# Application of the 3D Inverse Design Method in Reversible Pump Turbines and Francis Turbines

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## Abstract

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## 1. Introduction

## 2. Literature Overview of the Inverse Design Method

## 3. IDM and Suppression of Secondary Flows

#### 3.1. Secondary Flows: The Example of the “Jet–Wake” Phenomenon

#### 3.2. The IDM and the Suppression of Secondary Flows

## 4. IDM Applications in Hydraulic Turbines

#### 4.1. Application in Pump Turbines

#### 4.2. Applications in Francis Turbines

## 5. Conclusions

- ▪ To improve the efficiency under the two operating modes, a middle-loaded blade loading distribution in the hub and a back-loaded distribution on the shroud can be adopted. If designers want to concentrate on the turbine mode in order to increase its efficiency, a large positive blade lean angle on the high-pressure side is suggested, which means blades are leaning linearly against the direction of rotation in pump mode.
- ▪ Large blade lean angles may also induce a drop in the lowest pressure, negatively affecting the cavitation characteristics. To avoid this, the blade loading on the shroud should be reduced near the low-pressure side.
- ▪ Adopting a fore-loaded distribution at the shroud and an aft-loaded one at the hub and broadening the meridional section could effectively suppress the negative slope of the characteristic curve. Consequently, the runner may have a stable performance curve and hence may satisfy the requirements for safe operation in pumped storage plants.

- ▪ To guarantee a high hydraulic efficiency, it is recommended to load the hub forward and the shroud downward. Moreover, a high negative inlet lean angle is recommended for runners with a relatively short flow passage.
- ▪ The machine cavitation behavior can be improved by adopting an aft-loaded blade at the hub surface and a fore-loaded blade at the shroud surface at the leading edge to suppress pressure reductions, while the opposite situation occurs at the outlet section. Benefits are also derived from the use of positive stacking conditions.
- ▪ The adoption of a negative lean angle may also improve the machine cavitation behavior, resulting in a decrease in the pressure loading on the hub suction side and a pressure rise at the shroud surface. Furthermore, in the case of a limited flow passage length, the inlet blade lean also affects the pressure in the runner outlet section, with potential benefits in terms of pressure pulsation control.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$C$ | absolute velocity |

${C}_{p}$ | pressure coefficient |

$H$ | head |

$m$ | non-dimensional meridional distance |

PS | pressure side |

$p$ | pressure |

$r$ | radius |

$R$ | radius of curvature of the relative fluid trajectories |

$r{C}_{u}$ | flow angular momentum |

${R}_{i\omega}$ | rotational velocity Richardson number |

${R}_{ic}$ | curvature Richardson number |

${R}_{s}$ | splitter ratio |

SS | suction side |

$U$ | peripheral velocity |

$W$ | relative velocity |

y | coordinate perpendicular relative flow trajectories |

$Z$ | blades number |

Greek symbols | |

$\beta $ | blade angle |

$\theta $ | angular coordinate in meridional plan |

$\rho $ | density |

$\omega $ | angular velocity |

Subscripts | |

$m$ | meridional component |

$u$ | tangential component |

Superscripts | |

s | splitter |

+ | blade pressure side |

– | blade suction side |

* | static |

Abbreviations | |

CFD | computational fluid dynamics |

DOE | design of experience |

IDM | inverse design method |

LPS | Latin hypercube sampling |

MOGA | multi-objective genetic algorithm |

PHES | pumped hydro energy storage |

RBF | radial basis function |

RES | renewable energy sources |

RPT | reversible pump turbine |

RSM | response surface method |

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**Figure 1.**The parameters ${R}_{i\omega}$ and ${R}_{ic}$, studied along the blade-to-blade duct of a pump impeller.

**Figure 2.**Mixing of the relative velocity magnitude W due to the production of turbulent kinetic energy k.

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**MDPI and ACS Style**

Zanetti, G.; Siviero, M.; Cavazzini, G.; Santolin, A.
Application of the 3D Inverse Design Method in Reversible Pump Turbines and Francis Turbines. *Water* **2023**, *15*, 2271.
https://doi.org/10.3390/w15122271

**AMA Style**

Zanetti G, Siviero M, Cavazzini G, Santolin A.
Application of the 3D Inverse Design Method in Reversible Pump Turbines and Francis Turbines. *Water*. 2023; 15(12):2271.
https://doi.org/10.3390/w15122271

**Chicago/Turabian Style**

Zanetti, Giacomo, Monica Siviero, Giovanna Cavazzini, and Alberto Santolin.
2023. "Application of the 3D Inverse Design Method in Reversible Pump Turbines and Francis Turbines" *Water* 15, no. 12: 2271.
https://doi.org/10.3390/w15122271