Environmental Engineering Reference
In-Depth Information
v
Wi,Rot
= (
v
Wi,
1
+
v
Wi,
2
)/2). The mass flow rate
m
.
Wi
determined according to Equa-
tion (7.1) with regard to the rotor level allows the calculation of the theoretical
rotor power
P
Rot,th
or the power extracted by the rotor
P
Wi,ext
long before (
v
Wi,
1
) and
far behind the rotor (
v
Wi,
2
), the rotor circular surface
S
Rot
and the air density
ρ
Wi
.
v
+
v
1
⎛
⎞
Wi
,
Wi
,
2
2
2
P
=
P
=
ρ
⎜
⎝
⎟
⎠
S
(
v
−
v
)
(7.7)
Wi
,
ext
Rot
,
th
Wi
Rot
Wi
,
Wi
,
2
2
2
The theoretical power coefficient
c
p,th
expresses the respective maximum
physical conversion from wind into rotor power and thus the ratio of the maxi-
mum power to the power contained in undisturbed wind. It is defined as the ratio
of extractable power (
P
Wi,ext
; Equation (7.7)) to the theoretical maximum wind
power (
P
Wi
; Equation (7.4)). The power coefficient is calculated by Equation (7.8)
on the basis of the wind velocities long before and far behind the wind energy
converter (rotor). In this context, the velocity ratio (
v
Wi,
2
/v
Wi,
1
) is referred to as
wind velocity reduction factor.
⎛
v
+
v
⎞
⎛
2
2
⎞
⎛
+
⎞
⎛
−
2
⎞
P
v
−
v
1
v
v
⎜
⎝
Wi
,
Wi
,
2
⎟
⎠
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
c
=
Wi
,
ext
=
Wi
,
Wi
,
2
=
1
Wi
,
2
1
Wi
,
2
p
,
th
2
2
P
2
v
v
2
v
v
(7.8)
Wi
Wi
,
Wi
,
Wi
,
Wi
,
Wind power exploitation aims at extracting the maximum share of wind power.
Due to physical restrictions wind masses flowing through rotor level cannot be
entirely slowed down, as a complete slowdown would "clog" the rotor and impede
power extraction. On the other hand, wind velocity must be decreased if power is
to be extracted from flowing air masses. Consequently, there must exist a certain
ratio between the speed long before and far behind the rotor that corresponds to
the maximum power coefficient
c
p,th
.
To determine the maximum wind power that can be extracted from the wind by
means of a wind energy converter
P
Wi,ext
, Equation (7.7) needs to be differentiated
with respect to
v
Wi,
2
and zeroed (Equation (7.9)).
⎛
v
+
v
⎞
1
⎛
⎞
(
)
⎜
⎝
Wi
,
Wi
,
2
2
2
⎟
⎠
d
ρ
⎜
⎝
⎟
⎠
S
v
−
v
Wi
Rot
Wi
,
Wi
,
2
2
2
(7.9)
!
=
0
d
v
Wi
,
2
When resolving Equation (7.9) with respect to the rotor, the energetically most
favourable wind velocity appears to be obtained at one third of the wind speed be-
fore the rotor
v
Wi,
1
. The function shown in Fig. 7.3 also reveals this context. Ac-
cording to the curve the maximum (
c
p,ideal
) of the theoretical power coefficient
c
p,th
is achieved at a ratio of wind velocities of one third behind and before the rotor.
Search WWH ::
Custom Search