# Theoretical Analysis of Shrouded Horizontal Axis Wind Turbines

^{*}

## 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

- Foreman, K.; Gilbert, B.; Oman, R. Diffuser augmentation of wind turbines. Sol. Energy
**1978**, 20, 305–311. [Google Scholar] [CrossRef] - Betz, A. Energieumsetzungen in venturidüsen. Naturwissenschaften
**1929**, 17, 160–164. [Google Scholar] [CrossRef] - Lilley, G.; Rainbird, W. A Preliminary Report on the Design and Performance of Ducted Windmills; College of Aeronautics College of Aeronautics Cranfield: Bedfordshire, UK, 1956. [Google Scholar]
- Oman, R.; Foreman, K.; Gilbert, B. A progress report on the diffuser augmented wind turbine. In Proceedings of the 3rd Biennal Conference and Workshop on Wind Energy Conversion Systems, Washington, DC, USA, 9–11 June 1975; pp. 819–826.
- Hansen, M.O.L.; Sørensen, N.N.; Flay, R. Effect of placing a diffuser around a wind turbine. Wind Energy
**2000**, 3, 207–213. [Google Scholar] [CrossRef] - Ohya, Y.; Karasudani, T. A shrouded wind turbine generating high output power with wind-lens technology. Energies
**2010**, 3, 634–649. [Google Scholar] [CrossRef] - Lawn, C. Optimization of the power output from ducted turbines. Proc. Inst. Mech. Eng. Part A J. Power Energy
**2003**, 217, 107–117. [Google Scholar] [CrossRef] - Van Bussel, G.J. The Science of Making More Torque from Wind: Diffuser Experiments and Theory Revisited; IOP Publishing: Bristol, UK, 2007; Volume 75, p. 012010. [Google Scholar]
- Jamieson, P. Generalized limits for energy extraction in a linear constant velocity flow field. Wind Energy
**2008**, 11, 445–457. [Google Scholar] [CrossRef] - Werle, M.J.; Presz, W.M. Ducted wind/water turbines and propellers revisited. J. Propuls. Power
**2008**, 24, 1146–1150. [Google Scholar] [CrossRef] - Konijn, F.B.J.; Hoeijmakers, H.W.M. One Dimensional Flow Theory for Diffuser Augmented Wind Turbines; University of Twente: Enschede, The Netherlands, 2010; Volume 217, p. 9. [Google Scholar]
- Bergey, K. The Lanchester-Betz limit (energy conversion efficiency factor for windmills). J. Energy
**1979**, 3, 382–384. [Google Scholar] [CrossRef] - Okulov, V.L.; van Kuik, G.A. The Betz–Joukowsky limit: On the contribution to rotor aerodynamics by the british, german and russian scientific schools. Wind Energy
**2012**, 15, 335–344. [Google Scholar] [CrossRef] - Jamieson, P. Innovation in Wind Turbine Design; John Wiley & Sons: Hoboken, NJ, USA, 2011. [Google Scholar]
- Manwell, J.F.; McGowan, J.G.; Rogers, A.L. Wind Energy Explained: Theory, Design and Application; John Wiley & Sons: Hoboken, NJ, USA, 2010. [Google Scholar]
- Fox, R.W.; McDonald, A.; Pitchard, P. Introduction to Fluid Mechanics; John Wiley: Hoboken, NJ, USA, 2004. [Google Scholar]
- Phillips, D.G. An Investigation on Diffuser Augmented Wind Turbine Design. Ph.D. Thesis, ResearchSpace@ Auckland, University of Auckland, Auckland, New Zealand, 2003. [Google Scholar]
- Hansen, M.O. Aerodynamics of Wind Turbines; Routledge: London, UK, 2015. [Google Scholar]

**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 | ${V}_{\infty}\xb7\frac{4}{\psi +4}$ | ${V}_{\infty}\xb7\sqrt{\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(\right)open="("\; close=")">1-{\eta}_{N}}}$ |

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 ${C}_{P}$ | $\frac{64\psi}{{\left(\right)}^{\psi}}$ | $\psi {\left(\right)}^{\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(\right)open="("\; close=")">1-{\eta}_{N}}}\frac{3}{2}$ |

Thrust coefficient ${C}_{t}$ | $\frac{16\psi}{{\left(\right)}^{\psi}}$ | $\psi \left(\right)open="["\; close="]">\frac{1-{C}_{{p}_{b}}}{\psi +{\eta}_{N}+{\lambda}_{N}^{2}\left(\right)open="("\; close=")">1-{\eta}_{N}}$ |

**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 ${\eta}_{D}$ | $85\%$ |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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