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v Wi, 2
v Wi, Rot
v Wi, 1
S 1
.
S Rot
m Wi
S 2
.
.
m Wi
m Wi
Fig. 7.1 Flow through an idealised wind energy converter (for an explanation of symbols
see text; see /7-1/)
Regardless of the actual wind energy converter conditions, the same wind
pressure ( p Wi ,1 = p Wi ,2 ) and the same density ( ρ Wi ,1 = ρ Wi ,2 ) are assumed long before
and far behind the wind energy converter. If this assumption is considered for
Equation (7.2), the wind capacity P Wi,ext extracted by the wind energy converter
which, according to the law of power conservation corresponds to the theoretical
rotor capacity P Rot,th , is equal to the difference between the wind capacity before
( P Wi ,1 ) and behind ( P Wi ,2 ) the wind energy converter.
1
1
1
P
=
P
=
P
P
=
m
&
v
2
m
&
v
2
=
m
&
(
v
2
v
2
)
(7.3)
Wi
,
ext
Rot
,
th
Wi
,
Wi
,
2
Wi
Wi
,
Wi
Wi
,
2
Wi
Wi
,
Wi
,
2
2
2
2
Equation (7.3) reveals that from free wind flow, energy is only yielded if wind
speed is reduced. Hence, only kinetic wind energy can be harnessed.
According to the mass conservation law (Equation (7.1)) the stream-tube must
be extended during power extraction - as illustrated in Fig. 7.1 - due to the slow-
down of wind speed and power extraction. The stream-tube cross-section is stead-
ily enlarged since wind speed cannot be reduced abruptly.
For the free flow through the rotor surface S Rot, the kinetic wind capacity is cal-
culated according to Equation (7.4). m . Wi,free refers to the mass flow rate of the free
blown stream-tube without any energy extraction ( m . Wi,free = ρ Wi S Rot v Wi, 1 ) and as-
sumes that the surface S Rot is subject to an undisturbed wind speed v Wi, 1 .
1
1
= &
2
3
P
m
v
=
ρ
S
v
(7.4)
Wi
Wi
,
free
Wi
,
Wi
Rot
Wi
,
2
2
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