size Distribution Considering Coalescence and Breakup in Aero-Engine Bearing Chamber

In order to improve the inadequacy of the current research on oil droplet size distribution in aero-engine bearing chamber, the influence of oil droplet size distribution with the oil droplets coalescence and breakup is analyzed by using the computational fluid dynamics-population balance model (CFD-PBM). The Euler–Euler equation and population balance equation are solved in Fluent software. The distribution of the gas phase velocity field and the volume fraction of different oil droplet diameter at different time are obtained in the bearing chamber. Then, the influence of different initial oil droplet diameter, air, and oil mass flow on oil droplet size distribution is discussed. The result of numerical analysis is compared with the experiment in the literature to verify the feasibility and validity. The main results provide the following conclusions. At the initial stage, the coalescence of oil droplets plays a dominant role. Then, the breakup of larger diameter oil droplet appears. Finally, the oil droplet size distribution tends to be stable. The coalescence and breakup of oil droplet increases with the initial diameter of oil droplet and the air mass flow increasing, and the oil droplet size distribution changes significantly. With the oil mass flow increasing, the coalescence and breakup of oil droplet has little change and the variation of oil droplet size distribution is not obvious.

CFD-PB Modelling of Liquid–liquid Two-phase Flow in Pulsed Disc and Doughnut Column

ABSTRACT Computational fluid dynamics – population balance (CFD-PB) simulations are performed to predict Sauter mean drop diameter and dispersed phase holdup for liquid–liquid two-phase flow in a pulsed disc and doughnut column (PDDC). The continuity and momentum equations of the two-phases along with standard k–ε mixture model of turbulence and an appropriate model for interphase momentum exchange are solved to obtain temporal and spatial variations of dispersed phase holdup. Sauter mean drop diameter is obtained from Quadrature Method of Moments (QMOM)-based population balance model (PBM) which is solved along with the flow equations and takes into account coalescence and breakage of droplets. CFD-PB model is validated with the experimental data of Sauter mean drop diameter and dispersed phase holdup obtained for different pulsing velocities. Validated model is used to have detailed insights into the two-phase hydrodynamics prevalent inside the PDDC and to test the efficacy of scale-up methodology reported in literature.

Gas–Liquid Two-Phase Performance of Centrifugal Pump Under Bubble Inflow Based on Computational Fluid Dynamics–Population Balance Model Coupling Model

Abstract In this study, a numerical simulation method based on Eulerian–Eulerian model and population balance model (PBM) (i.e., computational fluid dynamics (CFD)–PBM coupling model) was developed to investigate the gas–liquid two-phase performance of centrifugal pump under bubble inflow. The realizable k–ε model turbulence model was implemented in ansysfluent solver. The air and water were employed as the working fluids, which was consistent with the experiment. The water head and pressure increment obtained by the experiment were used to validate the numerical method. The results show that the CFD–PBM coupling model is superior to the Eulerian–Eulerian model, particularly in the “surging” conditions. Using the CFD–PBM coupling model, the influences of parameters, such as inlet gas volume fraction, liquid phase flowrate, and rotational speed, on the head and efficiency of the centrifugal pump were investigated. Under the design condition, when the inlet gas volume fraction increases from 3% to 5%, the bubbles form air mass and stagnate in the impeller channel. The stagnated air mass can hardly be discharged with the liquid phase. Thus, the pump head drops suddenly, i.e., the surging occurs. The two-phase performance of centrifugal pump can be improved under the surging condition by increasing the liquid flowrate and the rotational speed to a certain value. The results contribute to an alternative simulation method to investigate the characteristics of bubble flow in pump and shed new lights on the understanding of the performance of centrifugal pumps under two-phase flow conditions.

Numerical Simulation of Gas–Liquid Two-Phase Flow Characteristics of Centrifugal Pump Based on the CFD–PBM

This work seeks to apply the computational fluid dynamics–population balance model (CFD–PBM) to investigate the gas distribution and flow mechanism in the gas–liquid two-phase flow of a centrifugal pump. The findings show that the numerical simulation accurately captures the bubble distribution characteristics in the process of coalescence and breakage evolution. In addition, comparing the CFD–PBM with the Double Euler, the hydraulic head of the pump are similar, but the efficiency using the Double Euler is much higher—even close to single-phase. This is in contrast to previous experimental research. Then, the unsteady flow usually led to the formation of bubbles with larger diameters especially where vortices existed. In addition, the rotor–stator interaction was a main reason for bubble formation. Generally, it was observed that the coalescence rate was greater than the breakage rate; thus, the coalescence rate decreased until it equaled the breakage rate. Thereafter, the average diameter of the bubble in each part tended to be stable during the process of bubble evolution. Finally, the average diameter of bubbles seemed to increase from inlet to outlet. The results of this study may not only enhance the gas–liquid two-phase internal flow theory of centrifugal pumps, but also can serve as a benchmark for optimizations of reliable operation of hydraulic pumps under gas–liquid two-phase flow conditions.

Simulation Study on Gas Holdup of Large and Small Bubbles in a High Pressure Gas–Liquid Bubble Column

The computational fluid dynamics-population balance model (CFD-PBM) has been presented and used to evaluate the bubble behavior in a large-scale high pressure bubble column with an inner diameter of 300 mm and a height of 6600 mm. In the heterogeneous flow regime, bubbles can be divided into “large bubbles” and “small bubbles” by a critical bubble diameter dc. In this study, large and small bubbles were classified according to different slopes in the experiment only by the method of dynamic gas disengagement, the critical bubble diameter was determined to be 7 mm by the experimental results and the simulation values. In addition, the effects of superficial gas velocity, operating pressure, surface tension and viscosity on gas holdup of large and small bubbles in gas–liquid two-phase flow were investigated using a CFD-PBM coupling model. The results show that the gas holdup of small and large bubbles increases rapidly with the increase of superficial gas velocity. With the increase of pressure, the gas holdup of small bubbles increases significantly, and the gas holdup of large bubbles increase slightly. Under the same superficial gas velocity, the gas holdup of large bubbles increases with the decrease of viscosity and the decrease of surface tension, but the gas holdup of small bubbles increases significantly. The simulated values of the coupled model have a good agreement with the experimental values, which can be applied to the parameter estimation of the high pressure bubble column system.