Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

where a – T = 0.3262.
Sharpe (1990) noted that for most practical, existent HAWTs, the value of a – rarely
exceeds 0.6.
Estimating the power output
Preliminary estimates of rotor diameter can easily be made using simple actuator
disc theory. A number of factors need to be taken into account, i.e. the wind regime in
which the turbine is to operate and the tip–speed ratio. Various losses must be allowed
for, the main ones being the mechanical transmission including gearbox losses and the
electrical generation losses. From the actuator disc theory the turbine aerodynamic
power output is
P= A C c
1 r
2 2 p x1
3
Under theoretical ideal conditions the maximum value of Cp = 0.593. According to
Eggleston and Stoddard (1987), rotor Cp values as high as 0.45 have been reported.
Such high, real values of Cp relate to very precise, smooth aerofoil blades and tip–speed
ratios above 10. For most machines of good design a value of Cp from 0.3 to 0.35 would
be possible. With a drive train efficiency, hd and an electrical generation efficiency, hg
the output electrical power would be
P = A C c
1 r h h
el 2 2 p g d x1
3
Example10.4. Determine the size of rotor required to generate 20kW of electrical
power in a steady wind of 7.5m/s. It can be assumed that the air density, r = 1.2kg/m 3 ,
Cp = 0.35, hg = 0.75 and hd = 0.85.
Solution. From the above expression the disc area is
A = 2 P ( rC h h c )= 2 ¥ 20 ¥ 10 1. 2 ¥ 0. 35 ¥ 0. 75 ¥ 0. 85 ¥ 7. 5
3 3 3
2 el p g d x1
= 354. 1 m2 Hence, the diameter is 21.2m.
Power output range
The kinetic power available in the wind is
1 P0 A cx
3
= r
2 2 1
( )
Wind Turbines 337
(10.10b)
where A2 is the disc area and cx1 is the velocity upstream of the disc. The ideal power
generated by the turbine can therefore be expected to vary as the cube of the wind
speed. Figure 10.9 shows the idealised power curve for a wind turbine, where the above
cubic “law” applies between the so-called cut-in wind speed and the rated wind speed
at which the maximum power is first reached. The cut-in speed is the lowest wind speed
at which net (or positive) power is produced by the turbine. The rated wind speed generally
corresponds to the point at which the efficiency of energy conversion is close to
its maximum.
At wind speeds greater than the rated value, for most wind turbines, the power output
is maintained constant by aerodynamic controls (discussed under “Control Methods”).