Harry Wertmuller's Doubt on Windmill Energy Proportionality

[HarryWertM]

Reading about renewable energy on the Internet a short time ago, I came across the statement that "energy is proportional to the cube of the velocity".

Huh? I thought energy was proportional to the _square_ of the velocity. After all, we're not talking relativity here.
-Harry Wertmuller

 

[tiny-tim]

Reading about renewable energy on the Internet …

Hi Harry!

Where on the internet?

I googled "energy is proportional to the cube of the velocity", but only got two hits, neither of them relevant.

 

[Celunas]

Maybe it's just an error ! :)

 

[russ_watters]

It's true. The reason is that an increase in velocity also involves and increase in pressure. Here are the fan laws: http://www.delhi-industries.com/Engineering/Tips/FanLaws.asp

1 CFM varies Directly with RPM (CFM 1 / CFM 2 = RPM 1 / RPM 2)
2 SP varies with the SQUARE of the RPM (SP 1 / SP 2) = (RPM 1 / RPM 2)2
3 HP varies with the CUBE of the RPM (HP 1 / HP 2 = (RPM 1 / RPM 2)3

You need to substitute the first equation into the third...

 

[LURCH]

Learned the same law in my Alternative Energy class at community college. It's true alright, and it's one of the factors that made our instalation of the school's windmill a lot easier. You see, as input (wind) increases by x, otuput increases by x³, and resistance (load) increases proportionally with output. So, as windspeed goes up, the blades' resistance to acceleration goes up exponentially, which keeps the blades from spinning too fast and ripping themselves apart.

 

[rcgldr]

Aerodynamic drag and lift forces vary with the square of the speed (assuming sub mach air speeds here). Since power = force x speed, then power varies with the cube of the speed (speed^2 for force component x speed).

 

[Redbelly98]

Reading about renewable energy on the Internet a short time ago, I came across the statement that "energy is proportional to the cube of the velocity".

Huh? I thought energy was proportional to the _square_ of the velocity. After all, we're not talking relativity here.
-Harry Wertmuller

Here is how I think of this surprising, but true, fact:

1. The energy per molecule (or mole, or gram, or whatever) of air is proportional to the square of the air velocity.

2. The number of air molecules per unit time, that pass through the windmill blades, is proportional to the air velocity.

Multiply the quantities in these two statements and we find:

The power (i.e. energy per unit time) in the air passing through the windmill is proportional to the cube of the air velocity.

 

[munajmurshid]

Reading about renewable energy on the Internet a short time ago, I came across the statement that "energy is proportional to the cube of the velocity".

Huh? I thought energy was proportional to the _square_ of the velocity. After all, we're not talking relativity here.
-Harry Wertmuller

the column of air in 1 second= v
the volume of air= va (a= cross area)
the mass of air=vad (d=density)
the kinetic energy= 1/2 m v v
=1/2 vad v v(vad=m)
= 1/2 ad (v v v)
hope you got the answer

 

[Bob S]

It is not theoretically possible to remove all the energy from wind, because the air has to go someplace after passing through the windmill. This limit is called the Betz limit, ~59%. Actual windmills achieve ~40 to 45% maximum. Furthermore, because the wind velocity distribution is approximated by the Rayleigh distribution
http://en.wikipedia.org/wiki/Rayleigh_distribution
windmills on average operate at only ~35% of their maximum design power. So a 1.5 MW windmill on average produces only about 500 kW.
Bob S.