Wind Turbines - Wind Energy Systems for Electric Power Generation - page 11

is the specific air mass which depends on air pressure and moisture; for
practical calculations it may be assumed

Here ρ

1 . 2kg / m 3 . The air streams in axial
direction through thea wind turbine, of which A is the circular swept area. The useful
mechanical power obtained is expressed by means of the power coefficient c p :
P = c p 2

ρ ≈

Av 1 3 (2.2)

In case of homogenous air flow the wind velocity, whose value before the tur-
bine plane is v 1 , suffers a retardation due to the power conversion to a speed v 3
well behind the wind turbine, see Fig. 2.1. Simplified theory claims that in the
plane of the moving blades the velocity is of average value v 2 =( v 1 + v 3 ) / 2.
On this basis Betz [Bet26] has shown by a simple extremum calculation that the
maximum useful power is obtained for v 3 / v 1 = 1 / 3; where the power coefficient
becomes c p = 16 / 27 0 , 59. In reality wind turbines display maximum values

c p , max = 0 , 4 ... 0 , 5 due to losses (profile loss, tip loss and loss due to wake ro-
tation). In order to determine the mechanical power available for the load machine
(electrical generator, pump) the expression (2.2) has to be multiplied with the effi-
ciency of the drive train, taking losses in bearings, couplings and gear boxes into
account.
An important parameter of wind rotors is the tip-speed ratio

≈

which is the ratio
of the circumferential velocity of the blade tips and the wind speed:

λ

= u / v 1 = D
2

v 1

·

λ (2.3)

is the angular rotor speed. Note that
the rotational speed n (conventionally given in min − 1 ) is connected with

Here D is the outer turbine diameter and Ω

(in s − 1 ) Ω

by n / 60.

Considering that in the rotating mechanical system the power is the product of
torque T and angular speed

Ω = 2 π

· Ω

Ω ( P = T ), the torque coefficient c T can be derived

from the power coefficient:

)= c p (

λ )

c T ( λ (2.4)

λ

Fig. 2.1 Idealized fluid model for a wind rotor (Betz)