Page 96 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Offshore Wind Chapter | 4 89

Consider a wind turbine that has a diameter D and a swept area A.Ifthe
wind speed is u, the available or ‘theoretical’ wind kinetic energy per unit time
(i.e. wind power, P t ) can be evaluated as
d 1 2 1 dm 2 1 2
P t = mu = u = ρQu (4.1)
dt 2 2 dt 2
where Q is the volumetric flow rate of the wind and ρ is the air density.
Replacing Q with Au, leads to
1 3
P t = ρAu (4.2)
2
Only part of the available/theoretical energy can be harvested by a wind turbine.
The power coefficient (C p ; or overall efficiency of a turbine) can be defined
as the ratio of the extracted power (generated electricity) to the available wind
power as follows:

P e
C p = (4.3)
P t
where P e is the extracted power. The power coefficient depends on a combina-
tion of aerodynamic, mechanical, and electrical efficiencies. It can be shown
theoretically that the maximum power coefficient of a horizontal axis wind
turbine is 59%, which is called the Betz limit/law.

Betz Limit
Fig. 4.6 shows a schematic of a streamtube that is classically used to analyse
wind turbines based on actuator disk theory. Because turbine blades partially

Upwind velocity Wind turbine Wake
, p
u w o
u , p o
o
u h
u t

A dis

u o u dis = (1 − a)u o u = (1 − 2a)u o
w

FIG. 4.6 Streamtube of a horizontal axis wind turbine.