Proof for Betz limits of 59%
The maximum power in wind is ½ x density x U^3 (Watt per unit
of area). Betz proved that a wind turbine would not extract more than
59% of this available power. Instead of looking at how much torque/power
can a rotor produce, Betz looks at how fast the wind can lose its' energy
to wind turbine. Due to the fact that the wind turbine retards wind far
before it reaches the rotor, the flowrate of the wind across the wind
turbine will be lowered if the turbine retards the wind too much in order
to extract maximum energy. In another word, the turbine can extract off
all the energy in the wind but at a slow rate. To optimise the rate of
energy extraction, the energy extraction must not be too high so that
a good flowrate can be achieved. The theory below proved that the maximum
power a wind can lose to a wind turbine is 59%, which occurs when the
wind lost a total of 67% of its original wind speed.
Plotting the power coefficient, Cp vs. induction factor, a graph, we
found Cp maximise at a = 1/3 with the maximum Cp of 59%. At a = 1/3, the
wind speed at rotor U’ decreases by 33% from the original wind speed
while downstream U” reduces by amount of 67%.
For Darrieus wind turbine, the disc is assumed as the whole cylindrical
volume (compare to circular plane of HAWT) as shown in the previous page
top figure. It can be argued that modeling Darrieus with two disks (one
upwind semi-cylinder surface and another downwind) is also a possibility,
which can elevate the Darrieus turbine from the 59% limits. But till now,
none of the wind turbines efficiency has yet to exceed the 59% limit.
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