**Wind Turbine Efficiency**

This table is set up to accompany
a blog
posting about wind turbine efficiency because the tabular data would not
display properly in the blog entry. The table shown below consists
of a range of common wind speeds in mph, since that's the easiest number for
Americans to grasp, followed by their equivalent speeds in m/s. The third
column represents the amount of energy present in the wind that moves through an
80 meter diameter wind turbine rotor (5027 m^{2}) at various speeds.
The 4th column represents the Betz Limit which is simply 59% of column three.
The 5th column is derived from the power curve of a Vestas V80 taken from the
bottom graph. The curves are comprised of a range of operating regimes of
the V80 turbine based on whether one is trying to optimize efficiency or noise.
I attempted to use the most efficient of these curves to compute numerical
values where possible. The last column is the conversion efficiency vs.
the Betz limit when converting the wind energy into electricity.

Wind Speed mph |
Wind Speed m/s |
Power (kW) of wind |
Power (kW) Betz limit |
Power (kW) Vestas V80 output |
Conversion Efficiency vs. Betz limit |

5 |
2.2 |
36 |
21 |
0 |
0% |

10 |
4.5 |
285 |
169 |
100 |
59% |

15 |
6.7 |
962 |
570 |
400 |
70% |

20 |
8.9 |
2280 |
1352 |
950 |
70% |

25 |
11.2 |
4453 |
2641 |
1600 |
61% |

28 |
12.5 |
6257 |
3710 |
2000 |
54% |

30 |
13.4 |
7695 |
4563 |
2000 |
44% |

35 |
15.6 |
12220 |
7246 |
2000 |
28% |

40 |
17.9 |
18241 |
10817 |
2000 |
18% |

45 |
20.1 |
25972 |
15401 |
2000 |
13% |

50 |
22.4 |
35626 |
21126 |
2000 |
9% |

55 |
24.6 |
47419 |
28119 |
2000 |
7% |

*56 |
25.0 |
50053 |
29681 |
2000 |
7% |

60 |
26.8 |
61563 |
36507 |
0 |
0% |

**Table of Theoretical, Betz Limit, and Actual output of a
80 M rotor turbine**

* cutoff speed

**Power Curve for
Vestas V80 Wind Turbine from Vestas Brochure**

Example calculation for power values in column 3 of the table:

**ρ = density of the air
A = swept area of turbine’s rotor
v = velocity of the wind**

Thus combining the equations for kinetic energy and wind speed, the power available in wind comes out to:

**½ ρAv ^{3}**

Air density (**ρ**) is about 1.2 kg/m^{3} at sea
level and a temperature of 20 °C. This number varies depending on temperature
and altitude. For example, in Colorado air density is about 1 kg/ m^{3 }
or 20% less than at sea level.

Using a Vestas V80
turbine with an 80 meter rotor as an example, the amount of wind energy
available in a 20 mph (9.8 m/s) wind for this turbine with 5027 m^{2} of
swept area is:

**½ × 1.2 kg/m ^{3 }×^{ }π × (40 m)^{2
}× (8.9 m/s)^{3 }= 2.3 × 10^{6 }kg·m^{2}/s^{3
} = 2.3 MW **