Theory of Wind Turbines

previous Ideal Wind Power calculations

A windmill extracts power from the wind by slowing down the wind. At stand still, the rotor obviously produces no power, and at very high rotational speeds the air is more or less blocked by the rotor, and again no power is produced.

Figure 3.2 Ideal Wind Energy Theory.

The Power produced (P_kin) by the wind turbine is the net kinetic energy change across the wind turbine (from initial air velocity of V₁ to a turbine exit air velocity of V₂) is given as:

The mass flow rate of wind is given by the continuity equation as the product of density, area swept by the turbine rotor and the approach air velocity as:

Hence the power becomes:

Since the rotor speed is the average speed (V_a) between inlet and outlet:

Hence, the power is

To find the maximum power extracted by the rotor, differentiate equation 11 with respect to V2 and equate it to zero

Since the area of the rotor (A) and the density of the air (r) cannot be zero, the expression in the bracket of equation 12 has to be zero. Hence, the quadratic equation becomes:
(3V₂ – V₁) (V₂ + V₁) = 0

Since V₂ = - V₁ is unrealistic in this situation, there is only one solution, equation 12 yields:

Substitution of equation 13 into equation 11 results in:

The theoretical maximum fraction of the power in the wind which could be extracted by an ideal windmill is, therefore the fraction 0.5925 is called the Betz Coefficient. Because of aerodynamic imperfections in any practical machine and of mechanical loses, the power extracted is less than that calculated above. Figure 3.7 demonstrates the effect of wind turbine design implications on the resulting power that can be harnessed from the incoming wind. Efficient wind turbines depend on the production of that optimum speed ratio giving the maximum or near the maximum power possible.

Equation 14 clearly shows that:

The power is proportional to the density (ρ) of the air which varies slightly with altitude and temperature
The power is proportional to the area (A) swept by the blades and thus to the square of the radius (R) of the rotor; and
the power varies with the cube of the wind speed (V³). This means that the power increases eightfold if the wind speed is doubled. Hence, one has to pay particular attention in site selection.

Distinction between rated and actual power output of the turbine

The world's largest wind turbine generator has a rotor blade diameter of 126 metres and is located on offshore, at sea-level and so we know the air density is 1.2 kg/m³. The turbine is rated at 5MW in 30mph (14m/s) winds,

Rotor Swept area A= (π. 126²)/4 = 12469 m²

Wind Power = 0.5 × A × ρ × V³ = 0.5 × 12469 × 1.2 × (14)³ = 20.5 MW

Why is the power of the wind (20MW) so much larger than the rated power of the turbine generator (5MW)?

The answer lies in the fact that the Betz limit and inefficiencies in the system seriously absorbs over 60% of the apparent power.

There are two further factors to be considered when estimating the power output from a turbine, the first is the mechanical transmission and the second is the generator’s efficiency, both of which are less than unity, hence the real power is proportionately less than the ideal value.

The capacity factor, Cf. Assuming a 5 kW wind turbine generates annually 10 MWh, if that same installation had run – theoretically – 24 hours a day and 365 days a year at full load, it would have generated 43.8 MWh. The capacity factor (Cf) is 10/43.8 = 0.23. Typical values for Cf between 0.2 and 0.4 in the united kingdom, depending on the exact location.