Abstract
Air bubble plume flow has been applied widely in the dredging, ice breaking, and pollution control at navigation projects. But the interaction regimes among bubbles or between bubbles and water are not quite clear. Especially in open channels, the bubble plume flow are significantly affected by the separation phenomenon which is caused by the cross flow velocity. According to the existing research, the interaction force of gas-liquid and the distribution of bubble size are the key parameters to simulate the hydrodynamic characteristics of bubble plume flow. In order to explore the mechanism of air bubbles entrained plumes in open channels, an Eulerian-Eulerian approach for air-water flows numerical model was introduced, and the population balance model (PBM) was included to describe the distribution of bubble size. The cross flow velocity of open channels has been discussed in the proposed numerical model. It shows that the separation of bubble plume is strongly influenced by the cross flow velocity. The influence of these parameters on the movement characteristics of air bubbles is studied. The results indicate that the cross flow velocity has great impact on bubble plume as well as the lifting effectiveness of pneumatic sluicing. This research provides references for bubble plume in engineering applications.
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1 Introduction
Pneumatic sluicing is an efficient dredging technology which can be used in approach channel, ports, or lake (Pan and Wu 2019). The air-injection of pneumatic sluicing can be employed to stir the bottom nutrients or sediments and take them to the downstream or other place (Ding et al. 2019). It is significant to study the rising process of bubble plume and air-water interaction in open channel in order to improve the transportation efficiency of sands or nutrients.
There are lots of scholars focusing on the movement of bubble plume in water, the gas-liquid characteristics such as gas holdup, bubble velocities, bubble size, and entrained plume flux (Qiang et al. 2018; Yao et al. 2019; Wan et al. 2017; Cheng et al. 2020). For example, Liang and Peng (2005) proposed a formula to calculate the velocity of rising bubble plume in quiescent water with gas discharge, the density of liquid-gas, and height. Song et al. (2011) studied the distribution characteristics of bubble void fraction by physical model experiment and image processing technology. The results show that when the aspect ratio of model’s height and width is 1.0, the bubble plume structure is less affected by the pressure, and the plume structure is stable. With the increase of aspect ratio and pressure, the plume structure is unstable.
With the development of multi-phase numerical simulation, there are lots of researchers adopt air-water two phase flow to study the air plumes in water. And Euler-Euler model is used wildly in multiphase flow modeling with high void fraction (Duguay et al. 2021). For example, Liu and Li (2018) study the bubbly flows in water, and the sensitivity of different turbulence models and the scale-adaption of Euler-Euler model had been conducted. It found that the mesh size should be taken account in the model. Fleck and Rzehak (2019) studied the dynamic flow phenomena of bubble plume with Euler-Euler two-fluid model, and the periodically oscillating bubble plume was been simulated, which shown a good precision about the plume oscillation period. Godino et al. (2020) studied the air-water dispersed and segregated multiphase flows with experiments and Euler-Euler numerical model. Five cases with different flow regimes were simulated using the same set of interfacial force models, and the local rheology of the flow in different interfacial models was considered by a linear blending method. And so on.
Existing studies have gained many achievements in bubble plume and its application. Nevertheless, most of the studies only focus on the rising characteristics of bubble plume and its influence on surrounding waters in quiescent waters. And the study of bubble plume in open channels is insufficient, which the bubble plume is obviously affected by cross flow (Qiang et al. 2018). Based on numerical simulation, this paper uses Eulerian-Eulerian approach for air-water flows, and the population balance model (PBM) is applied to describe the distribution of bubbles, then the gas holdup, size distribution of bubbles, and the flow field of aerated area are studied.
2 Simulation Model
In CFD-PBM model, the turbulent dissipation rate, gas holdup and flow field are calculated by the two-fluid CFD model. And the results are used to calculate the bubble coalescence and breaking rate. Then the PBM equations are solved for the bubble size distribution, flow pattern, interphase force and the turbulence source term caused by bubble turbulence in the improved two-fluid model.
2.1 Euler-Euler Model
Euler-Euler model is used to simulate gas-liquid two phase flow. The model regards the bubble as a continuous phase and runs through water phase. It can simulates the distribution of gas holdup well, and greatly promotes the calculation speed. The continuity equations and momentum equations are represented as (Ranganathan and Sivaraman 2011):
In these equations, ρ, αi, u, t stand for density, the ith group volume, fraction velocity vector, time, and the subscript i represents gas (g) or liquid (l). \( P^{\prime}\), F, μeff and g, are modified pressure, interphase force, effective viscosity, and acceleration of gravity, respectively. \( P^{\prime}\) can be calculated by Eq. (3):
The effective viscosity of liquid phase μeff,l can be calculated as
in which μl is the liquid viscosity, μtl and μtg represent the turbulence viscosity of liquid phase and the turbulence viscosity induced by gas phase, as given by Eq. (5) and (6):
where kl is turbulent kinetic energy, Cμb is empirical parameters with a value of 0.6 (Ranganathan and Sivaraman 2011).
The k–ε model is used for turbulence model. Its advantage is that this model has the wide applicability and is verified useful in CFD-PBM model.
2.2 The Drag Force
During the pneumatic sluicing process, momentum exchange occurs at the interface between gas and liquid phases, and the interphase force plays an important role in the accuracy of simulation results. The gas-liquid interphase forces include the drag force, lift force, virtue mass force, turbulent dispersion force, and wall lubrication force. Compared the drag force, the other interphase forces can be neglected because of their less significance in the interaction. The drag force represents the interfacial momentum transfer caused by the gas-liquid phase velocity slip. It can be described by Schiller-Naumann formulas shown in Eq. (7) (Olmos et al. 2001):
where the CD is the drag coefficient of a bubble size db.
2.3 PBM Model
PBM model is a general method to describe the distribution of dispersed phase size in multiphase flow system (Hulburt and Katz 1964). In gas-liquid multiphase flow system, PBM model can be used to consider the influence of bubble coalescence and breakup on the distribution of bubble size, so as to study the mechanism of two-phase interaction in gas-liquid system. For the gas-liquid system, the group equilibrium equation can be expressed as:
In which, v and v′ is the bubble volume, n(v, t) is the distribution function of bubble size, a(v, v′) is the bubble coalescence rate function, b(v) is the bubble breaking rate function, β(v|v′) denotes the probability density function of bubble breaking up into daughter bubbles with the volume from v to v′. In this paper, the discrete method is employed, and transport equation of the bubbles in the kth group can be expressed by Eq. (9):
In which, ρ is the density of gas phase, αk is the volume fraction of the bubbles in the kth group, as shown in Eq. (10). The source terms of bubble generation and extinction caused by coalescence and breakup are represented by Eq. (11)–(14).
In this study, Luo’s bubble breaking and coalescence model (Luo and Svendsen 1996) is used to investigate the distribution of bubble size in liquid.
The model mentioned above is employed in bubble columns and aeration tanks. Gas holdup, bubble breakup rate and the distribution of daughter bubble size are in good agreement with the tests data (Cheng et al. 2020).
2.4 Simulation Model
The sketch of simulation model is shown in Fig. 1. The model is 10 m long and 0.7 m high. The 0.1 m aeration zone is set in the middle bottom of the channel located at x is from 0.45 m to 0.55 m. The water depth is kept in 0.5 m. The left side of the model is velocity inlet and the right side is set as free outflow. The quadrilateral mesh is adopted with the size of 1 mm, and the total number is 70,000 (as shown in Fig. 1(a)). The gas enters from the inlet with the uniform velocity of 0.2 m/s and initial gas diameter of 0.03 mm, as shown in Fig. 1(b). The cross water flow velocity ranges from 0 to 1 m/s, as shown in Table 1. All the simulations are carried out on the ANSYS Fluent platform. SIMPLE algorithm is used for pressure-velocity equations. QUICK scheme is employed for momentum equations. The volume fraction equations are discretized by the first-order upwind format, and the relaxation factors are set with the default values. The time step in the calculation is 0.002s, and the process ends when air bubble plume is steady (Table 2).
3 Results
3.1 Gas Holdup
Figures 2 and 3 show the distribution contours of gas holdup at different flow velocities and gas holdup curves at different heights. It can be seen that gas holdup decreases gradually along the radial direction, and gas holdup in the middle of bubble plume is higher than that at the sides. With cross flow, the bubble plume is incline to the direction of velocity. And with the increase of cross velocity, the bubble plume is wider and gas hold up is lower at the same height than that with a small velocity.
3.2 Bubble Size Distribution
Figure 4 shows the distribution of bubble size with different flow velocities. It can be seen that with the uplifting of the bubbles, the diameter of bubbles gradually increases, and the diameter reaches the maximum at the water surface. The coalescence of bubbles is effected by cross flow velocity, the smaller the flow velocity is, and the larger the diameter of bubble flow is.
3.3 The Influence of Bubbles on Water Flow
The water velocity of the y direction in bubble plume is the main influencing factor for the pneumatic sluicing.
Figure 5 shows water vertical velocity of the y direction near bubbles at different heights with different flow velocities. Results show that in different conditions, the greater the cross flow velocity, the smaller the vertical velocity of water phase at the same height. Therefore, a smaller velocity of cross flow can bring the sediment or nutrients higher, but a larger flow velocity can take the masteries to the farther downstream.
4 Conclusions
The uplifting characteristics of bubble plume is a main influence factor of the pneumatic sluicing. There are little studies focus on the bubble plume with cross flow. Based on CFD-PBM model, this paper studied the bubble plume with different cross flow velocities, and the results show that:
-
(1)
The width of bubble plume is increasing with the increase of cross velocity, and the gas hold up is reducing with the cross velocity.
-
(2)
The bubble diameter is enlarged during the uplifting of bubbles and the bubble growth to maximum at the water surface. The larger the cross flow velocity, the smaller the bubble diameter at the same water level.
-
(3)
The uplifting force is decreased with the cross flow velocity, and the greater the cross flow velocity, the smaller the vertical velocity of water phase.
The results indicate that the cross flow velocity has great influence on the bubble plume as well as the lifting effectiveness of pneumatic sluicing, which provides references for the further studies.
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Acknowledgements
The authors appreciate the support of the National Key Research and Development Program of China (No. 2021YFC3201101), Project funded by China Postdoctoral Science Foundation (No. 2020T130309, 2019M651892), Jiangsu Water Resources Science and Technology Project (No. 2020022, 2021024), and Nanjing University of Information Science & Technology Research Foundation (No. 2017r097). The authors also want to thank the people for their helpful suggestions and corrections on the earlier draft of our study according to which we improved the content. There is no conflict of interest in our paper.
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Xu, J., Wang, X., Zhu, L., Zhou, D., Zhao, J. (2023). Study on Air Bubble Plume in Open Channel with CFD-PBM Coupling Model. In: Li, Y., Hu, Y., Rigo, P., Lefler, F.E., Zhao, G. (eds) Proceedings of PIANC Smart Rivers 2022. PIANC 2022. Lecture Notes in Civil Engineering, vol 264. Springer, Singapore. https://doi.org/10.1007/978-981-19-6138-0_110
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