# Theoretical Analysis of Shrouded Horizontal Axis Wind Turbines

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## Abstract

**:**

## 1. Introduction

## 2. Review of Previous Analytical Models

## 3. 1-D Theory of Nozzle Diffuser Augmented Wind Turbines

#### 3.1. Assumptions and Geometry

#### 3.2. 1-D Theory for Bare Wind Turbines

#### 3.3. 1-D Theory for Shrouded Turbines with Nozzle and Diffuser

- The largest possible negative value of the exit plane pressure coefficient (i.e., diffuser exit pressure is reduced well below atmospheric pressure).
- The largest possible diffuser and nozzle efficiencies.
- A unique relation of turbine disk loading to diffuser pressure recovery in which high recovery favors low power loading by inducing greater volume flow through the disk.

## 4. Validation

#### 4.1. Validation with Field Experimental Results

#### 4.2. Validation with CFD Results

## 5. Diffuser Only vs. Converging-Diverging Nozzle

## 6. Results and Discussions

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Wind turbine equipped with a brimmed diffuser [6].

**Figure 3.**Power and thrust coefficients of the NDAWT using the data in Table 2 and in comparison with a bare wind turbine.

**Figure 4.**Comparison between the theoretical and experimental results [6] for the NDAWT and bare wind turbine.

**Figure 5.**The duct shape used in Hansen’s study [18].

**Figure 6.**Theoretical calculations compared to the CFD results by Hansen et al. [5].

**Figure 7.**The effect of the pressure and loading coefficients on the velocity distribution of diffuser only and and converging-diverging (C-D) nozzle.

**Figure 8.**The effect of the pressure and loading coefficients on the power coefficient of diffuser only and C-D nozzle.

**Figure 9.**The effect of the pressure and loading coefficients on the thrust coefficient of diffuser only and C-D nozzle.

**Figure 10.**Plot of relative velocity ratios using ${D}_{i}=0.76$ m, ${D}_{1}=0.7$ m and ${D}_{e}=1.072$ m for different efficiencies using Equation (33).

**Figure 12.**Plot of power coefficient using ${D}_{i}=0.76$ m, ${D}_{1}=0.7$ m, and ${D}_{e}=1.072$ m for different efficiencies using Equation (36).

**Figure 14.**Plot of thrust coefficient using ${D}_{i}=0.76$ m, ${D}_{1}=0.7$ m and ${D}_{e}=1.072$ m for different efficiencies using Equation (34).

**Table 1.**Summary results of bare wind turbine and the current method. NDAWT, nozzle diffuser-augmented wind turbine.

Parameter | Bare Wind Turbine | NDAWT |
---|---|---|

Upstream wind speed | ${V}_{\infty}$ | ${V}_{\infty}$ |

Wind speed at rotor plane | ${V}_{\infty}\xb7\frac{4}{\psi +4}$ | ${V}_{\infty}\xb7\sqrt{\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(1-{\eta}_{N}\right)-{\eta}_{D}\left(1-{\lambda}_{D}^{2}\right)}}$ |

Far wake wind speed | ${V}_{\infty}\xb7\frac{4-\psi}{\psi +4}$ | ${V}_{\infty}\xb7\sqrt{{C}_{{p}_{b}}+{\lambda}_{D}^{2}\frac{{V}_{1}^{2}}{{V}_{\infty}^{2}}}$ |

Power coefficient ${C}_{P}$ | $\frac{64\psi}{{\left(\psi +4\right)}^{3}}$ | $\psi {\left[\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(1-{\eta}_{N}\right)-{\eta}_{D}\left(1-{\lambda}_{D}^{2}\right)}\right]}^{\frac{3}{2}}$ |

Thrust coefficient ${C}_{t}$ | $\frac{16\psi}{{\left(\psi +4\right)}^{2}}$ | $\psi \left[\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(1-{\eta}_{N}\right)-{\eta}_{D}\left(1-{\lambda}_{D}^{2}\right)}\right]$ |

**Table 2.**Parameters of a shrouded wind turbine with a brim [6].

Parameter | Value |
---|---|

Turbine diameter D | 0.7 m |

Diffuser diameter ${D}_{e}$ | 1.072 m |

Nozzle diameter ${D}_{N}$ | 0.78 m |

Diffuser length ${L}_{t}$ | 1.029 m |

The brim height h | 0.35 m |

Density ρ | 1.225 kg/m${}^{3}$ |

The back pressure of the diffuser ${C}_{{p}_{b}}$ | −0.6 |

The nozzle efficiency ${\eta}_{N}$ | $85\%$ |

The diffuser efficiency ${\eta}_{D}$ | $85\%$ |

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**MDPI and ACS Style**

Khamlaj, T.A.; Rumpfkeil, M.P. Theoretical Analysis of Shrouded Horizontal Axis Wind Turbines. *Energies* **2017**, *10*, 38.
https://doi.org/10.3390/en10010038

**AMA Style**

Khamlaj TA, Rumpfkeil MP. Theoretical Analysis of Shrouded Horizontal Axis Wind Turbines. *Energies*. 2017; 10(1):38.
https://doi.org/10.3390/en10010038

**Chicago/Turabian Style**

Khamlaj, Tariq Abdulsalam, and Markus Peer Rumpfkeil. 2017. "Theoretical Analysis of Shrouded Horizontal Axis Wind Turbines" *Energies* 10, no. 1: 38.
https://doi.org/10.3390/en10010038