Wind Energy-Basics

Wind Energy-where does it come from?

As was mentioned in the article “Renewable Energy-the present state of affairs”, wind energy is “in vogue”. Basically it uses the kinetic energy of moving air, and converts it into useful electrical or mechanical energy. We will get into the details of it soon, but for now let’s just say that it works, in principle, exactly opposite the way a fan works. You supply electricity to a fan, it provides flow of air. Reverse it, i.e. provide blowing air to a fan, it generates electricity. In reality, however, wind energy is a converted form of solar energy. The sun’s radiation heats different parts of the earth at different rates-most notably during the day and night, but also when different surfaces (for example, water and land) absorb or reflect at different rates. This in turn causes portions of the atmosphere to warm differently. Hot air rises, reducing the atmospheric pressure at the earth’s surface, and cooler air is drawn in to replace it. The result is wind.

Wind power production: National average values for year 2000 are shown, based upon BTM (2001) and an average capacity factor of 0.3. The world average for year 2000 is 0.92 W/cap. The growth in cumulated installed capacity from 2000 to 2001 was 35% (BTM, 2002). Some observers expect the growth to slow during the following years, for economic and political reasons, but then to resume growth (Windpower Monthly, 2003).

How much can we extract?

Wind energy is low quality energy, and hence converting it into a high quality form incurs some expenditure. What is that expenditure? First let’s calculate the energy contained in wind. Let the wind speed be v₁ flowing through a cross-section of area S. then the power contained is:

P = ½ .ρ. (S.v₁).v₁²

= ½.ρ.S.v₁³

where, ρ is the air density.

However, all of this cannot be converted into useful energy. In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a wind mill), imagine it to be replaced by a disc that withdraws energy from the fluid passing through it. At a certain distance behind this disc the fluid that has passed through flows with a reduced velocity.

Let v₁ be the speed of the fluid upstream and v₂ the speed downstream. The mean flow velocity through the disc representing the rotor is v_avg, where

V_avg = ½.(v₁ + v₂)

With the area of the disc equal to S, and with ρ = fluid density, the mass flow rate (the mass of fluid flowing per unit time) is given by:

m’ = ρ.S.V_avg = ½.ρ.S.(v₁ + v₂)

The power extracted is the difference between the kinetic energies of the flows approaching and leaving the rotor in unit time:

E’ = ½.m’. ( v₁² - v₂²)

= ¼.ρ.S. (v₁ + v₂). ( v₁² - v₂²)

= ¼. ρ.S. (1- v₁³ ((v₂ / v₁)² + (v₂ / v₁) – (v₂ / v₁)³))

By differentiating E’ with respect to (v₂ / v₁) for a given fluid speed v₁ and a given area S one finds the maximum or minimum value for E’. The result is that E’ reaches maximum value when (v₂ / v₁) reaches 1/3.

Substituting this value results in:

P = (16/27). (1/2).ρ.S.v₁³

The “power coefficient” C_p has a maximum value of: C_p.max = 16/27= 0.593. This number is called the Betz Limit, and it gives the maximum fractional wind energy that can be generated from a certain available wind energy.