WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–4
The mechanical power extracted is then the difference between the input and output power
in the wind:
P m,ideal = P 1 − P 4 = 1 2 ρ(A 1u 3 1 − A 4 u 3 4)= 1 2 ρ( 8 9 A 1u 3 1) W (7)
This states that 8/9 of the power in the original tube of air is extracted by an ideal turbine.
This tube is smaller than the turbine, however, and this can lead to confusing results. The
normal method of expressing this extracted power is in terms of the undisturbed wind speed
u 1 and the turbine area A 2 . This method yields
P m,ideal = 1 ρ[ 8 ( 2 A 2)u 3 1]= 1 16
ρ( A 2u 3 1) W (8)
2 9 3 2 27
The factor 16/27 = 0.593 is sometimes called the Betz coefficient. It shows that an actual
turbine cannot extract more than 59.3 percent of the power in an undisturbed tube of air
of the same area. In practice, the fraction of power extracted will always be less because of
mechanical imperfections. A good fraction is 35-40 percent of the power in the wind under
optimum conditions, although fractions as high as 50 percent have been claimed. A turbine
which extracts 40 percent of the power in the wind is extracting about two-thirds of the amount
that would be extracted by an ideal turbine. This is rather good, considering the aerodynamic
problems of constantly changing wind speed and direction as well as the frictional loss due to
blade surface roughness.
It is interesting to note that the total pressure difference across the turbine is rather small.
For a 6 m/s wind speed, p 2 will be about 12.6 Pa greater than p 1 , while p 3 will be about 7.6
Pa less. The pressure difference is then about 0.02 percent of the ambient pressure. Small
pressure differences are therefore able to provide rather substantial turbine power outputs.
2 AERODYNAMICS
Air flow over a stationary airfoil produces two forces, a lift force perpendicular to the air flow
and a drag force in the direction of air flow, as shown in Fig. 3. The existence of the lift
force depends upon laminar flow over the airfoil, which means that the air flows smoothly
over both sides of the airfoil. If turbulent flow exists rather than laminar flow, there will be
little or no lift force. The air flowing over the top of the airfoil has to speed up because of
a greater distance to travel, and this increase in speed causes a slight decrease in pressure.
This pressure difference across the airfoil yields the lift force, which is perpendicular to the
direction of air flow.
The air moving over the airfoil also produces a drag force in the direction of the air flow.
This is a loss term and is minimized as much as possible in high performance wind turbines.
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001