Originally Posted by PineyWoodsPete
"The energy content of the wind varies with the cube (the third power) of the average wind speed."
You seem to be confusing the POWER (kinetic energy over time), which varies with the cube of velocity, with the KINETIC ENERGY (transferred force of wind exerted against a surface) - which, of course, varies with the square of velocity:
E (kinetic) = 0.5 X (M x V-squared)
Okay, let me be of further assistance. I will recommend these references
so as to provide clarity of the output power being a factor of the cube of the velocity of the wind.
The Power available from the generator is 1/2pAv^3(Cp)
p = Density of the air
A = Swept area of the turbine
v = Velocity of the air [which velocity is a function taken at the third
Cp = Coefficient of power [i.e., the power conversion efficiency of the turbine system]
Copied below is an explanation in rather layman's terms and an exemplary output power graph thereof is attached below:
The Power of the Wind: Cube of Wind Speed
The Power of the Wind: Cube of Wind Speed
The wind speed is extremely important for the amount of energy a wind turbine can convert to electricity: The energy content of the wind varies with the cube (the third power) of the average wind speed, e.g. if the wind speed is twice as high it contains 2^3 = 2 x 2 x 2 = eight times as much energy.
Now, why does the energy in the wind vary with the third power of wind speed?
Well, from everyday knowledge you may be aware that if you double the speed of a car, it takes four times as much energy to brake it down to a standstill. (Essentially this is Newton's second law of motion) Similar to your conception of kinetic energy above
Power Content of the Wind
But in the case of the wind turbine we use the energy from braking the wind, and if we double the wind speed, we get twice as many slices of wind moving through the rotor every second, and each of those slices contains four times as much energy, as we learned from the example of braking a car.
The graph shows that at a wind speed of 8 metres per second we get a power (amount of energy per second) of 314 Watts per square metre exposed to the wind (the wind is coming from a direction perpendicular to the swept rotor area).
At 16 m/s we get eight times as much power, i.e. 2509 W/m 2
Because the power output increases by the cube of the velocity of the wind and a boat heels over significantly only when the wind speed is high [and the sails are large, not reefed] one can see that the generator machine typically reaches its thermal limits for power far before the tilting of the support structure will have any significant effect. When the boat tilts over the genny is lowered towards the water
surface a small amount and wind speeds nearer the surface are generally slower than they are higher up, so if the boat heels the generator will see less apparent wind then if it was held further up. A catamaran
will tilt less than a monohull so a catamaran may avail marginally superior power output potential, but the real limiter will be the power rating of the genny that is attained quickly as wind velocity increases. Cubic function is quite amazing in that regard.
About 20 years ago, here in Montana, I participated in the prototype design and building of a novel 3 megawatt, permanent magnet generator for a wind turbine. The generator operated at direct drive, couple to and rotating at the same speed as the blades [no expensive and heavy gear
box utilized to reduce torque and increase speed to the genny]. The generator machine's rated speed was only 13 rpm
. The generator machine was in one dimension, large at 12 meters [40 feet] in diameter, the turbine blades diameter was 100+ meters, and the nacelle would in turn be located on top of a tower 100+ meters above the ground. While the generator machine was large in diameter at 12 meters, it was axially quite short at about 6 feet at the junction of its rotor hub at the axle, and the rotor narrowed to less than 2 feet in axial length at the outer circumference of the 12 meter diameter. The stator of the generator was unique in architecture and materials as there was No Iron nor Wires in the construction of the Stator. The stator was only about 2 inches in axial length [cross thickness in the magnetic field] and only about 18 inches in radial length, extremely lightweight. The stator was not rigid axially, it could flop about a bit axially but obviously had tremendous torque applied to the stator. Not many motor
or generator engineers can even imagine a stator that has no iron nor wires in its architecture or bill of materials; they will draw a blank in their minds, kind of like as if talking to an architect and telling them about a building that has no floors, walls or ceilings. There be no frame of reference or prior experience with such architecture of electrical
I hope this helps in understanding. All the best.