# Fatigue Life Study of Francis Pump under Reverse Generation Condition Based on Fluid Solid Coupling

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Establishment and Experiments

#### 2.1. Modeling and Meshing

^{3}/s. The design head, H, was 7.8 m. The specific speed, n

_{s}= 657. An unstructured and hexahedral mesh was used with the ANSYS ICEM CFD, considering the irregular shape of the runner and guide vane area. The number of elements was 5,523,376, and of nodes, 950,675. The mesh quality was controlled by the skewness, which should be less than 0.4 [34]. The numerical model under the pump condition and the experimental site of the Francis pump employed are presented in Figure 1a,b.

#### 2.2. External Characteristics of the Francis Pump

^{3}/s), the head H (m) changed and the corresponding efficiency of the test model ($\eta $) was obtained in both pump and turbine generator.

^{3}/s and dropped subsequently. The prediction errors of the head and the efficiency were less than 5%, thus validating the accuracy of the method [36].

## 3. Numerical Simulation Method

#### 3.1. The Governing Equations

^{−4}, and the coupling convergence standard was 10

^{−3}. The governing equations for numerical simulation of the internal fluids in reverse power generation condition can be found in Equations (1) and (2),

^{2}), ρ is the density (kg/m

^{3}), ${\tau}_{ij}$ is the shear stress (N/m

^{2}) and s

_{ij}is the additional source term (N/m

^{2}). The elastic structural equation of the runner blades [37], then becomes Equation (3),

^{2}) and F is the force on nodes (N).

#### 3.2. Settings of Structural Field

#### 3.3. Settings of Flow Field

## 4. Analysis of the Numerical Results

#### 4.1. Comparison of the Streamline Graph and Blade Pressure Cloud Chart

#### 4.2. Force Analysis of the Blades

#### 4.3. Stress Analysis of the Runner

#### 4.4. Deformation Analysis of the Runner

#### 4.5. Estimation of Fatigue Life

#### 4.5.1. Theoretical Calculation of Fatigue Life

#### 4.5.2. Numerical Calculation of Fatigue Life

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Overall flow channel and test bench diagram under the pump condition. (

**a**) numerical model; (

**b**) experimental site.

**Figure 5.**Streamline graph. (

**a**) Pressure in pump condition; (

**b**) pressure in reverse power generation condition; (

**c**) velocity in pump condition; (

**d**) velocity in reverse power condition.

**Figure 6.**Blade pressure distribution. (

**a**) Suction surface for pump condition; (

**b**) pressure surface for pump condition; (

**c**) pressure surface for reverse generation condition; (

**d**) suction surface for reverse generation condition.

**Figure 8.**Maximum stress change diagram of runner. (

**a**) Stress distribution of reverse power generation; (

**b**) stress distribution of pump condition; (

**c**) maximum stress change in reverse power generation; (

**d**) maximum stress change in pump condition.

**Figure 9.**Maximum deformation change of the runner. (

**a**) Deformation distribution diagram of reverse power generation; (

**b**) deformation distribution diagram of pump condition; (

**c**) maximum deformation change in reverse power generation; (

**d**) maximum deformation of pump condition.

**Figure 10.**Fatigue life analysis under pump and reverse generation conditions. (

**a**) Fatigue life under pump condition; (

**b**) safety factor under pump condition; (

**c**) fatigue life under reverse power generation; (

**d**) safety factor under reverse power generation.

Density (kg/m^{3}) | Young’s Modulus (GPa) | Poisson’s Ratio (-) | Yield Strength (MPa) |
---|---|---|---|

7730 | 203 | 0.291 | 550 |

Condition | X-Direction Radial Force (N) | Y-Direction Radial Force (N) | Radial Resultant Force (N) | Axial Force (N) |
---|---|---|---|---|

Pump condition | 1226.2 | 700.1 | 1412.0 | 88,679.7 |

Reverse power generation condition | 1762.5 | 866.2 | 1963.9 | 67,654.7 |

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## Share and Cite

**MDPI and ACS Style**

Chen, X.; Zheng, Y.; Xu, J.; Zhang, Y.; Fernandez-Rodriguez, E.; Li, C.; Zhou, Y.; Jiang, T.
Fatigue Life Study of Francis Pump under Reverse Generation Condition Based on Fluid Solid Coupling. *Water* **2020**, *12*, 1162.
https://doi.org/10.3390/w12041162

**AMA Style**

Chen X, Zheng Y, Xu J, Zhang Y, Fernandez-Rodriguez E, Li C, Zhou Y, Jiang T.
Fatigue Life Study of Francis Pump under Reverse Generation Condition Based on Fluid Solid Coupling. *Water*. 2020; 12(4):1162.
https://doi.org/10.3390/w12041162

**Chicago/Turabian Style**

Chen, Xiaocui, Yuan Zheng, Junhui Xu, Yuquan Zhang, E. Fernandez-Rodriguez, Chengyi Li, Ying Zhou, and Tao Jiang.
2020. "Fatigue Life Study of Francis Pump under Reverse Generation Condition Based on Fluid Solid Coupling" *Water* 12, no. 4: 1162.
https://doi.org/10.3390/w12041162