Environmental Engineering Reference
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λ opt
Tip speed ratio λ
λ
Figure 1.39 Power coefficient C p as a function of the tip-speed ratio
the gap between the rotor blades without doing any work; if the rotor turns too quickly, the
blurring blades block the wind like a solid wall. Hence, wind turbines are designed to operate
at optimal tip-speed ratios so that as much power as possible can be extracted.
The power coefficient C p is a highly non-linear function of
λ
and
β
. For wind turbines with
a fixed pitch angle
often has the shape
shown in Figure 1.39. The power coefficient reaches its maximum point at the optimum tip-
speed ratio
β
, the relationship between C p and the tip speed ratio
λ
λ opt , which depends on the number of blades in the wind turbine rotor. The fewer the
number of blades, the faster the wind turbine rotor needs to turn to extract the maximum power
from the wind. A two-bladed rotor has an optimum tip-speed ratio of around 6, a three-bladed
rotor around 5, and a four-bladed rotor around 3 (Burton 2001). According to (1.7), the curves
of C p against different wind speeds have similar shapes, as shown in Figure 1.40(a), where six
C p curves are shown for six different wind speeds with
v w 1 >v w 2 >v w 3 >v w 4 >v w 5 >v w 6 .
The corresponding power P m is shown in Figure 1.40(b). Therefore, the operational points
of a wind turbine at different wind speeds are different, which are determined by the optimal
tip-speed ratio
λ opt and the wind speed. See (Heier 2006) for more details. It is worth noting
that the power coefficient C p of a wind turbine is limited by 1 27
.
0
593, according to the
Betz law. 5
1.4.2 Wind Turbines
A wind turbine is a device that captures the kinetic energy of wind. Historically, a wind turbine
was frequently used as a mechanical device with a number of blades to drive machinery.
Nowadays, it is often used to drive a generator so that the kinetic energy is converted to
electricity. The main types of wind turbines are shown in Figure 1.41 (Heier 2006). Most
modern wind turbines use a horizontal axis configuration with two or three blades, operating
either downwind or upwind (Manwell et al . 2009). The typical structure of a horizontal axis
5 http://en.wikipedia.org/wiki/Betz%27 law
 
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