Environmental Engineering Reference
In-Depth Information
0.16
0.16
0.14
0.14
0.12
0.12
0.10
0.10
F R
F R
0.08
0.08
F D
F D
0.06
0.06
v s
v s
v Wi,Rot
v Wi,Rot
0.04
0.04
0.02
0.02
S
S
0.00
0.00
0.0
0.0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.0
Speed ratio v s /v Wi,Rot
Speed ratio v s /v Wi,Rot
Fig. 7.9 Flow conditions and forces of airfoil drag (left) and shape of the corresponding
power coefficient (right) (for an explanation of symbols see text; see /7-4/, /7-5/, /7-6/)
The relation of power extracted from moving air masses to the power contained
in the airflow P Wi is expressed by Equation (7.4). Power coefficient c p is thus de-
duced from Equation (7.20) (i.e. the relation of extracted power to power con-
tained in the wind).
(
)
2
P
c
v
v
v
Wi
,
ext
c
=
=
d
Wi
S
S
(7.20)
p
3
P
v
Wi
Wi
Differentiating and zeroing the formula with respect to v S enables to determine
the maximum extractable wind power according to the drag principle. Also these
mathematical operations reveal that the maximum power coefficient c p is achieved
if the wind-blown surface is moved with one third of the wind velocity (Fig. 7.9),
since, according to Betz, derivation is independent of the wind converter type. For
this speed relation a maximum power coefficient c p,max of 14.8 % of the drag coef-
ficient c d is determined by Equation (7.21).
4
c
=
c
(7.21)
p
,
max
d
27
For instance, an infinitely large plate has a drag coefficient of 2.01; under these
circumstances the maximum power coefficient c p,max would amount to about 0.3.
For the rotor blades of drag-type wind turbines maximum drag coefficients of 1.3
are achieved, whereas the corresponding maximum power coefficient amounts to
0.2 or 20 %. The drag principle thus only allows for exploitation of roughly one
third of the ideal Betz value of 0.593.
This theoretical principle is even more restricted in practice due to special de-
sign requirements. Generally, rotations are required to drive a generator. Conse-
quently, the wind-blown surface needs to rotate around an axis (Fig. 7.9). Usually,
several star-type rotor blades are applied. However, wind rotation only advances
 
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