WHAT IS THE MAXIMUM POWER THAT CAN BE EXTRACTED FROM THE WIND?

Part B Module 1 – Wind Energy

MOVING AIR

• Engine drives propeller that accelerates air – increases kinetic energy of air passing through it

• Moving air gives up kinetic energy to turn wind turbine

BASICS

• For wind turbine having blades of length R:

𝐴 = 𝜋𝑅2 where A = area swept by the blades

R

Modified “Wind turbine 2”

BASICS (2) Flow through a stream tube having area A (shaded section):

�̇�𝑎𝑎𝑎 = 𝜌𝐴𝜌 where ρ = density of air Axial locations: • Plane 0 = upstream of wind turbine • Plane 1 = plane swept by wind turbine

blades • Plane 2 = downstream of wind turbine

After Burton et al., 2001

1

2

0

POWER AVAILABLE IN MOVING AIR • Kinetic energy of moving air for wind speed V:

12𝜌2 = 𝐾𝑎𝐾𝐾𝐾𝑎𝐾 𝐾𝐾𝐾𝑎𝑒𝑒

𝑢𝐾𝑎𝐾 𝑚𝑎𝑚𝑚 𝑜𝑜 𝑎𝑎𝑎

• Power = mass rate at which air moves through wind turbine x kinetic energy/unit mass

𝑃 = �̇�𝑎𝑎𝑎 ∙12𝜌2 = 𝜌𝐴𝜌 ∙

12𝜌2 =

12𝜌𝐴𝜌3

AIR FLOW ACROSS TURBINE • Turbine extracts some kinetic energy from the flow, reducing

wind speed from V0 upstream to V1 at the turbine and then V2 downstream

𝑃𝐾𝑢𝑎𝑡𝑎𝐾𝐾 = �̇�𝑎𝑎𝑎 ∙ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑎𝑟𝑟 𝑘𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑒𝑒

𝑃𝐾𝑢𝑎𝑡𝑎𝐾𝐾 = 𝜌𝐴𝜌1 ∙𝜌02

2−𝜌22

2

𝑃𝐾𝑢𝑎𝑡𝑎𝐾𝐾 = 12𝜌𝐴𝜌1 ∙ 𝜌02 − 𝜌22

DECELERATING AIR UNDERGOES PRESSURE RISE

Bernoulli’s equation along a streamline:

𝑒ℎ0 +𝑃0𝜌

+12𝜌02 = 𝑒ℎ2 +

𝑃2𝜌

+12𝜌22

Solving for the pressure rise gives: 𝑃2 − 𝑃0 = 1

2𝜌 𝜌02 − 𝜌22

Flow through stream tube:

�̇�𝑎𝑎𝑎 = 𝜌𝐴𝜌 After Burton et al., 2001

0

1

2

THRUST FORCE ACROSS TURBINE • Pressure difference between planes 0 and 2 creates an axial

thrust force:

𝐹𝑎𝑎𝑎𝑎𝑎 𝐾𝑡𝑎𝑢𝑚𝐾 = 𝑃2 − 𝑃0 𝐴 =12𝜌𝐴 𝜌02 − 𝜌22

• Axial thrust force is also equal to the change in momentum

𝐹𝑎𝑎𝑎𝑎𝑎 𝐾𝑡𝑎𝑢𝑚𝐾 = �̇�𝑎𝑎𝑎 ∙ (𝜌0 − 𝜌2) = 𝜌𝐴𝜌1 ∙ (𝜌0 − 𝜌2)

WIND TURBINE POWER OUTPUT • Equating the two expressions for thrust force:

12𝜌𝐴 𝜌02 − 𝜌22 = 𝜌𝐴𝜌1 𝜌0 − 𝜌2

• Solving for V1:

𝜌1 =𝜌0 + 𝜌2

2

Wind turbine power output can be expressed in terms of V0 and V2:

𝑃𝑤𝑎𝐾𝑤 𝐾𝑢𝑎𝑡𝑎𝐾𝐾 =12𝜌𝐴

𝜌0 + 𝜌22 𝜌02 − 𝜌22

MAXIMUM POWER FROM WIND TURBINE • Express wind turbine power output expressed in terms of V2/V0:

𝑃𝑤𝑎𝐾𝑤 𝐾𝑢𝑎𝑡𝑎𝐾𝐾 =12𝜌𝐴𝜌0

31 + 𝜌2

𝜌0

21 − 𝜌2

𝜌02

• To find maximum,

𝑟1 + 𝜌2

𝜌0

21 − 𝜌2

𝜌02

𝑟 𝜌2𝜌0

= 0

MAXIMUM POWER FROM WIND (2)

•1+𝑉2𝑉0

21−𝑉2𝑉0

2 maximized when 𝜌2 = 𝑉0

3,

• when 𝜌2 = 𝑉03

, 1+𝑉2𝑉0

21−𝑉2𝑉0

2 = 16/27 = 0.593

• Betz limit

𝑃𝑚𝑎𝑎 𝑤𝑎𝐾𝑤 𝐾𝑢𝑎𝑡𝑎𝐾𝐾 =12𝜌𝐴𝜌03

1627

@ MAXIMUM POWER CONDITION

Upstream

• 𝐾𝐾𝑚 𝑤𝑎𝐾𝑤 𝑢𝑢𝑚𝐾𝑎𝐾𝑎𝑚

= 12𝜌02

Downstream

• 𝐾𝐾𝑚 𝑤𝑎𝐾𝑤 𝑤𝑜𝑤𝐾𝑚𝐾𝑎𝐾𝑎𝑚

= 12

𝑉03

2

_______________________________________________________________ 𝐾𝐾𝑚 𝑤𝑎𝐾𝑤 𝑤𝑜𝑤𝐾𝑚𝐾𝑎𝐾𝑎𝑚

=19

𝐾𝐾𝑚 𝑤𝑎𝐾𝑤 𝑢𝑢𝑚𝐾𝑎𝐾𝑎𝑚

Wind turbine has extracted 8/9 of initial kinetic energy

WHAT IS THE MAXIMUM POWER THAT CAN BE EXTRACTED FROM THE WIND?

SUMMARY • Power = mass rate at which air moves through wind turbine x

kinetic energy/unit mass

𝑃 = �̇�𝑎𝑎𝑎 ∙12𝜌

2 = 𝜌𝐴𝜌 ∙12𝜌

2 =12𝜌𝐴𝜌

3

• Betz limit is max power that can be extracted from the wind

𝑃𝑚𝑎𝑎 𝑤𝑎𝐾𝑤 𝐾𝑢𝑎𝑡𝑎𝐾𝐾 =12𝜌𝐴𝜌0

3 1627

PRACTICE EXERCISES

REFERENCES AND PHOTO CREDITS References • Burton, Tony, David Sharpe, Nick Jenkins and Ervin Bossanyi, “Handbook of

Wind Energy, John Wiley & Sons, Chichester, 2001. Photo credits • “Airplane3” (Public domain) free from http://www.clipartlord.com/free-

propeller-aircraft-clip-art/, http://www.clipartlord.com/wp-content/uploads/2013/03/airplane3-300x84.png

• “Wind turbine 2”, (public domain) free from: http://www.clipartlord.com/free-wind-turbine-clip-art-2/, http://www.clipartlord.com/wp-content/uploads/2014/04/wind-turbine2.png