# Optimization of Hydrokinetic Swept Blades

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Blade Element Momentum Theory for Swept Blades

#### 2.1. Axial Momentum Theory with Sweep Effect

#### 2.2. Blade Element Momentum Theory for Turbines with Swept Blades

#### 2.3. Optimization Model for the Turbine Swept Blade

## 3. Results and Discussion

#### 3.1. Validation

#### 3.2. Performance Analysis of the Proposed Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin Symbols | |

a, ${a}^{\prime}$ | Axial and tangential induction factors at the rotor |

${a}_{opt}$ | Optimal axial induction factor |

B | Number of blades |

c | Chord (m) |

${C}_{D}$ | Drag coefficient |

${C}_{L}$ | Lift coefficient |

${C}_{M}$ | Torque coefficient |

${C}_{n}$ | Normal force coefficient |

${C}_{P}$ | Power coefficient |

${C}_{Popt}$ | Optimal power coefficient |

${C}_{t}$ | Tangential force coefficient |

${C}_{T}$ | Thrust coefficient |

$dA$ | Elementary area (m^{2}) |

F | Prandtl tip-loss factor |

${p}_{0}$ | Pressure in the external flow (Pa) |

${p}_{2}$ | Pressure at the turbine upstream (Pa) |

${p}_{3}$ | Pressure at the diffuser outlet (Pa) |

P | Output power (W) |

r | Radial position at the rotor plane (m) |

R | Radius of the rotor (m) |

${V}_{0}$ | Freestream wind velocity (m/s) |

${V}_{3}$ | Axial velocity at the diffuser outlet (m/s) |

w | Total induced velocity (m/s) |

W | Relative velocity (m/s) |

x | Local speed ratio |

Greek Symbols | |

$\alpha $ | Angle of attack (rad) |

$\beta $ | Twist angle (rad) |

$\lambda $ | Tip speed ratio |

$\rho $ | Density of the fluid (kg/m^{3}) |

$\sigma $ | Solidity of the turbine |

$\varphi $ | Flow angle (rad) |

$\Omega $ | Angular speed of the turbine (rad/s) |

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**Figure 1.**Simplified illustration of the velocities at the rotor plane and in the wake for a swept radius [20].

**Figure 5.**(

**a**) SG6040 airfoil for the section of the rotor blade. (

**b**) ${C}_{l}/{C}_{d}$ ratio for the SG6040 foil (Reynolds number of 150,000).

Parameters | Value |
---|---|

Turbine diameter, m | 0.8 |

Hub diameter, m | 0.08 |

Number of blades | 4 |

Stream velocity, m/s | 1.0 |

water density, kg/m^{3} ^{(1)} | 997 |

Angular velocity, rad/s | 10.5 |

Swept angle, degrees | 30 |

^{(1)}at 25 °C.

Swept Blades | Straight Blades | |
---|---|---|

${C}_{Q}$ | 0.13 | 0.11 |

${C}_{P}$ | 0.529 | 0.46 |

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**MDPI and ACS Style**

Gemaque, M.L.A.; Vaz, J.R.P.; Saavedra, O.R. Optimization of Hydrokinetic Swept Blades. *Sustainability* **2022**, *14*, 13968.
https://doi.org/10.3390/su142113968

**AMA Style**

Gemaque MLA, Vaz JRP, Saavedra OR. Optimization of Hydrokinetic Swept Blades. *Sustainability*. 2022; 14(21):13968.
https://doi.org/10.3390/su142113968

**Chicago/Turabian Style**

Gemaque, Miriam L. A., Jerson R. P. Vaz, and Osvaldo R. Saavedra. 2022. "Optimization of Hydrokinetic Swept Blades" *Sustainability* 14, no. 21: 13968.
https://doi.org/10.3390/su142113968