# Investigation of Structural Strength and Fatigue Life of Rotor System of a Vertical Axial-Flow Pump under Full Operating Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Fluid Governing Equations

#### 2.2. Structural Governing Equations

## 3. Numerical Settings

#### 3.1. Geometry of Pump Device

#### 3.2. Fluid Field

#### 3.2.1. Spatial Discretization

#### 3.2.2. Solution Method and Boundary Conditions

#### 3.2.3. Validation of External Characteristics

#### 3.3. Structural Field

## 4. Results and Discussion

#### 4.1. Analysis of Deformation and Stress of Blades

#### 4.1.1. Total Deformation of Blades

#### 4.1.2. Equivalent Stress of Blades

#### 4.2. Analysis of Fatigue Life of Impeller

#### 4.2.1. Number of Stress Cycles

#### 4.2.2. Blade Safety Factor

## 5. Conclusions

- (1)
- When the axial-flow pump device operates in both pump mode and PAT mode, the maximum blade deformation increases with the increasing flow rate. The deformation at the blade root can be neglected and the radial deflection gradient is small, while the deflection gradient increases gradually near the top of the blade. In comparison, the maximum deformation in the PAT mode is generally higher than that in the pump mode at all flow rates.
- (2)
- Under all operating conditions, the stress concentration phenomenon mainly occurs at the blade root, with the maximum equivalent stress at the impeller root occurring at the blade suction surface, and the stress gradually decreasing from the blade root to the blade edge. Comparing different flow rate conditions, the maximum equivalent stress of both modes occurs at a flow rate of $1.2{Q}_{\mathrm{BEP}}$. Under the same flow rate conditions, the equivalent stress in the PAT mode is relatively higher than in the pump mode.
- (3)
- The number of cycles of the impeller exceeds ${10}^{6}$ under all operating conditions, indicating that the load carrying capacity of the impeller is within the safe allowable range when the axial-flow pump device operates in both modes. The simulated impeller safety factor in the pump mode and the pump-turbine mode is slightly higher than the theoretical calculation value, which is reasonable. The minimum safety factor appears at the blade root, and the safety factor is smaller when operating in the PAT mode. Therefore, the blade root of the blade needs to be strengthened during processing to ensure the safe and stable operation of the pump device.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational fluid dynamics |

FEM | Finite element method |

FSI | Fluid–structure interaction |

FFT | Fast Fourier transform |

PAT | Pump as turbine |

RNG | Re-normalization group |

RANS | Reynolds-averaged Navier–Stokes |

BEP | Best efficiency point |

PS | Pressure surface |

SS | Suction surface |

LE | Leading edge |

TE | Trailing edge |

${u}_{i}$ and ${u}_{j}$ | Velocity components in x, y and z direction |

i and j | Directional indices for x, y and z |

t | Physical time |

$\rho $ | Fluid density |

$\mu $ | Fluid dynamic viscosity |

${f}_{i}$ | External force |

k | Turbulence kinetic energy |

$\u03f5$ | Turbulence dissipation rate |

${C}_{\u03f51\mathrm{RNG}}$, ${C}_{\u03f52}$, ${\sigma}_{k}$ and ${\sigma}_{\u03f5}$ | Constants of the RNG model |

$\mathit{M}$ | Matrix of structure mass |

$\mathit{C}$ | Matrix of structural damping |

$\mathit{K}$ | Matrix of structural rigidity |

$\ddot{\mathit{q}}$ | Nodal acceleration vector |

$\dot{\mathit{q}}$ | Nodal velocity vector |

$\mathit{q}$ | Nodal displacement vector |

$\mathit{Q}$ | Fluid load vector |

$\eta $ | Efficiency |

H | Head |

D | Impeller diameter |

Q | Flow rate |

${Q}_{\mathrm{r}}$ | Rated flow rate |

n | Rotation speed |

${n}_{\mathrm{r}}$ | Rated rotation speed |

$\delta $ | Deformation |

$\sigma $ | Equivalent stress |

${\sigma}_{1}$, ${\sigma}_{2}$ and ${\sigma}_{3}$ | Principle stresses |

${n}_{\sigma}$ | Safety factor |

${\sigma}_{-1}$ | Fatigue limit |

${K}_{\sigma D}$ | Fatigue reduction coefficient |

${\sigma}_{a}$ | Stress amplitude |

$\left[n\right]$ | Safety allowance coefficient |

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**Figure 2.**Schematic diagram of (

**a**) integral flow channel and (

**b**) pump section of axial-flow pump device.

**Figure 6.**Comparison between simulation and experiment values of head and efficiency for axial-flow pump under (

**a**) Pump mode and (

**b**) PAT mode.

**Figure 7.**Details of structural field setting including (

**a**) grid division of structural solid domain and (

**b**) structural field constraint.

**Figure 8.**Total deformation of blades under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in pump mode.

**Figure 9.**Total deformation of blades under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in PAT mode.

**Figure 10.**Equivalent stress of blades under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in pump mode.

**Figure 11.**Equivalent stress of blades under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in PAT mode.

**Figure 12.**Blade safety factor under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in pump mode.

**Figure 13.**Comparison between theoretical value and numerical simulation value of minimum safety factor of impeller fatigue in pump mode.

**Figure 14.**Blade safety factor under the flow rate of (

**a**) $0.8{Q}_{\mathrm{BEP}}$, (

**b**) $1.0{Q}_{\mathrm{BEP}}$ and (

**c**) $1.2{Q}_{\mathrm{BEP}}$ in PAT mode.

**Figure 15.**Comparison between theoretical value and numerical simulation value of minimum safety factor of impeller fatigue in PAT mode.

Impeller Diameter D/(mm) | Rated Rotational Speed ${\mathit{n}}_{\mathit{r}}$/(r/min) | Rated Flow Rate ${\mathit{Q}}_{\mathit{r}}$/(m${}^{3}$/s) | Number of Impeller Blades | Number of Guide Vanes |
---|---|---|---|---|

2350 | 166.7 | 16.67 | 5 | 8 |

Scheme | Grid Size at Blade/(mm) | Number of Units | Number of Nodes | Ultimate Stress/(Mpa) | Maximum Offset/(mm) |
---|---|---|---|---|---|

I | 45 | 45,557 | 83,243 | 38.87 | 0.398 |

II | 35 | 74,511 | 133,740 | 43.86 | 0.412 |

III | 25 | 145,001 | 255,622 | 47.92 | 0.413 |

IV | 15 | 322,237 | 557,320 | 49.93 | 0.408 |

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## Share and Cite

**MDPI and ACS Style**

Li, H.; Cai, Z.; Zheng, Y.; Feng, J.; Xu, H.; Chen, H.; Binama, M.; Kan, K.
Investigation of Structural Strength and Fatigue Life of Rotor System of a Vertical Axial-Flow Pump under Full Operating Conditions. *Water* **2023**, *15*, 3041.
https://doi.org/10.3390/w15173041

**AMA Style**

Li H, Cai Z, Zheng Y, Feng J, Xu H, Chen H, Binama M, Kan K.
Investigation of Structural Strength and Fatigue Life of Rotor System of a Vertical Axial-Flow Pump under Full Operating Conditions. *Water*. 2023; 15(17):3041.
https://doi.org/10.3390/w15173041

**Chicago/Turabian Style**

Li, Haoyu, Zhizhou Cai, Yuan Zheng, Jiangang Feng, Hui Xu, Huixiang Chen, Maxime Binama, and Kan Kan.
2023. "Investigation of Structural Strength and Fatigue Life of Rotor System of a Vertical Axial-Flow Pump under Full Operating Conditions" *Water* 15, no. 17: 3041.
https://doi.org/10.3390/w15173041