# Numerical Study of Sediment Erosion Analysis in Francis Turbine

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Erosion Phenomena

#### 2.1. Erosion

_{1}= coefficient of sediment concentration, S

_{2}= coefficient of sediment hardness, S

_{3}= coefficient of sediment particle size, S

_{4}= coefficient of sediment particle shape, M

_{r}= coefficient of water aversion of material, and ${V}^{n}$ = relative velocity (n = 3) [27,28]. Equation (2), on the other hand, is the most often cited statement of wear and velocity of a particle.

_{f}), of a particle impacting on a material surface.

^{3}, which is equal to 1000 ppm (0.1% of volume).

#### 2.2. Erosion Model

_{p}is the mass of the particle. The overall erosion of the turbine wall is the cumulative of overall solid particles, which provides an erosion rate (kg/s) and erosion rate density (kg/s/m

^{2}) [35,36].

## 3. Computer Simulation

#### 3.1. Geometrical Modeling and Meshing

^{−6}at a design operating condition (gate valve angle of 20

^{0}). It is difficult to control the grids of the whole geometry simultaneously. Therefore, the whole geometry was segmented by three parts: casing, runner, and draft tube. To decrease the effect on grid sensitivity on the numerical values, a grid independency test is essential to check the grid convergence. The error from the computational simulation is well-accepted by the CFD which is not entirely from the grid convergence error but also from many other error sources. Nevertheless, one can minimize the total error by reducing the error due to grid sensitivity and this must be done in a systematic fashion. To observe the variation of numerical values according to the number of grids, an independent grid test was carried out at design operating conditions (GV-20°) for the prototype turbine as illustrated in Figure 3. The grid independency test was conducted based on the undertaken grid convergence index (GCI) technique [37,38,39,40,41,42,43]. The approximate and extrapolated relative errors can be written as:

#### 3.2. Numerical Method

^{+}. According to conventional theory, near a no-slip wall, there are negative gradients in dependent variables. Furthermore, viscous effects on the transport processes are relatively high; these computations are enlarged across the viscosity-affected sublayer adjacent to the wall. The low-Re approach essentials a very fine grid in the near-wall zone and correspondingly greater number of nodes. Computer performances and ability demands are greater than those of the wall-function, and care may be taken to make a certain good computational resolution in the near-wall region to apprehend the quick difference in variables [35]. To minimize the resolution necessities, an automatic wall treatment function was established by CFX, which enables a gradual switch between wall functions and low-Reynolds number grids, without a loss of precision [35]. Wall functions are the well-accepted way to account for wall effects. In CFX, scalable wall functions are applied for all turbulence models based on the ε - equation. For k-ω–based models (including the SST model), an automatic near-wall treatment function was accounted in the near wall region [35].

^{−5}controlled by convergence criteria.

^{3}, and a molar mass of 60.08 kg/kmol is used [9,35]. The mean diameter of the particles value is of 0.1 mm. Pressure boundary conditions were considered at the inlet (931,630 Pa) and outlet (0 Pa) for the whole passage of the turbine. Quartz particles are consistently injected at the inlet, and the particles will follow through the whole passage at the outlet. The turbulence dissipation force is initiated, and the Schiller–Naumann model estimates the drag force comply with the particles [35,36]. The coupling between the water and particles is categorized into two sets: one-way coupled and fully coupled. The particles inflow rates diversified from 1 kg/s to 100 kg/s and had given constant particle number from 500 to 5000. The steady noncavitating results were accounted as the initial conditions for the homogeneous multiphase flow simulation for the numerical stability. The reference pressure was set 1 atm. The water temperature was considered as standard temperature and pressure (STP) during the simulation, and the saturated vapor pressure was set 3169 Pa. Rayleigh–Plesset cavitation model and Tabakoff–Grant erosion model was adapted for estimation of cavitation and sediment erosion respectively.

## 4. Results and Discussion

#### 4.1. Validation of Numerical Results

^{3}/s.

_{Q})

_{sys}, specific hydraulic energy (f

_{E})

_{sys}, torque (f

_{T})

_{sys}, rotational speed (f

_{n})

_{sys}, and density of water (f

_{ρ})

_{sys}:

_{nh})

_{total}was calculated and found to lie within a band of ±1.224%. The measurement was carried out only one operating point (GV-20°). Therefore, only one measurement data could not illustrate the 95% confidence limit and error propagation from the result (Table 4).

#### 4.2. Erosion Effects on Performance Characteristics

#### 4.2.1. Effect of Sediment Inflow Rates

#### 4.2.2. Influence of Sediment Particle Shape Factor

#### 4.3. Cavitation-Erosion Effects

#### 4.4. Comparison of Performances at Different Sand Inflow Rates

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

CFD | Computational Fluid dynamics |

GCI | Grid convergence index |

KIMM | Korea Institute of Machinery and Materials |

SSU | Soongsil University |

C | Concentration of particle, kg/m^{3} |

c | Velocity of particle, m/s |

df | Degree of freedom |

E | Erosion rate, mm/yr |

f(α) | Function of impingement angle |

f(V_{pn}) | Function of velocity of particle |

f | Uncertainty |

K_{mat} | Material constant |

K_{env} | Environmental constant |

k_{1}, k_{2}, k_{3}, k_{4}, k_{12} | Variable constant |

${m}_{p}$ | Mass of the particle |

M_{r} | Coefficient of water resistance |

$\dot{N}$ | Number rate |

n | Exponent number |

P_{f} | Abrasive power |

r | Grid ratio |

R_{θ} | Tangential restitution factor |

R | Radius of curvature on surface |

S_{1} | Coefficient of sediment concentration |

S_{2} | Coefficient of sediment hardness |

S_{3} | Coefficient of sediment particle size |

S_{4} | Coefficient of sediment shape |

V | Velocity, m/s |

$\overline{V}$ | Volume of particle |

V_{p} | Particle impact velocity, m/s |

W | Erosion rate, mm/year |

Greek Symbols | |

α | Impingement angle, degree |

β | Impact angle, rad |

μ | coefficient of friction between particle and surface |

ρ | Density, kg/m^{3} |

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**Figure 2.**3-D unstructured grids by parts of the Francis turbine (

**a**) spiral casing, stay vane, and guide vane, (

**b**) runner, and (

**c**) draft tube.

**Figure 3.**Grid sensitivity test of the Francis turbine (vertical dotted line represents the used grid).

**Figure 5.**Computed efficiencies and powers as a function of flow rates of the numerical simulation results for the prototype turbine. SSU: Soongsil University; KIMM: Korea Institute of Machinery and Materials.

**Figure 7.**Effect of sand flow rates on the sand rate density (

**a**) best efficiency (GV-20°) and (

**b**) full load (GV-25°) conditions.

**Figure 9.**Erosion rate densities versus particle inflow rates at three guide vane opening angles (GV-16°, GV-20°, and GV-25°).

**Figure 10.**Sediment erosion rate density for different particle shape factors (0.5, 1.0) (

**a**) best efficiency and (

**b**) full load conditions.

**Figure 11.**Water vapor volume fraction contours on the runner at different sand inflow rates (10 kg/s, 50 kg/s).

**Figure 12.**Sand volume fraction contours on the runner at different sand inflow rates (10 kg/s, 50 kg/s).

**Figure 14.**Comparison of efficiency differences (water versus sand erosion, water versus cavitation-erosion, and sand erosion versus cavitation-erosion) with different particle inflow rates at best efficiency (GV-20°).

**Figure 15.**Comparison of sand erosion and cavitation–erosion rate densities with different particle inflow rates at best efficiency (GV-20°).

Description | Dimension |
---|---|

Runner inlet diameter (mm) | 2728 |

Runner outlet diameter (mm) | 2546 |

Head (m) | 95 |

Flow rate (m^{3}/s) | 59.4 |

Max. power (MW) | 51.6 |

Rotational speed (rpm) | 257 |

Runner blade | 13 |

Guide vane | 24 |

Stay vane | 23 |

Description | Elements | Nodes | Min. Y^{+} | Max. Y^{+} |
---|---|---|---|---|

Spiral casing | 14,552,267 | 246,6738 | 0.81 | 476 |

Runner | 15,868,719 | 268,2908 | 2.71 | 274 |

Draft tube | 4,368,719 | 785,142 | 1.58 | 66 |

Total | 34,789,303 | 5,934,788 |

No. | Nodes | Grid Ratio, r | Efficiency (%) | Error, e_{a} (%) | GCI |
---|---|---|---|---|---|

1 | 963,652 | 2.788 | 90.82 | 0.1515 | 0.02844 |

2 | 2,686,965 | 1.337 | 90.96 | 0.9344 | 1.48009 |

3 | 3,594,116 | 1.133 | 91.81 | 0.1742 | 0.76761 |

4 | 4,072,292 | 1.245 | 91.97 | 0.3153 | 0.71562 |

5 | 5,071,234 | 1.170 | 92.26 | 0.1409 | 0.47659 |

6 | 5,934,788 | 1.020 | 92.39 | 0.0324 | 0.98898 |

7 | 6,055,348 | 0.102 | 92.36 | 0.0866 | 0.10943 |

Description | Efficiency (%) | Difference (%) |
---|---|---|

Pre-experiment (Fuji Electric Co.) | 93.292 (Exp.) | 0 |

Reverse design (KIMM) | 92.324 (Sim.) | 1.0485 |

Simulation (KIMM) | 92.724 (Sim.) | 0.6088 |

Simulation (SSU) | 92.398 (Sim.) | 0.9579 |

Experiment (KIMM) | 92.781 (Exp.) | 0.5477 |

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**MDPI and ACS Style**

Rakibuzzaman, M.; Kim, H.-H.; Kim, K.; Suh, S.-H.; Kim, K.Y.
Numerical Study of Sediment Erosion Analysis in Francis Turbine. *Sustainability* **2019**, *11*, 1423.
https://doi.org/10.3390/su11051423

**AMA Style**

Rakibuzzaman M, Kim H-H, Kim K, Suh S-H, Kim KY.
Numerical Study of Sediment Erosion Analysis in Francis Turbine. *Sustainability*. 2019; 11(5):1423.
https://doi.org/10.3390/su11051423

**Chicago/Turabian Style**

Rakibuzzaman, Md, Hyoung-Ho Kim, Kyungwuk Kim, Sang-Ho Suh, and Kyung Yup Kim.
2019. "Numerical Study of Sediment Erosion Analysis in Francis Turbine" *Sustainability* 11, no. 5: 1423.
https://doi.org/10.3390/su11051423