Geoscience Reference
In-Depth Information
Fig. 6.2 Wind-speed
dependent turbine thrust
coefficient [see Eq. (
6.13
)]
used in the simple analytical
model park model
relation for the thrust coefficient (taken from Fig. 9 in Magnusson
1999
) and
additionally consider the maximal value at Betz's limit
1
:
C
T
¼
min
ð
max
ð
0
:
25; 0
:
5
þ
0
:
05
ð
14
u
h
ÞÞ
; 0
:
89
Þ
ð
6
:
13
Þ
Due to (
6.13
), c
t
depends on u
h
(see Fig.
6.2
) and we have to iterate at least
once when we want to solve for u
h
later.
The reduction of wind speed in hub height h in the park interior does not only
depend on the turbine drag coefficient c
t
but also on the roughness of the surface
underneath the turbines. This surface roughness can be described by a surface drag
coefficient, c
s,h
observed at height h by rearranging (
6.11
):
W
2
h
z
0
h
L
c
s
;
h
¼
u
2
=
u
h
¼
j
2
ln
ð
6
:
14
Þ
Turbine drag and surface drag can be combined in an effective drag coefficient:
c
teff
¼
c
t
þ
c
s
;
h
:
ð
6
:
15
Þ
There are two ratios describing the wind reduction in the wind park. The
reduction of the wind speed at hub height compared to the undisturbed wind speed
aloft is denoted by R
u
:
R
u
¼
u
h
u
0
ð
6
:
16
Þ
The reduction of the wind speed at hub height compared to the undisturbed
wind speed upstream of the wind park in the same height h, u
h0
is denoted by R
t
:
1
The thrust coefficient is the ratio of resistance force T to the dynamic force 0.5qu
2
0
D (rotor area
D). The resistance force of an ideal turbine is given by T = 0.5qu
2
0
A[4r(1-r)] with r=(u
o
-u*
h
)/
u
0
. u*
h
is the mean of u
h
and u
0
. We have u*
h
=u
0
(1-r). Thus, C
T
=[4r(1-r)]. For u
h
=0it
follows u*
h
= 0.5u
0
, r = 0.5 and C
T
= 1. For u
h
=u
0
follows u*
h
=u
0
, r=0 and C
T
= 0. The yield
is P = Tu*
h
0.5= qu
3
A[4r(1-r)
2
] and the yield coefficient is C
P
=[4r(1-r)
2
]. For optimal yield at
the Betz's limit is r = 1/3 (calculated from qC
P
(r)/qr = 0) and C
T
= 8/9 (Manwell et al.
2009
)
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