# Constraints of Parametrically Defined Guide Vanes for a High-Head Francis Turbine

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometry Description

_{d}is the design rotational speed, Qd is the design flow rate, Hn is the design net head, Br

_{1}is the runner inlet height, and g is the gravitational acceleration. According to this, the runner inlet velocity triangle is obtained. The angle ${\alpha}_{o}$ in Figure 1 is the cascade outlet flow angle which needs to be developed, and it slightly differs from the absolute velocity angle ${\alpha}_{1}$ at the runner inlet, due to the free-vortex effect in the vaneless space. The radial cascade outflow conditions are guided from the outlet velocity and its components in a radial (which represents the flow rate) and the tangential (which represents the circulation) manner. The free-vortex law which preserves in the vaneless space is transferred from the runner inlet to the guide vane outlet, with respect to the change of the radial distance as:

_{i}are the adopted parameters for thickness distribution development of the blade.

## 3. Turbine Inputs and Developed Guide Vane Configurations

## 4. CFD Results

_{o}and calculated in accordance with the Euler turbine equation, the relation can be presented as:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Radial cascade geometry parameters: (

**a**) Geometrical scheme representation; (

**b**) description of geometrical parameter symbols.

**Figure 3.**Development of a hydrofoil blade: (

**a**) Calculated velocity flow angles enclosed with the chord line (scheme); (

**b**) Bezier thickness distribution and camber-line used for developing a hydrofoil.

**Figure 4.**Developed guide vane configurations: (

**a**) Developed configuration geometrical data; (

**b**) schematic representation of the geometry changes and analysis of opening space and overlap section with examples of several developed configurations in MATLAB.

**Figure 5.**Numerical mesh: (

**a**) Mesh at the hubs; (

**b**) overall mesh preview; (

**c**) guide vane zone mesh independence test.

**Figure 6.**Relative turbine efficiency for each of the tested combinations described in Figure 4 and the highest and lowest efficiencies for the tested combinations.

**Figure 7.**Results interpretation for Zgv = 32 blades: (

**a**) Cascade density vs. relative turbine efficiency; (

**b**) relative opening vs. relative turbine efficiency; (

**c**) contour plot of (

**a**,

**b**).

**Figure 8.**Results interpretation for Zgv = 28 blades: (

**a**) Cascade density vs. relative turbine efficiency; (

**b**) relative opening vs. relative turbine efficiency; (

**c**) contour plot of (

**a**,

**b**).

**Figure 9.**Results interpretation for Zgv = 26 blades: (

**a**) Cascade density vs. relative turbine efficiency; (

**b**) relative opening vs. relative turbine efficiency; (

**c**) contour plot of (

**a**,

**b**).

**Figure 10.**Results interpretation for Zgv = 24 blades: (

**a**) Cascade density vs. relative turbine efficiency; (

**b**) relative opening vs. relative turbine efficiency; (

**c**) contour plot of (

**a**,

**b**).

**Figure 11.**Results interpretation for guide vane diameters ratios: (

**a**) For Zgv = 32; (

**b**) for Zgv = 28; (

**c**) for Zgv = 26.

**Figure 13.**CFD post-process of velocity streamlines: (

**a**) Combination 16—streamlines (Cri = 1.1; Cro = 1.075; L/t = 1.456; a/L = 0.071; fi = 1.4; Zgv = 28); (

**b**) combination 21—streamlines (Cri = 1.1; Cro = 1.75; L/t = 1.138; a/L = 0.155; fi = 1.09; Zgv = 26).

Description | Symbol | Unit | Value |
---|---|---|---|

Net head | H_{n} | (m) | 11.2 |

Design flow rate | Q_{d} | (m^{3}/s) | 0.2 |

Design rotational speed | n_{d} | (rpm) | 333.33 |

Runner inlet diameter | D_{r1} | (m) | 0.62 |

Runner outlet diameter | D_{r2} | (m) | 0.349 |

Guide vane height | B_{gv} | (m) | 0.06 |

Speed factor (IEC60193) | n_{ed} | (–) | 0.185 |

Discharge factor (IEC60193) | Q_{ed} | (–) | 0.1567 |

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**MDPI and ACS Style**

Stojkovski, F.; Lazarevikj, M.; Markov, Z.; Iliev, I.; Dahlhaug, O.G.
Constraints of Parametrically Defined Guide Vanes for a High-Head Francis Turbine. *Energies* **2021**, *14*, 2667.
https://doi.org/10.3390/en14092667

**AMA Style**

Stojkovski F, Lazarevikj M, Markov Z, Iliev I, Dahlhaug OG.
Constraints of Parametrically Defined Guide Vanes for a High-Head Francis Turbine. *Energies*. 2021; 14(9):2667.
https://doi.org/10.3390/en14092667

**Chicago/Turabian Style**

Stojkovski, Filip, Marija Lazarevikj, Zoran Markov, Igor Iliev, and Ole Gunnar Dahlhaug.
2021. "Constraints of Parametrically Defined Guide Vanes for a High-Head Francis Turbine" *Energies* 14, no. 9: 2667.
https://doi.org/10.3390/en14092667