Capturing coalescence and break-up processes in vertical gas–liquid flows: Assessment of population balance methods

Gas–liquid flows are commonly encountered in industrial flow systems. Numerical studies have been performed to assess the performances of different population balance approaches – direct quadrature method of moments (DQMOMs), average bubble number density (ABND) model and homogeneous MUlti-SIze-Group (MUSIG) model – in tracking the changes of gas void fraction and bubble size distribution under complex flow conditions and to validate the model predictions against experimental measurements from medium- and large-sized vertical pipes. Subject to different gas injection method and flow conditions, bubble size evolution exhibited a coalescence dominant trend in the medium-sized pipe; while bubble break-up was found to be dominant in large-sized pipe. The two experiments were therefore strategically selected for carrying out a thorough examination of existing population balance models in capturing the complicated behaviour of bubble coalescence and break-up. In general, predictions of all the different population balance approaches were in reasonable agreement with experimental data. More importantly, encouraging results have been obtained in adequately capturing the dynamical changes of bubbles size due to bubble interactions and transition from wall peak to core peak gas void fraction profiles. As a compromise between numerical accuracy and computational time, DQMOM has performed rather well in capturing the essential two-phase flow structures within the medium- and large-sized vertical pipes when compared to those of ABND and homogeneous MUSIG models. From a practical perspective, the ABND model may still be considered as a more viable approach for industrial applications of gas–liquid flow systems.     

Comparison and correction of the drop breakup models for stochastic dilute spray flow

Spray–gas interaction is common in many industrial applications that use a liquid jet injection system. Numerous liquid drops interact with the surrounding gas as they travel through the air. During such a travel, aerodynamic interaction between a drop and the surrounding gas flattens the drop and ultimately, breaks up the drop. The TAB (Taylor Analogy Breakup) model was proposed by O’Rourke and Amsden (1987) [6] for the KIVA spray code, but the use of this model has been controversial because the original paper that proposed this model has typographical errors. Another well-known drop breakup model, such as the DDB (Drop Deformation Breakup) model of Ibrahim et al. (1993) [8], has been widely used. However, although numerical solutions of the DDB model ostensibly make it appear superior to those of other previous breakup models, they contain errors that need to be amended. This paper aims to clarify the error controversies of both models; the typographical errors and the erroneous numerical solutions. The complete mathematical derivation of the TAB model is presented, and its correct numerical solutions are compared against the experimental data. We found that the TAB model was superior to other breakup models, such as Clark (1988) [7] and DDB.     

A numerical model for predicting bubble formation in a 3D fluidized bed

Fluidized bed systems have the potential to be widely used in the power generation, mineral processing and chemical industries. One factor limiting their increased use is the lack of adequate design techniques for scaling such systems. A model has been developed for simulating gas–solid fluidized bed plant. The model uses a multiphase Eulerian–Eulerian technique to predict the transient behaviour of fluidized bed systems. The commercial CFD code CFX is used as the computational framework for solving the discretized equations. To overcome the problem of accurate geometrical representation experienced in previous models a body fitted grid system is employed. The model is used to predict isothermal flow in a three-dimensional bubbling fluidized bed. Predictions of the three-dimensional model show bubble formation with gas bubbles or voids preferentially moving along the centre of the bed. Predicted behaviour is qualitatively consistent with experimental observations.     

Pendant drop formation of shear-thinning and yield stress fluids

A volume-of-fluid numerical method is used to predict the dynamics of shear-thinning liquid drop formation in air from a circular orifice. The validity of the numerical calculation is confirmed for a Newtonian liquid by comparison with experimental measurements. For particular values of Weber number and Froude number, predictions show a more rapid pinch-off, and a reduced number of secondary droplets, with increasing shear-thinning. Also a minimum in the limiting drop length occurs for the smallest Weber number as the zero-shear viscosity is varied. At the highest viscosity, the drop length is reduced due to shear-thinning, whereas at lower viscosities there is little effect of shear-thinning. The evolution of predicted drop shape, drop thickness and length, and the configuration at pinch-off are discussed for shear-thinning drops. The evolution of a drop of Bingham yield stress liquid is also considered as a limiting case. In contrast to the shear-thinning cases, it exhibits a plug flow prior to necking, an almost step-change approach to pinch-off of a “torpedo” shaped drop following the onset of necking, and a much smaller neck length; no secondary drops are formed. The results demonstrate the potential of the numerical model as a design tool in tailoring the fluid rheology for controlling drop formation behaviour.     

Modelling gas injection of a Peirce–Smith-converter

A commercial CFD-code PHOENICS was used to solve isothermal flow field of gas and liquid in a Peirce–Smith-converter. An Euler–Euler based algorithm was chosen for modelling fluid dynamics and evaluating controlling forces of a submerged gas injection. Predictions were made with a k–ε turbulence model in the body fitted coordinate system. The model has been verified with a 1/4 scale water model, and a parametric study with the mathematical model of submerged gas injection was made for the PS-process and the ladle injection processes. Limits of the modelling technique used were recognised, but calculated results indicate that the present model predicts the general flow field with reasonable accuracy. Predicted bubble distribution, pattern of the flow field and magnitude of flow velocities were used to evaluate scaling factors of physical models and general flow conditions of an industrial PS-converter.     

Numerical study of dynamics of single bubbles and bubble swarms

A systematic computational study of the dynamics of gas bubbles rising in a viscous liquid is presented. Two-dimensional simulations are carried out. Both the dynamics of single bubbles and small groups of bubbles (bubble swarms) are considered. This is a continuation of our previous studies on the two-bubble coalescence and vortex shedding [A. Smolianski, H. Haario, P. Luukka, Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach, Appl. Math. Model. 29 (2005) 615–632]. The proposed numerical method allows us to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining the mass conservation. The computed time-evolution of bubble’s position and rise velocity shows a good agreement with the available experimental data. At the same time, the results on the dynamics of bubble interface area, which are, up to our knowledge, presented for the first time, show how much the overall mass transfer would be affected by the interface deformation in the case of the bubble dissolution. Another set of experiments that are of interest for chemical engineers modelling bubbly flows concerns the bubble swarms and their behavior in different bubble-shape regimes. The ellipsoidal and spherical shape regimes are considered to represent, respectively, the coalescing and non-coalescing bubble swarms. The average rise velocities of the bubble swarms are computed and analyzed for both regimes.     

壁面效应对剪切稀化流体内气泡上浮特性的影响

数值研究了壁面效应对剪切稀化流体内气泡上浮运动特性的影响,气液两相的界面捕捉采用流体体积(VOF)法,剪切稀化流体流变特性和气液相间表面张力的计算分别采用Carreau模型和连续表面张力模型.详细研究了不同流变指数下,壁面效应对气泡形状、液相流场和气泡终端速度的影响.结果表明,强的壁面效应或弱的剪切稀化程度会限制气泡的变形和尾涡的形成,使气泡的终端速度减小;气泡终端速度最易受壁面效应的影响;强的壁面效应和强的剪切稀化程度会导致高剪切速率区域出现在壁面附近,引起壁面附近液相表观黏度大幅度的下降.     

Computational modelling of bubbles,droplets and particles in metals reduction and refining

A multi-phase framework is typically required for the CFD modelling of metals reduction processes. Such processes typically involve the interaction of liquid metals, a gas (often air) top space, liquid droplets in the top space and injection of both solid particles and gaseous bubbles into the bath. The exchange of mass, momentum and energy between the phases is fundamental to these processes. Multi-phase algorithms are complex and can be unreliable in terms of either or both convergence behaviour or in the extent to which the physics is captured. In this contribution, we discuss these multi-phase flow issues and describe an example of each of the main “single phase” approaches to modelling this class of problems (i.e., Eulerian–Lagrangian and Eulerian–Eulerian). Their utility is illustrated in the context of two problems – one involving the injection of sparging gases into a steel continuous slab caster and the other based on the development of a novel process for aluminium electrolysis. In the steel caster, the coupling of the Lagrangian tracking of the gas phase with the continuum enables the simulation of the transient motion of the metal–flux interface. The model of the electrolysis process employs a novel method for the calculation of slip velocities of oxygen bubbles, resulting from the dissolution of alumina, which allows the efficiency of the process to be predicted.     

Modeling and simulation of severe slugging in air-water systems including inertial effects

A mathematical model and numerical simulations corresponding to severe slugging in air-water pipeline-riser systems are presented. The mathematical model considers continuity equations for liquid and gas phases, with a simplified momentum equation for the mixture. A drift-flux model, evaluated for the local conditions in the riser, is used as a closure law. In many models appearing in the literature, propagation of pressure waves is neglected both in the pipeline and in the riser. Besides, variations of void fraction in the stratified flow in the pipeline are also neglected and the void fraction obtained from the stationary state is used in the simulations. This paper shows an improvement in a model previously published by the author, including inertial effects. In the riser, inertial terms are taken into account by using the rigid water-hammer approximation. In the pipeline, the local acceleration of the water and gas phases are included in the momentum equations for stratified flow, allowing to calculate the instantaneous values of pressure drop and void fraction. The developed model predicts the location of the liquid accumulation front in the pipeline and the liquid level in the riser, so it is possible to determine which type of severe slugging occurs in the system. A comparison is made with experimental results published in literature including a choke valve and gas injection at the bottom of the riser, showing very good results for slugging cycle and stability maps. Simulations were also made assessing the effect of different strategies to mitigate severe slugging, such as choking, gas injection and increase in separation pressure, showing correct trends.     

Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

Electromagnetic flow rate measurement in turbulent liquid metal channel flow

Flow control in hot aggressive melts is a big challenge for metallurgical and glass technology applications. For instance, during the production of float glass, the glass melt solidifies on the top of a liquid tin bath. To achieve a spatially homogeneous glass thickness, both global and local control of the thermofluiddynamic transport phenomena in the tin bath are crucial. The concept a global flow control involves the coupling of the non-contact electromagnetic flow rate measurement technique of Lorentz force velocimetry and the non-contact flow driving technique of electromagnetic pumps based on rotating permanent magnets. The paper presents a model experiment using the low-melting test melt GaInSn as well as numerical simulations using the commercial code FLUENT/MHD. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of internal flow fields of swirl coaxial injector in a hot environment

Based on the fractional volume of fluid (VOF), a pure Eulerian model for defining and capturing the gas/liquid interface is developed in this paper. This model can describe gas/liquid interface in high refinement, which is better than the original VOF methodology. To validate the proposed model and the algorithm, the computational code is employed to predict the flow performance in a cylindrical swirl injector under cold-flow condition, and the predicted results agree well with experimental measurements. Furthermore, the proposed model is used to simulate gas-liquid reacting flows inside a gas/liquid coaxial swirl injector operating in a hot environment. The turbulent combustion process is simulated with the k−ε−f−g model. The numerical simulation is carried out under actual operating condition of the coaxial injector. The injector performances, such as liquid film thickness, liquid film injection velocity, spray angle, pressure drop, are obtained based on the detailed information of the internal flow field. The predicted results also show that droplets are shed from the liquid film in the recess cup of the coaxial injector because of the large velocity gradient between the gas and liquid streams, and a burning area, which is characterized by high temperature, is present inside the injector.     

Bubble simulations with an interface tracking technique based on a partitioned fluid-structure interaction algorithm

Numerical techniques frequently used for the simulation of one bubble can be classified as interface tracking techniques and interface capturing techniques. Most of these techniques calculate both the flow around the bubble and the shape of the interface between the gas and the liquid with one code. In this paper, a rising axisymmetric bubble is simulated with an interface tracking technique that uses separate codes to determine the position of the gas-liquid interface and to calculate the flow around the bubble. The grid converged results correspond well with the experimental data.The gas-liquid interface is conceived as a zero-mass, zero-thickness structure whose position is determined by the liquid forces, a uniform gas pressure and surface tension. Iterations between the two codes are necessary to obtain the coupled solution of both problems and these iterations are stabilized with a fluid-structure interaction (FSI) algorithm. The flow around the bubble is calculated on a moving mesh in a reference frame that rises at the same speed as the bubble. The flow solver first updates the mesh throughout the liquid domain given a position of the gas-liquid interface and then calculates the flow around the bubble. It is considered as a black box with the position of the gas-liquid interface as input and the liquid forces on the interface as output. During the iterations, a reduced-order model of the flow solver is generated from the inputs and outputs of the solver. The solver that calculates the interface position uses this model to adapt the liquid forces on the gas-liquid interface during the calculation of the interface position.     

Mathematical and physical simulation of the interaction between a gas jet and a liquid free surface

In this work a two phase 3D mathematical model was developed using the volume of fluid (VOF) algorithm, which is able to accurately describe the cavity geometry and size as well as the liquid flow patterns created when a gas jet that impinges on a liquid free surface. These phenomena are commonly found in steelmaking operations such as in the Electric Arc Furnace (EAF) and the Basic Oxygen Furnace (BOF) where oxygen jets impinge on a steel bath and they control heat, momentum and mass transfer. The model was successfully validated with measurements made on a physical model through velocity fields obtained by Particle Image Velocimetry (PIV) and high speed camera images of the cavity. Agreement between model predictions and experimental measurements is excellent in both x-velocity component of the liquid and cavity sizes. The cavity formed in the liquid by the impinging jet depends on a force balance at the free surface where the inertial force of the jet governs this phenomena, while the liquid circulation depends on also the jet inertial force of the jet, but its angle plays an important role, being the lowest angle the best choice to shear the bath and promote stronger circulation and better mixing in the liquid.     

Numerical simulation of Casson fluid flow through differently shaped arterial stenoses

The present investigation deals with a mathematical model of blood flow through an asymmetric (about its narrowest point) arterial constriction obtained from casting of a mildly stenosed artery. The flowing blood is represented as the suspension of all red cells (erythrocytes) in plasma assumed to be Casson fluid, while the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method in order to compute the physiologically significant quantities with desired degree of accuracy. The necessary checking for numerical stability has been incorporated in the algorithm for better precision of the results computed. The quantitative analyses have been carried out finally with the inclusion of the respective profiles of the flow field over the entire arterial segment as well. The key factors such as the wall shear stress, the pressure drop and the velocity profiles are exhibited graphically and examined thoroughly for qualitative insight into blood flow phenomena through arterial stenosis.     

Vortex shedding behind a rising bubble and two-bubble coalescence: A numerical approach

In this work, we present the computational results on the wake instability in wobbling bubble regime as well as on the coalescence of two bubbles in different shape regimes. This is a continuation of our previous studies on the dynamics of a single gas bubble rising in a viscous liquid (see [A. Smolianski, H. Haario, P. Luukka, Computational Study of Bubble Dynamics, Research Report 86, Lappeenranta University of Technology, Finland]), and we use the same, finite-element/level-set/operator-splitting method that was proposed in [A. Smolianski, Numerical Modeling of Two-Fluid Interfacial Flows, Ph.D. Thesis, University of Jyväskylä, 2001]. The numerical method allows to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining a good mass conservation. Due to the highly unstable and small-scale nature of the considered problems there are very few experimental investigations, but the comparison with available experimental data confirms a good accuracy of our numerical predictions. Our studies show that plausible results can be obtained with two-dimensional numerical simulations, when a single buoyant bubble or a coalescence of two bubbles is considered.     

Influence of nonisothermal effects on gas production in northern regions

The influence of mathematical model parameters on the dynamics of pressure and temperature fields at nonisothermal gas filtration is investigated in a numerical experiment. A nonlinear system of partial differential equations obtained from the energy and mass conservation laws and the Darcy law are used to describe the process, and physical and caloric equations of state are used as closing relations. The boundary conditions correspond to a given pressure drop at the bottomhole. It is shown that the influence of the temperature field on such integral characteristics as cumulative gas production is most pronounced at moderate pressure drops. The size of the zone of possible hydrate formation in a gas reservoir is determined in a particular example.     

Non-isothermal,compressible gas flow for the simulation of an enhanced gas recovery application

In this work, we present a framework for numerical modeling of CO₂ injection into porous media for enhanced gas recovery (EGR) from depleted reservoirs. Physically, we have to deal with non-isothermal, compressible gas flows resulting in a system of coupled non-linear PDEs. We describe the mathematical framework for the underlying balance equations as well as the equations of state for mixing gases. We use an object-oriented finite element method implemented in C++. The numerical model has been tested against an analytical solution for a simplified problem and then applied to CO₂ injection into a real reservoir. Numerical modeling allows to investigate physical phenomena and to predict reservoir pressures as well as temperatures depending on injection scenarios and is therefore a useful tool for applied numerical analysis.     

Transient thermal model for the hot chamber injection system in the pressure die casting process

This paper describes a three-dimensional numerical model that is used to predict the transient thermal behaviour of the metal injection system of a hot chamber pressure die casting machine. The behaviour of the injection system is considered in conjunction with that of the die. The Boundary Element Method (BEM) is used to model the transient thermal behaviour of the injection system elements and the die blocks. A perturbation approach is adopted. By adopting this approach, only those surfaces over which a significant transient variation in temperature occurs need be considered. The model assumes that a corresponding steady-state analysis has first been performed so that time-averaged thermal information is available. A finite element based technique is used to model the phase change of the liquid metal in the die cavity and in the injection system. At injection the nozzle and die are assumed to be instantly filled with liquid metal, however, a procedure is presented that attempts to model the heat transfer associated with the flow through the nozzle, gate, and runner regions during injection. Model predictions are compared against thermocouple readings and thermal images obtained from experimental tests. Good agreement is obtained between predicted and measured temperatures. The transient thermal behaviour of an existing hot chamber injection system is investigated in detail and recommendations for improved performance are made. In an attempt to improve the solidification pattern of the casting and the thermal behaviour of the injection system, a redesign of the experimental die is considered. The numerical predictions indicate that the redesign will have a beneficial effect on the solidification pattern of the casting, and on the performance of the injection system.     

First stages of drop impact on a dry surface: asymptotic model

After impact of a viscous liquid drop on a dry wall surrounded by a gas, the drop surface is highly deformed, leading to the formation of an axisymmetrical lateral lamella along the wall. A local asymptotic model for the potential flow and unsteady boundary layer flow is developed to describe the lamella dynamics at early stages after impact. The second-order potential flow displaced by the unsteady boundary layer is taken into account. The lamella shape, its velocity and pressure are calculated with this model in parametrical forms. The three model parameters are evaluated here by fitting with recent experimental findings.