Two exponents, and only one of them is for sale
The power carried by moving air through an area A is:
P = ½ ρ A v³
Density in kilograms per cubic metre, area in square metres, speed in metres per second, and the answer in watts. For a conventional rotor the area is the disc the blades sweep, πd²/4, so the formula carries a square on diameter and a cube on speed.
Those two exponents are the whole subject. Diameter is something you can buy: a rotor twice as wide sweeps four times the area and takes four times the power. Wind speed is something the site either has or does not, and it is worth three powers instead of two. The arithmetic falls out cleanly:
| Change | Output multiplier | Equivalent in the other variable |
|---|---|---|
| Wind 26% faster | 2.0× | same as a 41% bigger rotor |
| Wind 50% faster | 3.375× | same as an 84% bigger rotor |
| Wind twice as fast | 8× | same as a rotor 2.83× the diameter |
| Rotor twice as wide | 4× | same as wind 1.59× as fast |
The break-even is exact and worth carrying around: a rotor of twice the diameter is beaten by a site whose wind is 58.7 per cent faster, because 4 is 1.587 cubed. Rotor diameter is a purchase; that speed difference is a location. It is why the honest first step in small wind is a recording anemometer at hub height for a season rather than a catalogue, and why two sites on the same property can give completely different answers for the same machine.
The Betz limit is a ceiling, not an efficiency
A rotor cannot take all the energy out of the air passing through it, because if it did the air would have to stop, and stopped air cannot get out of the way of the air behind it. Working through the momentum balance gives a maximum of 16/27, or 59.3 per cent, of the power in the free stream. That is the Betz limit, and it applies to any device in an open flow regardless of how it is built.
What it is not is a rating. A real machine loses more at the blade tips, in the bearings, in whatever gearing there is, in the generator, and in the rectifier or inverter downstream. The coefficient on this page is deliberately the whole chain, wind to wires, so the number you can compare against your loads is the number the page prints. A whole small system landing well under the Betz figure is normal and not a sign of anything wrong.
If a specification quotes a large output at a wind speed, that is a rated point rather than a coefficient. Divide it by the power in the wind at that speed and that rotor area and you will get the actual system Cp, which is often a surprise.
Air density is not a constant, and it moves the answer
Power is directly proportional to density. Air at 7,000 feet on a 59 degree day comes out around 77 per cent of the sea level value, so the same rotor in the same wind produces close to a quarter less. Temperature works the other way and matters less, but cold winter air is measurably denser than hot summer air at the same site.
The calculator works density from the standard atmosphere at your elevation and then applies the ideal gas law at the temperature you enter, so both effects are in the answer. A mountain site is not disqualified by this — mountain sites are often windy enough to more than make it up — but running the calculation at sea level density and installing at 8,000 feet builds in a systematic overestimate.
Why an average wind speed understates the energy
This page gives power at one speed. Annual energy is a different question, and it cannot be answered by putting an average speed into a cube, because the cube is convex: the average of the cubes is always at least the cube of the average, and usually much more.
A blunt illustration. Take a site that is dead calm half the time and blows 20 mph the other half. Its average speed is 10 mph. Cubing the average gives 1,000. Averaging the cubes gives (0 + 8,000)/2 = 4,000 — four times as much. Real wind is not that extreme, but the direction of the error is always the same, and factors of two are ordinary.
The number that actually settles a wind decision is a measured distribution of speeds at hub height over at least a season, not a mean. Everything short of that is a feasibility sketch, which is what this page is.
Where wind sits against the alternatives
Wind is intermittent in a way that is uncorrelated with solar, which is the argument for it: the windiest weather is often the cloudiest. Both feed the same battery, and the storage question does not change — the battery bank sizing calculator and the off-grid load audit calculator are the same pages whatever is charging the bank. Where wind loses to solar is maintenance: a solar array has no moving parts at eighty feet in the air.
The tower is usually the larger half of the cost and the whole of the risk. Wind loading on a mast is a structural calculation of its own, and the antenna mast wind load calculator covers the force side for a mast, which is the other thing the same wind is doing while the rotor turns.
Anything that ties into house wiring is a licensed electrician job with a permit behind it. Back-feeding a line that the utility believes is dead can kill a lineman working on it, which is why the transfer arrangement is not a detail you improvise. Battery banks store real energy and can vent hydrogen. This page produces a number of watts and gives no wiring guidance whatsoever.
Questions people ask
Why does a small increase in wind speed change the output so much?
Because power depends on the cube of the speed. There are three separate reasons and they multiply: faster air carries more kinetic energy per unit mass, which is a square, and more mass arrives per second, which is another factor of speed. Put together that is a cube. A site averaging 12 mph carries twice the power of one averaging 9.5, and roughly eight times the power of one averaging 6. This is why nearly all the useful effort in small wind goes into finding a better location and a taller tower rather than a bigger rotor.
What is the Betz limit and can anything beat it?
It is the maximum share of the power in a free airstream that any rotor can extract, 16/27 or 59.3 per cent, and it follows from the fact that the air has to keep moving to get out of the way. Nothing in an open flow beats it, including designs that claim to. Occasionally a device is quoted above it because the coefficient has been divided by the wrong area, usually a shroud or duct that is smaller than the flow the device actually disturbs. It is not an efficiency rating either. A real machine has to give up more to blade losses, bearings, gearing and the generator, so the whole system coefficient wind to wires is considerably lower.
Can I use the wind speed from my local airport?
As a rough regional indication, not as a site figure. Airport anemometers sit at a standard height over deliberately open flat ground, which is neither your height nor your terrain. Your site can be sheltered by trees or a ridge, or accelerated over one, and the difference is easily a factor of two in speed, which is a factor of eight in power. The only figure worth designing against is a recording anemometer at hub height at your site over at least a season. Regional data is useful for deciding whether the survey is worth doing.
Does a vertical axis turbine change the arithmetic?
Not the physics. The power in the wind is still half rho A v cubed and the Betz limit still applies. What changes is the swept area, which for a vertical axis rotor is height times width rather than a circle, so use the area mode on this page and enter it directly. Published coefficients for vertical axis machines are generally lower than for a good horizontal axis rotor, which is the trade for the things they are chosen for: they take wind from any direction without yawing, and they can be mounted lower. Mounted lower is usually the problem, since wind at low height is both slower and more turbulent.
How do I turn this into kilowatt-hours a year?
You cannot do it properly from an average speed, and this page does not pretend to. Because the cube is convex, the average of the cubes always exceeds the cube of the average, so any annual estimate built on a mean speed comes out low, frequently by a factor of two. What is needed is the distribution of hourly speeds at hub height, from a recording anemometer over at least a season, with the power worked out for each speed bin and summed. The hours field on this page is deliberately narrow: it gives the energy for hours spent at exactly the speed entered, which is useful for reasoning about one condition and is not an annual figure.