# Investigations of the Mass Transfer and Flow Field Disturbance Regulation of the Gas–Liquid–Solid Flow of Hydropower Stations

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## Abstract

**:**

^{−3}m

^{2}·s

^{−2}. The higher whirl regulation improved the material transport process and conveying efficiency and enhanced the particle mixing effect in the reaction space. Relevant research results can provide theoretical references for material mass transfer mechanisms, dynamic regulation strategies, and particle flow pattern identifications and can also provide technical support for hydropower energy conversion.

## 1. Introduction

## 2. Mathematical Models and Solving Methods

#### 2.1. Flow Field Dynamic Model

**u**

_{f}denotes the velocity vector, p represents fluid pressure,

**g**denotes gravity acceleration,

**Q**denotes the surface tension, and ρ and μ are fluid density and viscosity. Two immiscible fluids consist of two incompressible Newtonian fluids and an interface, so the momentum equation should include surface tension. This section couples the continuous surface tension (CSF) model [21,22] into a numerical model to calculate surface tension.

_{I}, u

_{i}, and ρ

_{i}are the ith phase volume fraction, speed, and density, S denotes the source item, and ${\dot{m}}_{ij}$ and ${\dot{m}}_{ji}$ represent the quality transport of two phases, respectively.

_{k,}and σ

_{ε}denote Prandtl numbers, ρ denotes fluid density, G

_{k}, G

_{b,}and Y

_{M}are the related terms of the k-ε model, v is fluid velocity, μ

_{t}represents turbulent viscosity [37,38,39,40], and E

_{1}denotes the modulus of the mean strain rate tensors. Viscosity μ

_{t}denotes the critical variable at turbulent computation and is expressed as

_{μ}denotes a variable, A

_{0}and A

_{s}denote feature constants:

_{μ}denotes an essential variable of μ

_{t}and is a parameter function including rotational speed, time-averaged strains, and turbulent intensity. This model suits various flow types, including shear flows, cavity flows, boundary layer flows, and separated flows [41,42,43]. It can more accurately simulate the plane and circular jet’s diffusion velocity. Meanwhile, calculating the rotating flow and boundary layer with directional pressure gradients is consistent with actual situations. Therefore, the turbulence model suits three-phase mixing flows.

#### 2.2. Discrete Element Method

**F**

_{n}and

**F**

_{t}denote normal contact forces and tangential contact forces,

**F**

_{d},

**F**

_{p},

**F**

_{s},

**F**

_{m}, and

**F**

_{b}are the drag forces, pressure gradient forces, Saffman lift forces, Magnus forces, and buoyant forces, m

_{i}, and

**u**

_{i}denote mass and velocity,

**G**is particle gravity, and I

_{i},

**ω**

_{i}and

**T**

_{i}are rotary inertia, angular speed, and total torques, respectively.

**G**is the relative velocity between particles i and j,

_{r}**n**is the unit normal vector of the spherical center from particle i to particle j, k

_{n}, and η

_{n}are the normal elastic coefficient and normal damping coefficient of particles respectively, k

_{t}and η

_{t}are the tangential elastic coefficient and tangential damping coefficient respectively, and

**G**is the relative slip velocity of the contact point.

_{t}_{p}) is one, and the porosity is described as

_{c}represents the summation of the sampling point. The particles are located in a bounding box, and the sampling point can be counted. As the sampling point is located in the particles and cells, they can be recorded, and the volume fractions of the particles can be obtained.

## 3. Implementation of the Calculation Model

#### 3.1. Calculation Model

#### 3.2. Boundary Conditions

#### 3.3. Grid Independence Study

## 4. Results and Discussion

#### 4.1. Transport Dynamic Characteristics of Mixing Flows

#### 4.2. Effect of the Initial Swirling Intensity on the Critical Pumping State

#### 4.3. Particle Flow Patterns

^{−3}m

^{2}·s

^{−2}. In Figure 13b, particles have the maximum potential energy in the initial evolution course. At this time, particles are in a higher position, and the peak potential energy is 0.155 m

^{2}·s

^{−2}. With the evolution of the mixing field, the potential energy of particles decreases, and there is an energy transition process when the kinetic energy is at its maximum and particle potential energy is small. In Figure 13c, the total energy of particles suddenly increases when t = 3 s. The total energy is dominated by potential energy. With the development of the mixing field, the total energy decreases to between t = 3 s and t = 20 s, which is consistent with the potential energy characteristics of Figure 13b. At t = 20 s, the total energy is mainly dominated by kinetic energy, resulting in a significant increase in the total energy. Subsequently, the total energy is dissipated during the swirl transport process. The above process shows the particle pumping course’s dynamic energy transformation mechanism.

## 5. Conclusions

- Based on the CFD-DEM coupling method and particle porous model, a three-phase mixing flow mass transfer model is put forward to obtain the distribution of relevant physical variables (volume fraction, velocity, and streamline). The critical formation time denotes the crucial transition state of fluid mediums. The fluid composition transition affects quality and production efficiency in the metallurgy, chemical industry, and other industrial processes.
- Initial disturbance speeds are the critical factor of mixing pumping formation. Under the influence of initial disturbance, the fluid microcluster on the surfaces has different disturbance patterns, inducing different speed gradients. As the mixing flow reaches the inspiratory state, the disturbance factors are in equilibrium with the inertia forces and viscous resistance. The macroscopic motions of mixing flows enhance higher transfer efficiencies of mass, momentum, and energy and make vortices penetrate the drain pipe.
- The particles in the mixing flow center have high randomness and nonlinearity under the intense suction action of the water nozzle. The aggregation and dissipation of turbulence energies lead to the weakening of particle pumping effects. The suction of the water inlet can influence the flow pattern evolution of particles in the mixing course, and the particle’s total energy is dissipated during the swirl transport process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Objective model and boundary conditions. (

**a**) Physical model. (

**b**) Numerical model and boundary condition.

**Figure 3.**Volume fraction cloud diagram of the gas phase (v = 3.0π rad/s). (

**a**) t = 5.0 s. (

**b**) t = 10.0 s. (

**c**) t = 20.1 s. (

**d**) t = 26.6 s. (

**e**) t = 29.1 s. (

**f**) t = 30.2 s. (

**g**) t = 31.4 s. (

**h**) t = 32.2 s. (

**i**) t = 33.3 s.

**Figure 4.**Streamline distribution diagrams. (

**a**) t = 26.6 s. (

**b**) t = 29.1 s. (

**c**) t = 30.2 s. (

**d**) t = 31.4 s.

**Figure 6.**Characteristic diagrams of the mixing flow field. (

**a**) Dynamic pressure. (

**b**) Radial velocity. (

**c**) Turbulent energy. (

**d**) Total pressure.

**Figure 7.**Characteristic curves of the mixing flow field. (

**a**) Dynamic pressure. (

**b**) Radial velocity. (

**c**) Turbulent energy. (

**d**) Total pressure.

**Figure 8.**Volume fraction cloud diagrams of the gas phase in the pumping state. (

**a**) t = 49.2 s. (

**b**) t = 53.7 s. (

**c**) t = 60.9 s. (

**d**) t = 40.0 s. (

**e**) t = 45.1 s. (

**f**) t = 49.6 s. (

**g**) t = 33.0 s. (

**h**) t = 37.1 s. (

**i**) t = 43.0 s.

**Figure 9.**Radial velocity curves of vortices under different initial disturbances. (

**a**) z = 0.18 m. (

**b**) z = 0.24 m.

**Figure 10.**Particle flow patterns under initial disturbances. (

**a**) t = 0.0 s. (

**b**) t = 5.0 s. (

**c**) t = 15.0 s. (

**d**) t = 20.6 s.

**Figure 11.**Particle flow patterns under the initial disturbances. (

**a**) t = 1.0 s. (

**b**) t = 5.0 s. (

**c**) t = 15.0 s. (

**d**) t = 20.6s.

**Figure 12.**Particle velocity and force at the pumping transport processes. (

**a**) Particle velocity. (

**b**) Particle force.

**Figure 13.**Particle energy variation at the pumping transport processes. (

**a**) Particle kinetic energy. (

**b**) Particle potential energy. (

**c**) Particle total energy.

Item | Parameter |
---|---|

Inlet | Pressure inlet |

Outlet | Pressure outlet |

Wall | No-slip wall |

Gravity/(N) | 9.81 |

Pipeline length/(m) | 0.15 |

Pipeline diameter/(m) | 0.010 |

Gas phase elevation/(m) | 0.15 |

Water phase elevation/(m) | 0.40 |

Container elevation/(m) | 0.55 |

Container diameter/(m) | 0.5 |

Parameter | Value |
---|---|

Water density (kg/m^{3}) | 980 |

Gas density (kg/m^{3}) | 1.225 |

Particle diameter (mm) | 20 |

Particle density (kg/m^{3}) | 950 |

Restitution coefficient, e | 0.9 |

Time step of CFD, (s) | 1.01 × 10^{−5} |

Time step of DEM, (s) | 2.36 × 10^{−7} |

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## Share and Cite

**MDPI and ACS Style**

Yan, Q.; Fan, X.; Li, L.; Zheng, G.
Investigations of the Mass Transfer and Flow Field Disturbance Regulation of the Gas–Liquid–Solid Flow of Hydropower Stations. *J. Mar. Sci. Eng.* **2024**, *12*, 84.
https://doi.org/10.3390/jmse12010084

**AMA Style**

Yan Q, Fan X, Li L, Zheng G.
Investigations of the Mass Transfer and Flow Field Disturbance Regulation of the Gas–Liquid–Solid Flow of Hydropower Stations. *Journal of Marine Science and Engineering*. 2024; 12(1):84.
https://doi.org/10.3390/jmse12010084

**Chicago/Turabian Style**

Yan, Qing, Xinghua Fan, Lin Li, and Gaoan Zheng.
2024. "Investigations of the Mass Transfer and Flow Field Disturbance Regulation of the Gas–Liquid–Solid Flow of Hydropower Stations" *Journal of Marine Science and Engineering* 12, no. 1: 84.
https://doi.org/10.3390/jmse12010084