In accordance with section 2.3.1 the kinetic energy of a mass in motion is
2
2 1 mv
E . (5.1)
The power in the wind flowing through a control area A can be calculated as
13 wind 2
2 1 2
1mv Av
E
P U (5.2)
since the mass flow rate is m = U A dx/dt = U A v1, see Fig. 5-1.
As already illustrated in chapter 2, the wind’s power cannot be extracted com- pletely. Betz [1] analysed a heavily idealised wind turbine that extracts the power from the wind at an ‘active plane’ by retarding its speed, without any losses. The physical process that extracts the kinetic energy from the air flow is not yet con- sidered at this stage.
As shown, in Fig. 5.2, Betz assumed a homogeneous air flow v1 that is retarded by the turbine to the velocity v3 far downstream. That means that, for the reasons of continuity, he assumes a stream tube with divergent streamlines
U v1 A1 = U v2 A = Uv3A3 . (5.3)
x
v1
A
dx
Fig. 5-1 Air flow in a stream tube of velocity v through a control area A (from chapter 2)
v1
v2 v3
v1 v1
Fig. 5-2 Air flow through an ideal Betz wind turbine (from chapter 2)
As the changes in the air pressure are minimal, the air density U can be assumed to be constant.
The extracted kinetic energy Eex is upstream energy minus downstream energy
2 23
1
ex -v
2 1 mv
E . (5.4)
The power extracted from the wind is therefore
2 23
1
ex -v
2 1 mv
E . (5.5)
If the wind were not retarded at all (v3 = v1), no power would be extracted. If the wind is retarded too much, the mass flow rate m becomes very small. Taken to the
5.1 How much power can be extracted from the wind?
170
extreme (m = 0), this would lead to a ‘congestion’ of the stream tube (v3 = 0) such that once again no power can be extracted. There must be a value between v3
= v1 and v3 = 0 for maximum power extraction. This value can be determined if the velocity v2 at the rotor plane is known. The mass flow rate will then be
m = U A v2 . (5.6)
At this point, the plausible assumption v2 = 2
3 1 v v
(5.7) is made, which will be proved later on (Froude-Rankine Theorem). If the mass
flow rate of equation (5.6) and the rotor plane velocity v2 of equation (5.7) are inserted into the power-extraction equation (5.5), we obtain
ằằ
ẳ º
ôô
ơ ê
áá
á
ạ
ã
¨¨
¨
©
§
ááạ
ăă ã
© §
áá
ạ
ăă ã
©
§
2 1 3 1
3 13
ex 1 1
2 1 2
1
v v v
v v A
E U (5.8)
Power in the wind Power coefficient cP
Thus, the power in the wind is multiplied by a factor cP which depends on the ratio v3 / v1 .
The maximum power coefficient is cP, Betz =
27
16 = 0.59 . (5.9)
It occurs when the wind velocity v1 (upstream of the rotor) is retarded to v3 = (1/3)
60% of the power in the wind is extractable by an ideal wind turbine! In this case, the velocity in the rotor plane is 2v1/3 and far downstream v1/3.
The diagram, in Fig. 5.4, illustrates how much power can be extracted in Betz’s ideal case of cP, Betz = 0.59. Of course, it depends on the wind velocity and the diameter of the wind turbine, Due to additional losses which will be analysed later on, the actual power of modern wind turbines is somewhat smaller. However, values up to cP § 0.50 are being achieved by some modern wind turbines.
v1 (downstream). This can be found either by drawing the curve, Fig. 5-3, or, mathematically, by setting the first derivation of equation (5.8) to zero. Nearly
0.6 0.5 0.4 0.3 0.2 0.1
1.0 0.8 0.6 0.4 0.2 0.0
v3 v1 cP
0.0
0.5 1/3 1/4 a 0.0 cP.max= 16/27 0.6
0.5 0.4 0.3 0.2 0.1
1.0 0.8 0.6 0.4 0.2 0.0
v3 v1 cP
0.0
0.5 1/3 1/4 a 0.0 cP.max= 16/27
Fig. 5-3 Power coefficient cP versus the ratio of wind velocity v3 far downstream and the up- stream wind velocity v1 ; cP,max = 0,593 at v3/v1 = 1/3, additional axis: induction factor a Now, using the principle of linear momentum, we can determine the thrust acting on the wind turbine at the rotor plane when maximum power is extracted. The thrust equation
T = m(v1v = U A
3 3 1
1 (
2v v v
v (5.10)
yields for the given ratio v3 / v1 = 1/3 T = cT á
ạ
ă ã
©
§
Uv
1 A ; cT = 8/9 = 0.89 . (5.11)
The term in brackets is the dynamic pressure on the area A. Comparing this result with the drag D of a circular disk in the wind flow
D = cD á
ạ
ă ã
©
§ Uv
1 A ; cD = 1.11, (5.12)
it is found that in the case of optimum power extraction the thrust is nearly 20%
lower than that of a circular disk.
5.1 How much power can be extracted from the wind?
172
Wind speedin m/s
Diameter of wind wheel in m Power in kW
Power in kW Wind speedin m/s
Diameter of wind wheel in m Power in kW
Power in kW
Fig. 5-4 “Betz power” of wind turbines as a function of wind velocity and diameter, Betz [1]
Remark: In the English literature, the power coefficient and the thrust are often shown versus the “induction factor a”. This is based on the idea that the wind tur- bine superimposes a sort of headwind av1 on the wind velocity v1. For the active rotor plane this gives
v2 = v1 (1- a) and far downstream of the rotor
v3 = v1 (1- 2 ã a).
We show this induction factor on an additional scale in Fig. 5-3.
5.1.1 Froude-Rankine Theorem
In the following it will be proved that the velocity v2 in the rotor plane according to Betz’ theory is actually the arithmetic mean of the far-upstream and far- downstream velocities.
The thrust can be expressed using the principle of linear momentum in equation (5-10)
v1 v3
m
T .
Alternatively, it can be derived from the Bernoulli equation (energy balance), which we apply for both the plane left and the plane right of the rotor plane, Fig.
5-5.
p1 +
U ãv12 = 2 22
2
v
p U
and (5.13)
22
2 2
v
p U
= p3 + v32
U . (5.14)
The subscript -2 denotes the plane immediately before the rotor, and +2 immedi- ately after the rotor.
To preserve continuity, the velocity immediately left and right of the rotor must be equal, v-2 = v+2. Moreover, the static pressure far upstream and far down- stream are also equal, p1 = p3. Hence, the difference of equation (5.13) and (5.14) gives
2 32
2 v1 v
U = p-2 p+2 (5.15)
According to these (energetic) considerations, the thrust at the tower results from the difference of the static pressure before and after the rotor plane
T = A ã (p-2 p+2) . (5.16)
5.1 How much power can be extracted from the wind?
174
(x) v
p0 3
p+2 p(x)
p v2
v1 1
p-2 2 2 - +2
(x) v
x 3
Fig. 5-5 Development of velocity and static pressure p along the stream tube
Putting the mass flow rate m = ȡ A v2 into equation (5.10) and introducing this into the equations (5.16) and (5.15), we obtain the expression used in section 5.1 for the velocity v2 in the rotor plane
v2 = 2
) (v1v3
.