Lesson 4 of the Mechanical Engineering Offshore Windfarm Design course. Actuator Disk Theory Turbine Energy Yield Calculation

1. Offshore
Windfarm
Design
Structure, Park, Planning, Installation and Maintenance
Offshore en Constructie Minor
Leo Hulspas en Johan Antonissen
Don’t throw my advice in the wind

2. AGENDA
▸ Les 1 – History, Current State of Art, North Sea
▸ Les 2 – Site Conditions
▸ Les 3 – The Turbine
▸ Les 4 – Actuator Disk Theory and Energy Yield
▸ Les 5 – Structure Design and Load Calculations
▸ Les 6 – Structure Design and Load Calculations
▸ Les 7 – Park Design
▸ Les 8 – Vessels
▸ Les 9 – Installation and Comissioning
▸ Les 10 – Operations and Maintenance

3. What you’ll learn...
▸ Conservation laws
▸ Actuator disk theory
▸ Powercurve
▸ Energy yield of turbine

4. Conservation Laws
▸ Conservation of Mass
▸ Bernoulli (Conservation of Energy)
▸ Conservation of Shit
ϕM inlaat=ϕM uitlaat
p+ρ⋅g⋅h+
1
2
⋅ρ⋅v
2
=constant
.=.
inlaat uitlaat

5. Horn’s Rev...
What do we see?
Wake effects

6. The Actuator Disk...
1
2
Stream
Tube
Actuator
Disk
Assumptions:
▸ Rotor is assumed a permeable
disk that blocks flow
▸ Flow inside a stream tube
▸ Laminar flow (no turbulance)
▸ Constant air density
Let’s apply some magic ...

7. The Actuator Disk...
1
2
Stream
Tube
Actuator
Disk
Assumptions:
▸ Rotor is assumed a permeable
disk that blocks flow
▸ Flow inside a stream tube
▸ Laminar flow (no turbulance)
▸ Constant air density
Let’s apply some magic ...
I mean math...

8. The Actuator Disk… and Mass Conservation...
1
2
▸ Conservation of Mass
ϕm1=ϕm 2→ A1< A2→v1>v2
Stream
Tube
Actuator
Disk

9. The Actuator Disk… and Bernoulli….
1
2
▸ Conservation of Mass
▸ Bernoulli
ϕm1=ϕm 2→ A1< A2→v1>v2
Stream
Tube
Actuator
Disk
p1+ρ⋅g⋅h1+
1
2
⋅ρ⋅v1
2
=p2+ρ⋅g⋅h2+
1
2
⋅ρ⋅v2
2

10. The Actuator Disk...
1
2
▸ Conservation of Mass
▸ Bernoulli
ϕm1=ϕm 2→ A1< A2→v1>v2
Stream
Tube
Actuator
Disk
p1+ρ⋅g⋅h1+
1
2
⋅ρ⋅v1
2
=p2+ρ⋅g⋅h2+
1
2
⋅ρ⋅v2
2
p1=p2=patm
h1=h2

11. The Actuator Disk...
1
2
▸ Conservation of Mass
▸ Bernoulli
ϕm1=ϕm 2→ A1< A2→v1>v2
Stream
Tube
Actuator
Disk
p1+ρ⋅g⋅h1+
1
2
⋅ρ⋅v1
2
=p2+ρ⋅g⋅h2+
1
2
⋅ρ⋅v2
2
p1=p2=patm
h1=h2
+pturbine
pturbine=
1
2
⋅ρ⋅(v1 ²−v2 ²)→v1>v2→ profit !
v2=v1⋅
A1
A2
But what is A1
and A2
?

12. Let’s rewrite that a bit….
1
2
pturbine=
1
2
⋅ρ⋅(v1 ²−v2 ²)
Pturbine=pturbine⋅ϕv=
1
2
⋅ρ⋅(v1 ²−v2 ²)⋅ϕv
▸ Pressure energy extracted from air:
▸ Energy extracted from air:
▸ Change in momentum air (Thrust experienced by RNA):
Tturbine=ρ⋅ϕv⋅(v1−v2)
Tturbine
Pturbine

13. Let’s rewrite that a bit….
1
2
Pturbine=pturbine⋅ϕv=
1
2
⋅ρ⋅(v1 ²−v2 ²)⋅ϕv
▸ Thrust :
▸ Power :
Tturbine=ρ⋅ϕv⋅(v1−v2)
Tturbine
Pturbine
Now…
we can measure v1
…
but v2
is affected by the turbine...

14. Let’s talk about Betz’s limit….
Thisguy...
…
andhiswindtunnel

15. Let’s talk about Betz’s limit….
Thisguy...
…
andhiswindtunnel a = induction factor
a = 0,3 (optimal)
for v2
= 1/3 v1
Betz optimum.

16. Let’s rewrite that a bit…. again...
Pturbine=pturbine⋅ϕv=
1
2
⋅ρ⋅(v1 ²−v2 ²)⋅ϕv
Tturbine=ρ⋅ϕv⋅(v1−v2)
v2=
v1
3
=v1⋅(1−2⋅a)
a = induction factor
a = 0,3 (optimal)
for v2
= 1/3 v1
Betz optimum.

17. Let’s rewrite that a bit…. again...
Pturbine=
1
2
⋅ρ⋅v1 ³⋅A⋅4 a(1−a)²
Tturbine=
1
2
⋅ρ⋅v1 ²⋅A⋅4 a(1−a)
a = induction factor
a = 0,3 (optimal)
for v2
= 1/3 v1
Betz optimum.

18. Let’s rewrite that a bit…. again...
Pturbine=
1
2
⋅ρ⋅v ³⋅A⋅4 a(1−a)²
Tturbine=
1
2
⋅ρ⋅v²⋅A⋅4 a(1−a)
a = induction factor
a = 0,3 (optimal)
for v2
= 1/3 v1
Betz optimum.
Tturbine=
1
2
⋅ρ⋅v²⋅A⋅Ct →Ct=0,89
Pturbine=
1
2
⋅ρ⋅v ³⋅A⋅Cp→Cp=0,59

19. What you need to remember...
Tturbine=
1
2
⋅ρ⋅v²⋅A⋅Ct →Ct=0,89
Pturbine=
1
2
⋅ρ⋅v ³⋅A⋅Cp→Cp=0,59
▸ For load calculations:
▸ For power calculations:
OW
D
You can forget the rest...

20. However…
What happens if wind speeds become too high ?
U OK ?
What do
you think
bro?

21. Introducing the Power Curve...
Pturbine=
1
2
⋅ρ⋅v ³⋅A⋅Cp
Pwind=
1
2
⋅ρ⋅v ³⋅A
Pturbine=Pmax
Pturbine=0
Pturbine=0

22. Introducing Blade Pitch...
Cp≈0,59
θ=0̊
v<vrated
Ct≈0,89
Cp≪0,59
θ>0̊
v>vrated
Ct≪0,89
So maximum Thrust at Vrated
, thus very important loadcase!

23. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

24. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

25. Lets Compare...
pdf (v10)=
k
A
⋅(
v10
A
)
k−1
⋅exp(−(
v10
A
)
k
) pdf (v10)=
k
A
⋅(
v10 ²/vrna
A
)
k−1
⋅exp(−(
v10 ²/vrna
A
)
k
)
vrna(Hrna)=v10⋅
log(
Hrna
z0
)
log(
Href
z0
)

26. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E=∑
vcutin
vcutout
( pdf (vi)⋅P(vi)⋅T )
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

27. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∑
i=vcutin
vcutout
( pdf (vi)⋅P(vi)⋅T)
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

28. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∑
i=vcutin
vcutout
( pdf (vi)⋅P(vi)⋅T)
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

29. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∑
i=vcutin
vcutout
( pdf (vi)⋅P(vi)⋅T)
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

30. Annual Energy Production Turbine
Annual Energy =
x
x 24 x 365
E= ∑
i=vcutin
vcutout
( pdf (vi)⋅P(vi)⋅T)
E= ∫
vcutin
vcutout
pdf (vrna)⋅P(vrna)⋅T dv

31. Annual Energy Production Turbine
▸ Annual Energy = 42133,8 Gwh
▸ Annual Electricity = 6583 Persons
▸ Annual Electricity = 10533 Households
▸ 17000000 persons in Holland
▸ 3 persons per Household
▸ Only 537 turbines required for electricity for all households in Holland!
▸ ~ 5 large windfarms …
▸ Why then is only ~8% of our energy consumption said to be renewable?
▸ What are we missing ?

32. Homework
▸ Bereken de maximale Power van je turbine (slide 19)
▸ Bereken de maximale Thrust van je turbine (slide 19)
▸ Bereken de nieuwe Weibull @ RNA hoogte (slide 25)
▸ Bereken een powercurve van je turbine (slide 21)
▸ Bereken jaarlijkse energieopbrengst van je turbine (slide 26)

33. Thank you!