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Modelling of gas-liquid bubbly flows is achieved by coupling a population balance equation with the three-dimensional, two-fluid, hydrodynamic model. For gas-liquid bubbly flows, an average bubble number density transport equation has been incorporated in the CFD code CFX 5.7 to describe the temporal and spatial evolution of the gas bubbles population. The coalescence and breakage effects of the gas bubbles are modeled. The coalescence by the random collision driven by turbulence and wake entrainment is considered, while for bubble breakage, the impact of turbulent eddies is considered. Local spatial variations of the gas volume fraction, interfacial area concentration, Sauter mean bubble diameter, and liquid velocity are compared against experimental data in a horizontal pipe, covering a range of gas (0.25 to 1.34 m/s) and liquid (3.74 to 5.1 m/s) superficial velocities and average volume fractions (4% to 21%). The predicted local variations are in good agreement with the experimental measurements reported in the literature. Furthermore, the development of the flow pattern was examined at three different axial locations of

Gas-liquid, two-phase flow in horizontal pipes is encountered often in a number of industrial processes. Common applications include chemical plants, evaporators, oil wells and pipelines, fluidized bed combustors, and evaporators. Horizontal bubbly flows have received less attention in the literature than vertical flows, even though this flow orientation is equally important in industrial applications such as hydrotransport, an important technology in bitumen extraction. Experimental observations are also difficult in this case, as the migration of dispersed bubbles towards the top of the pipe, due to buoyancy, causes a highly nonsymmetric volume distribution in the pipe cross-section. This density stratification is not often accompanied by a strong secondary flow. Gas volume fraction, interfacial area concentration, and mean bubble diameter are the three characterizing field variables that characterize the internal flow structure of two-phase, gas-liquid flows in horizontal pipe. In various industrial processes, the gas volume fraction parameter is required for hydrodynamic and thermal design. The interfacial transport of mass, momentum, and energy is proportional to the interfacial area and the driving forces. This is an important parameter required for a two-fluid model formulation. The mean bubble diameter serves as a link between the gas volume fraction and interfacial area concentration. An accurate knowledge of local distributions of these three parameters is of great importance to the eventual understanding and modelling of the interfacial transfer processes [

In the past two decades, significant developments in the modeling of two-phase flow processes have occurred since the introduction of the two-fluid model. In the volume averaged, two-fluid model, the interfacial transfer terms are strongly related to the interfacial area concentration and the local transfer mechanisms such as the degree of turbulence near the interfaces. Fundamentally, the interfacial transport of mass, momentum, and energy are proportional to the interfacial area concentration (

In this work, an attempt has been made to demonstrate the possibility of combining population balance with computational fluid dynamics (CFD) for the case of gas-liquid bubbly flow in the horizontal pipe. The MUSIG model has been implemented in CFX-5.7 to account for the nonuniform bubble size distribution in a gas-liquid flow [

Population balance modelling is used in computing the size distribution of the dispersed phase and in accounting for the breakage and coalescence effects in bubbly flows. A general form of the population balance equation is

The breakup of bubbles in turbulent dispersions employs the model developed by Luo and Svendsen [

The coalescence of two bubbles is assumed to occur in three steps. The first step where the bubbles collide and trap a layer of liquid between them, a second step where this liquid layer drains until it reaches a critical thickness, and a last step during which this liquid film disappears and the bubbles coalesce. The collisions between bubbles may be caused by turbulence, buoyancy, or laminar shear. Only the first cause of collision (turbulence) is considered in the present model. Indeed collisions caused by buoyancy cannot be taken into account as all the bubbles from each class move at the same speed. The coalescence rate considering turbulent collision taken from Prince and Blanch [

When bubbles collide, a small amount of liquid is entrapped between them, forming a small circular lens or film of radius ^{−4} m [^{−8} m [

The turbulent collision rate

The numerical simulations presented are based on the two-fluid, Eulerian-Eulerian model. The Eulerian modelling framework is based on ensemble-averaged mass and momentum transport equations for each phase. Regarding the liquid phase (

Continuity equation of the gas phase

Momentum equation

In (

The origin of the drag force is due to the resistance experienced by a body moving in the liquid. Viscous stress creates skin drag, and pressure distribution around the moving body creates form drag. The formulation of the drag force is a key issue in multiphase flows. Clift et al. [

The lift force considers the interaction of the bubble with the shear field of the liquid. It acts perpendicular to the main flow direction and is proportional to the gradient of the liquid velocity field. The lift force in terms of the slip velocity and the curl of the liquid phase velocity can be modelled as [

The wall lubrication force arises because the liquid flow rate between bubble and the wall is lower than between the bubble and the main flow. This results in a hydrodynamics pressure difference driving bubble away from the wall. This force density is approximated as [

The turbulent dispersion force, derived by Lopez de Bertodano [

For the continuous liquid phase, a

The multiple size group (MUSIG) model (CFX 5.7 from ANSYS) used in this study combines the population balance method with the break-up [

In this present study, bubbles ranging from 1 mm to 10 mm diameter are equally divided into 10 classes (see Table

Diameter of each bubble class tracked in the simulation.

Class index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Bubble diameter, | 1.45 | 2.35 | 3.25 | 4.15 | 5.05 | 5.95 | 6.85 | 7.75 | 8.65 | 9.55 |

Solution to the coupled sets of governing equations for the balances of mass and momentum of each phase was obtained using CFX 5.7. The conservation equations were discretized using the control volume technique. Computational grid is based on the unstructured set of blocks each containing structured grid. The structured grid within each block is generated using general curvilinear coordinates ensuring accurate representation of the flow boundaries. In order to select an adequate grid resolution, the effect of changing grid size was investigated. Several simulations were carried out using progressively larger number of grid points of 87156, 152482, 257670, 303245, and 341612. Sample grid sensitivity results are shown in Figure ^{−4} s, followed by 300 steps at 5 × 10^{−4} s, 400 steps at 1 × 10^{−3} s, 1400 steps at 5 × 10^{−3} s, and 8000 steps at 1 × 10^{−2} s. Underrelaxation factors between 0.6 and 0.7 were adopted for all flow quantities, and pressure was never underrelaxed. The hybrid-upwind discretization scheme was used for the convective terms. At the inlet, gas, liquid, and the average volume fraction have been specified. At the pipe outlet, a relative average static pressure of zero was specified. For initiating the numerical solution, average volume fraction and parabolic liquid velocity profile are specified as initial conditions. The operating conditions are summarized in Table

Operating conditions.

Geometry | 50.3 mm ID |
---|---|

Gas phase | Air at 25°C |

Liquid phase | Water at 25°C |

Gas superficial velocity | 0.25–1.34 m/s |

Liquid superficial velocity | 3.74–5.1 m/s |

Average gas volume fraction | 0.04–0.205 |

Effect of grid size on gas volume fraction and axial liquid velocity.

The CFD simulations are carried out for the experimental conditions reported by Kocamustafaogullari and Wang [

Accurate prediction of developing bubbly flows in horizontal pipes cannot be carried out without sufficient knowledge of a transverse lift force acting on the bubbles, the force that governs the transverse migration of a bubble in a shear field. It has been clarified through a number of experiments that the lateral migration strongly depends on bubble size, that is, small bubbles tend to migrate toward the pipe wall which results in a peak in the bubble volume fraction distribution near the wall. Just like the functional form of the drag coefficient for a single particle interaction is extended to multiparticle systems, the functional form of the lift force that captures the lateral migration phenomenon is given by (

Lift coefficient (

The difference between the model predictions and experimental data on the spatial variation of field quantities such as liquid velocity profiles, volume fraction profiles, and interfacial area measurements is minimized by the tuning of this parameter. Bubbly horizontal pipe flow experiments by Kocamustafaogullari and Wang [^{3 }+ ^{2 }+ ^{−10}, ^{−7}, ^{−4}, and

Figures

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

Comparison of predicted and experimental data of Kocamustafaogullari and Wang [

The current simulation results and the experimental results of Kocamustafaogullari and Wang [^{2}/m^{3} towards the top of the pipe in horizontal two-phase flow. These values are quite high compared to vertical bubbly flows. This will result in increasing the intensity of the interfacial transport of mass, momentum, and heat near the top of the pipe. In addition, it can be observed that increasing the superficial gas velocity or decreasing the superficial liquid velocity would increase the local and overall interfacial area concentration and tend to flatten the interfacial area concentration profile. The model prediction of interfacial area concentration shows relative mean and maximum errors are ±8% and ±22%, respectively.

The comparison of predicted and experimental data of the local Sauter mean bubble diameter distribution is shown in Figures

Figures

From the simulation, we can get much more additional information, while some of these quantities are more difficult to measure in an experiment. One such quantity is the slip velocity between the two phases. The variation in the vertical direction of the slip velocity is shown in Figure

Slip velocity at different superficial gas and liquid velocities: (a)

While we have used the data on the spatial variation of quantities such as volume fraction and interfacial area to tune the model parameters, from a practical view point one is often interested only in a quantity that is averaged over the pipe cross-section. Hence, area averaged gas volume fraction, interfacial area concentration, and mean bubble diameter at the exit plane are shown in Figure

Effect of superficial gas and liquid velocity on (a) average gas volume fraction; (b) average interfacial area concentration (c) average mean bubble diameter: (1)

The bubble size distribution is determined by bubble coalescence and breakup. In a given system, bubble coalescence and breakup are primarily influenced by the local gas volume fraction and kinetic energy dissipation rate. Because of the nonuniform profiles of the gas volume fraction and dissipation rate, the bubble size distribution varies with the position as well. The spatial evolution of bubble size distribution between the inlet and the outlet of the pipe is shown in Figure

The bubble class volume-based p.d.f at inlet and exit of the pipe for superficial gas velocity is 0.25 m/s, superficial liquid velocity is 4.67 m/s, and average volume fraction is 0.043.

To see the development of flow pattern in the axial direction, several three-dimensional simulations were carried out using the coupled two-fluid and population balance models. The flow evolution is shown in Figure

Gas volume fraction development in axial direction for superficial gas velocity is 1.21 m/s, superficial liquid velocity is 4.67 m/s, and average volume fraction is 0.205. At vertical position, (a)

Interfacial area concentration (IAC) development in axial direction for superficial gas velocity is 1.21 m/s, superficial liquid velocity is 4.67 m/s, and average volume fraction is 0.205. At vertical position, (a)

Liquid velocity development in axial direction for superficial gas velocity is 1.21 m/s, superficial liquid velocity is 4.67 m/s, and average volume fraction is 0.205. At vertical position, (a)

A two-fluid model coupled with population balance approach is presented in this paper to handle gas-liquid bubbly flows in horizontal pipe. To demonstrate the application of the population balance approach, the average bubble number density transport equation was formulated and implemented for gas-liquid bubbly flows in the CFD code CFX 5.7 to determine the temporal and spatial geometrical changes of the gas bubbles. Population balance combined with coalescence and break-up models were taken into consideration. A detailed comparison has been presented between the CFD simulation and the experimental data reported by Kocamustafaogullari and Wang [

The authors gratefully acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Syncrude Canada Ltd. for this project.