Siting a wind turbine

wind resource assessmentwind turbine power calculationhub height wind speedcapacity factor

Siting a wind turbine: shearing the mast wind up to hub height, the power in it, what the rotor takes, and the capacity factor over a year.

Wind Shear Power Law

v_2 = v_1 \left( \frac{h_2}{h_1} \right)^{\alpha}

Wind speed at one height from a measurement at another. Ground friction slows the lowest air, so the wind at a 100 m hub is meaningfully stronger than the wind at the 10 m mast that measured it — and the cube law turns that difference into a large one.

Wind Power Density

\frac{P}{A} = \tfrac{1}{2} \rho v^{3}

The power carried past every square metre of open air by a wind of speed v. The figure that ranks one site against another before any turbine is chosen, and the place the cube law first bites.

Rotor Swept Area

A = \frac{\pi d^{2}}{4}

The area of the disc a horizontal-axis rotor sweeps, from its diameter. Every wind power equation on this site takes this area, and it is the circle the blade tips trace — not the surface of the blades themselves.

Betz Limit — Maximum Extractable Wind Power

P_{max} = \frac{16}{27} \cdot \tfrac{1}{2} \rho A v^{3}

The most power any device can take from an open stream of moving air: 16/27, or 59.26%, of the kinetic flux through its swept area. Betz proved it in 1920 from momentum and mass conservation alone, with no reference to blades.

Wind Turbine Power Output

P = \tfrac{1}{2} \rho A v^{3} C_p

The mechanical power a rotor of swept area A takes from a wind of speed v, where the power coefficient Cp is the fraction of the available kinetic flux it actually captures.

Tip-Speed Ratio

\lambda = \frac{\omega R}{v}

How fast the blade tips travel compared with the wind coming at them. The single number that decides whether a rotor is running at its best power coefficient, and the reason large turbines turn slowly while small ones spin fast.

Capacity Factor

CF = \frac{E}{P_r \, t}

Energy actually generated over a period, divided by the energy the plant would have made running flat out at its rated power for the whole of it. The number that separates a nameplate from a power station.

How they fit together

Everything here is downstream of one number, and that number is almost never the one you measured. The wind shear power law comes first for that reason: the anemometer is at 10 m and the hub is at 80, and wind power goes as the cube of speed, so a 25% correction in speed is nearly double the power. Get the shear exponent from the site rather than the textbook. About 1/7 over open water or short grass, 0.20 to 0.25 over crops and hedgerows, 0.30 and up over forest or suburbs — and using the open-country 0.14 on a wooded site is the single most common way a resource assessment ends up optimistic.

Wind power density is the site's merit figure and it deliberately mentions no turbine at all: watts per square metre in the undisturbed wind, which is how sites are compared and classed. Bring the machine in with swept area, and note that area goes as diameter squared while power goes as the cube of speed — a 10% larger rotor is 21% more power, but a 10% windier site is 33%. Site beats machine, every time, which is the whole argument for spending the money on the met mast.

The Betz limit and turbine power are the same equation with different coefficients, and they are best read as a pair. Betz sets 16/27, about 59.3%, as the physical ceiling for any device in an open flow, for the plain reason that extracting all the energy would mean stopping the air, and stopped air cannot leave to let more through. Real machines reach 0.35 to 0.45 in Cp, so quoting Betz as an expected output overstates a good turbine by about a third. Tip-speed ratio is the reason Cp is not a constant: a rotor only achieves its rated coefficient near its design λ, typically 6 to 8 for a modern three-blade machine, so a turbine held off-design by a fixed-speed generator gives less than the sheet says. Capacity factor closes the year, and it is worth ending on what it is not — it is not availability, and it is not efficiency. A 35% capacity factor does not mean the machine was broken 65% of the time; it means the wind was not at rated speed, which it usually is not.