Page 43 - Fluid Power Engineering

Basics of W ind Energy and Power 21

of the 59.3% of total wind energy that is extracted from wind is trans-
ferred to the turbine, but some of it is used to overcome viscous drag
on blades and create vortices in the wake. Within the turbine, most of
the energy is converted into useful electrical energy, while some of it
is lost in gearbox, bearings, generator, power converter, transmission
and others. Most practical rotors with three blades reach an overall
efficiency of about 50%. From a theoretical standpoint, Okulov et al. 2
show that a rotor reaches the Betz limit with a large number blades
operating at a very high tip speed ratio (the ratio of tip speed to wind
speed).
Often, inventors claim to have created a rotor that achieves an
efficiency that is greater than 59.3%; such claims are suspect and must
be analyzed. There are turbine configurations that may violate the Betz
3
limit. A shrouded rotor that augments axial velocity is an example.
No large turbine has been created, to the best of author’s knowledge,
with a shroud; however, there are a few small horizontal and vertical
axis turbines with shroud. The following are illustrative examples of
the Betz limit.

Example 1
As an example, consider 1-MW rated turbine with rotor diameter =
70 m and power curve, as shown in Fig. 2-8. Power curve is a key
performance indicator of a turbine that is provided by the turbine
manufacturer. It describes the relationship between power produced
and wind speed. A Betz limit curve is also plotted in Fig. 2-9. It is
alwaysabovethepowercurve,whichimpliesthattheturbineiswithin
the Betz limit at all wind speeds.
16 ρA r v 3 0 16 ρA r v 3 0
P Betz (v 0 ) = = (2-33)
27 2 27 2
To show calculations at one point in the curve, ν 0 = 12 m/s with air
3
density of 1.225 kg/m :
2 3
70
16 1.225π 2 12
P Betz (v 0 ) = = 2.4 MW (2-34)
27 2
Note the power curve indicates a power production capacity of
slightly less than 1 MW, which is less than P Betz .
Example 2
As a second example, consider a turbine with rotor diameter = 2 m
and power rating of 2 KW at 12 m/s.

2 3
2
3
ρA r v 0 1.22π 2 12
P ideal = = = 3.3 kW (2-35)
2 2
P Betz = 0.59 P ideal = 1.953 kW (2-36)