# Analysis of Sediment and Water Flow and Erosion Characteristics of Large Pelton Turbine Injector

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Turbulence Model

_{k}is the turbulent kinetic energy k-generation term, G

_{ω}is the ω-generation term, Γ

_{k}is the effective diffusion phase of k, and Γ

_{ω}is the effective diffusion phase of ω, where: ${\Gamma}_{k}=\mu +\frac{{\mu}_{t}}{{\sigma}_{k}}$, ${\Gamma}_{\omega}=\mu +\frac{{\mu}_{t}}{{\sigma}_{\omega}}$, Y

_{k}is the divergence phase of k, Y

_{ω}is the divergence phase of ω, D

_{ω}is the orthogonal divergence phase, and S

_{k}and S

_{ω}are user-defined source terms. The subscripts i and j are tensor coordinates, μ

_{t}is the turbulent vortex dynamic viscosity coefficient, σ

_{k}is the turbulent Prandtl number of k, and σ

_{ω}is the turbulent Prandtl number of ω, where: ${\sigma}_{k}=\frac{1}{0.85{F}_{1}+(1-{F}_{1})}$, ${\sigma}_{\omega}=\frac{1}{0.5{F}_{1}+0.856(1-{F}_{1})}$, F

_{1}is the value of the wall function, ${G}_{k}={\mu}_{t}{S}^{2}$, $S=\sqrt{2{S}_{ij}{S}_{ij}}$, ${S}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{j}}{\partial {x}_{i}}+\frac{\partial {u}_{i}}{\partial xj}\right)$, ${G}_{\omega}=\frac{\rho {G}_{k}}{{\mu}_{t}}$, Y

_{k}= ρβ

^{*}kω, where: β

^{*}= 0.09. Y

_{ω}= ρβkω

^{2}, where: β = 0.075F

_{1}+ 0.0828(1 − F

_{1}). ${D}_{\omega}=2(1-{F}_{1})\rho {\sigma}_{\omega ,2}\frac{1}{\omega}\frac{\partial k}{\partial {x}_{j}}\frac{\partial \omega}{\partial {x}_{j}}$, where σ

_{ω}

_{,2}= 1.168 [24].

#### 2.2. Solid–Liquid Two-Phase Flow Model

_{i}is the velocity of water (m/s), V

_{i}is the velocity of sediment (m/s), ρ is the density of phase material (kg/m

^{3}), g is the acceleration of gravity (m/s

^{2}), ν is the coefficient of kinematic viscosity of phase material, P is the pressure (Pa), x

_{i}is the coordinate component, $B=18\left(1+{B}_{0}\right){\rho}_{f}{\nu}_{f}/{d}^{2}$ is the interphase action coefficient, d is the sediment particle diameter; the B

_{0}term is introduced to consider other action factors other than the Stokes linear drag action. In general, B

_{0}is a constant, ϕ is the phase volume fraction and has the relational equation, ϕ

_{f}+ ϕ

_{p}= 1, the subscript f is the liquid phase, p is the solid phase, and I, j, and k are tensor coordinates [25].

#### 2.3. Particle Trajectory Model

_{p}is the velocity component of the particle (m/s), K

_{m}is the virtual mass force coefficient, K

_{m}≈ 0.5. $\overline{\rho}$ is the ratio of particle density ρ

_{p}to fluid density ρ, C

_{D}is the particle drag coefficient, C

_{D}= 0.44. d

_{p}is the particle size, K

_{B}is the Basset force coefficient, K

_{B}≈ 6.0. ν is the fluid motion viscosity coefficient, K

_{S}is the Saffman lift coefficient, K

_{S}≈ 1.615. C

_{M}is the Magnus lift coefficient, C

_{M}≈ 1.0. ${\Omega}_{i}={\omega}_{pi}-0.5\nabla \times {u}_{i}$, ω

_{p}is the angular velocity of the particle’s own rotation, p is the pressure, g is the acceleration of gravity, and sgn is the sign function.

#### 2.4. Sediment Erosion Model

_{e}is the wall erosion rate, N

_{p}is the total number of particles, m

_{p}is the particle mass flow rate, c

_{(dp)}is the particle size function, ƒ(α) is the impact angle function (α is the angle at which the particles impact the wall), b(v) is the function of the relative velocity of the particles (v is the relative velocity of the particles to the wall), and A

_{face}is the wall area (m

^{2}). The erosion model parameters were set as follows: the normal bounce coefficient was defined as Equation (9):

^{−9}, and the velocity index function was set to 2.6.

## 3. Geometric Physical Model and Boundary Conditions

#### 3.1. Geometric Modeling and Meshing

_{0}of the injector is the main factor that determines whether the impressed turbine runner can efficiently generate electricity. Usually, the cylindrical jet velocity V

_{0}is defined using the following Equation (11):

_{v}is the jet velocity coefficient considered as 0.98, H is the turbine design head, and g is the local acceleration (m/S

^{2}). The design head of the power station was 671 m, and the design value of the cylindrical jet velocity V

_{0}was calculated to be 112.39 m/s.

#### 3.2. Boundary Conditions and Calculation Settings

^{3}; the median particle size was 0.0142 mm, simplified as spherical particles and shot vertically from the inlet. The incidence velocity was the same as the water flow velocity, and the walls were all set to bounce in contact. Transient calculations were performed for the solid–liquid two-phase flow. The time step was set to 7.77726 × 10

^{−5}s, and each time step was iterated 20 times.

## 4. Numerical Calculation Results and Analysis

#### 4.1. Analysis of the Flow Field in the Injector

#### 4.2. Analysis of the Velocity Deficit Phenomenon in the Injector

#### 4.3. Effect of Sediment Particle Diameter on the Injector Erosion Analysis

^{3}and the d

_{p}= 0.0142 mm was 0.767 and 0.243 mm, respectively (1 year of operation). With reference to the erosion assessment standards and profile assessment standards of the Pelton turbine, the sediment erosion of the large Pelton turbine injector was high, and the sediment erosion phenomenon was serious. Long-term operation will lead to a reduction in the jet quality of the injector and reduce the operating life of the unit. Therefore, anti-erosion measures should be taken to address the wear and tear and erosion of the injector.

## 5. Conclusions

- (1)
- The pressure at the nozzle outlet is minimum. The direction of the pressure gradient changes, and the velocity is maximum. The boundary layer on the surface of the injector causes a velocity deficit, which affects the velocity distribution of the jet and the quality of the jet. Water shoots out of the nozzle, and the maximum jet velocity continues to increase, before decreasing again.
- (2)
- The sediment particle diameters will affect the erosion of the needle. The smaller the particle size, the more serious the erosion of the needle rod and the head. The erosion of the lower needle guide is more serious than that of the upper needle guide. The erosion of the needle rod and needle guide is groove-shaped. The erosion of the needle is mainly point-like and exhibits asymmetrical distribution.
- (3)
- The particle size has little effect on the erosion location of the nozzle port ring; however, it has an effect on the erosion amount. The erosion of the nozzle port ring exhibits symmetrical distribution. The erosion of the nozzle port ring is greater than that of the needle with the same sediment particle diameter.
- (4)
- The sediment erosion of the power station is very serious, and anti-erosion measures are necessary.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Velocity distribution of the flow path from the tail end of the deflector to the S1 section.

Point | Angle | Value |
---|---|---|

1 | 0 | 0 |

2 | 20 | 0.8 |

3 | 30 | 1 |

4 | 45 | 0.5 |

5 | 90 | 0.4 |

Name | Nozzle Inlet Diameter/mm | Nozzle Outlet Diameter/mm | Needle Stroke/mm | Number of Needle Guide/Number |
---|---|---|---|---|

Parameter | 1502 | 518 | 316.3 | 2 |

Option | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Numbers of grid cells | 445,814 | 800,121 | 1,788,149 | 2,288,892 |

Sediment Particle Diameter d_{p} (mm) | Needle Maximum Erosion Rate R_{en} (mm/s) | Nozzle Maximum Erosion Rate R_{es} (mm/s) |
---|---|---|

0.001 | 3.243 × 10−^{8} | 7.718 × 10^{−8} |

0.005 | 1.57 × 10^{−8} | 4.454 × 10^{−8} |

0.0142 | 7.701 × 10^{−9} | 2.432 × 10^{−8} |

0.05 | 4.761 × 10^{−9} | 1.064 × 10^{−8} |

0.1 | 2.832 × 10^{−9} | 9.837 × 10^{−9} |

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**MDPI and ACS Style**

Liu, J.; Pang, J.; Liu, X.; Huang, Y.; Deng, H. Analysis of Sediment and Water Flow and Erosion Characteristics of Large Pelton Turbine Injector. *Processes* **2023**, *11*, 1011.
https://doi.org/10.3390/pr11041011

**AMA Style**

Liu J, Pang J, Liu X, Huang Y, Deng H. Analysis of Sediment and Water Flow and Erosion Characteristics of Large Pelton Turbine Injector. *Processes*. 2023; 11(4):1011.
https://doi.org/10.3390/pr11041011

**Chicago/Turabian Style**

Liu, Jitao, Jiayang Pang, Xiaobing Liu, Yu Huang, and Huiming Deng. 2023. "Analysis of Sediment and Water Flow and Erosion Characteristics of Large Pelton Turbine Injector" *Processes* 11, no. 4: 1011.
https://doi.org/10.3390/pr11041011