created at August 24, 2021

# Wind turbine basic power calculation

Let's see how we could estimate the power, produced by wind turbine. Wind turbines are basically divided into two large groups - with horizontal axis - a, and vertical axis (carousel - b, Savonius rotor - c, Darrieus rotor - d, etc.).

Each turbine has the frontal Area, swept by turbine. For horizontal axis turbines the area is a circle, which is envelope for the turbine blades. For horizontal axis turbines the area is basically the width times the height.

There is an air flux through this frontal area, and this air has some kinetic energy. Consider a cubic shape (parallelepiped) containing some amount of air, which is flowing through the area A during the time t.

the air mass inside the shape is:

$m = \rho \cdot V;\quad V = A\cdot L = A\cdot v\cdot t\quad (1)$

Air's kinetic energy:

$E_k = \frac{m\cdot v^2}{2}\quad (2)$

Substituting (1) into (2) yields:

$E_k = 0.5\cdot A\cdot t\cdot \rho \cdot v^3$

Since the power is the energy per time, then:

$N = \frac{E_k}{t} = 0.5\cdot A\cdot\rho\cdot v^3$

This could be the power of the wind turbine with the frontal area A, if the one could consume the whole air's energy. This is not the case in real life of course. If we divide the real turbine mechanical power by N, we will get an air usage coefficient ξ, which is always smaller than 1. For horizontal axis turbines it is usually reasonable higher than for the ones with vertical axis.

Here are some approximate values for ξ:

 wind turbine type ξ carousel 0.1 Savonius rotor 0.18 Darrieus rotor 0.4 Horizontal axis high speed turbine 0.46

Besides that, there are always some energy loose in the electric generator and transmission. We could assume these looses by introducing efficiency η.

Finally, the output power of the wind turbine is:

$P = \xi\cdot \eta \cdot N = 0.5\cdot \xi\cdot \eta \cdot A\cdot\rho\cdot v^3$

where A - the frontal area [m2], ρ - air density [kg/m3], v - wind speed. We can take a year-average wind speed value in our region if we are interested in long-perpective energy production.