Wind turbine design according to Betz and Schmitz
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Transcript Wind turbine design according to Betz and Schmitz
1
Wind turbine design according
to Betz and Schmitz
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Energy and power from the wind
• Power output from
wind turbines:
A
3
v
Power A cp
2
• Energy production
from wind turbines:
Energy Power Time
v
3
Stream Tube
V
4
Extracted Energy and Power
1
2
2
E ex m v1 v 3
2
2
2
E 1 m
v1 v 3
ex
2
Where:
E• ex = Extracted Energy
Eex = Extracted Power
m = Mass
•
m = Mass flow rate
v = Velocity
[J]
[W]
[kg]
[kg/s]
[m/s]
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Extracted Energy and Power
• If the wind was not retarded, no power would be extracted
• If the retardation stops the mass flow rate, no power would be
extracted
• There must be a value of v3 for a maximum power extraction
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Extracted Energy and Power
• The retardation of
the wind cause a
pressure difference
over the wind
turbine
p p2 p2
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We assume the following:
•
There is a higher pressure right upstream the turbine (p-2) than the
surrounding atmospheric pressure
•
There is a lower pressure right downstream the turbine (p+2) than the
surrounding atmospheric pressure
•
Since the velocity is theoretically the same both upstream and downstream
the turbine, the energy potential lies in the differential pressure.
•
The cross sections 1 and 3 are so far away from the turbine that the
pressures are the same
A3
A2
A1
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Continuity
(We assume incompressible flow)
v1 A1 v2 A2 v3 A3
A3
A2
A1
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Balance of forces:
(Newton's 2. law)
Because of the differential pressure over the
turbine, it is now a force F = (p-2 – p+2)∙A2 acting
on the swept area of the turbine.
A3
A2
A1
F v1 A1 v1 (p2 p2 ) A2 v3 A3 v3
Impulse force
Pressure force
Impulse force
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Energy flux over the wind turbine:
(We assume incompressible flow)
v12
v32
E v1 A1 p1 v1 A1 v3 A 3 p3 v3 A 3
2
2
v1 A1 v2 A2 v3 A3
v32
v3 A 3
2
v12
E v1 A1
2
A3
A2
A1
p1 p3
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Energy flux over the wind turbine:
(We assume incompressible flow)
v22
v22
E v2 A 2 p 2 v 2 A 2 v 2 A 2 p 2 v 2 A 2
2
2
E v 2 A 2 p 2 p 2
A3
A2
A1
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Energy flux over the wind turbine:
(We assume incompressible flow)
E v2 A 2 p 2 p 2
A3
A2
A1
v32
v12
v1 A1 v3 A3
2
2
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Continuity:
v1 A1 v2 A2 v3 A3
Balance of forces:
v1 A1 v1 (p2 p2 ) A2 v3 A3 v3
Energy flux:
v32
v12
v2 A 2 p2 p2 v1 A1 v3 A3
2
2
If we substitute the pressure term; (p-2-p+2) from the equation for the balance of
forces in to the equation for the energy flux, and at the same time use the continuity
equation to change the area terms; A1 and A3 with A2 i we can find an equation for
the velocity v2:
v32
v12
v1 A1 v3 A3 v 2 v12 A1 v32 A3
2
2
v12
v2 A 2
2
v32
v2 A 2
2
1
v12 v32 v 2 v1 v3
2
v 22 A 2 v1 v3
v v3
1
v1 v3 v1 v3 v 2 1
2
2
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Power Coefficient
Rankine-Froude theorem
We define the Power Coefficient: cp
v32
v12
v1 A1 v3 A3
2
2
2
v1
v1 A
2
In the following, we assume that the velocity v3 can be
expressed as v3=x·v1, where x is a constant.
v3 x v1
We substitute:
From continuity:
A1 v2
A2
v1
A2
A3 v 2
v3
A1
v1 v3 A 2 A 2 v1 v3 A 2
1 x
2
v1
2
v1
2
v1 v3 A 2 A 2 v1 v3 A 2 1 A 2 1 x
A3
1
2
v3
2
v3
2 x 2 x
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Power Coefficient
Rankine-Froude theorem
We insert the expressions for A1 and A3 in to the equation for the power coefficient.
We will end up with the following equation:
v32
v12
v1 A1 v3 A3
2
2
cp
2
v
1 v1 A 2
2
cp
v12 v1 A1 v32 v3 A 3
v12 v1 A 2
v
v32 v1 A1
v12
v1 A 2
2
1
cp
cp
v12 v32 A1
2 2
v1 v1 A 2
1
1 x x 2 x 3
2
1 x A1
2
2
A2
1 x
2
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Maximum Power Coefficient
Rankine-Froude theorem
cp
x
3
x 2 x 1
2
Maximum power coefficient:
cp
x
3 x
0
2
2 x 1
2
x 1
cpmax
x 1
3
3
2
1 1 1 1
3
3
3
32
0,59
2
54
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Power Coefficient
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The Betz Power
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Thrust
1
1
2
2
2
2
T m v3 v1 A 2 v1 v3 A 2 v1 x v1
2
2
1
1
2
2
T A 2 v1 1 x A 2 v12 cT
2
2
At maximum power coefficient we
have the relation: x =1/3
1 8
cT 1 x 1
9 9
2
v2
T
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Example
Find the thrust on a wind
turbine with the following
specifications:
v1 =
D =
cT =
20 m/s
100 m
8/9
1
D
2
2
T A 2 v1 cT
v1 cT
2
2 4
2
1, 2 100
2 8
T
20 1676,5 kN
2
4
9
2