Ideal Wind Power Calculations
The value of the ideal power is limited by what is know as Betz coefficient with a value of Cp = 0.59 as the highest possible conversion efficiency possible.
In practice, most wind turbines have efficiencies well below 0.5, depending on the type, design and operational conditions. In the operational output range, wind power generated increases with wind speed cubed. In other words, at a wind speed of 5 m/s, the power output is proportional with 5 cubed = 125, whereas at a wind speed of 10 m/s, the power output is proportional to 1000. This shows that doubling the speed from 5 to 10 m/s resulted in a power increase of 8 folds. This highlights the importance of location when it comes to install wind turbines. The effect of the rotor diameter affect the power output in a square manner, i.e, doubling the rotor diameter results in increasing the power output by four times.
On the other hand, since power generated is related to wind speed by a cubic ratio. That means if your turbine is rated at producing 1KW at 12m/s then it will produce 125W at 6m/s and 15W at 3m/s.
Theory of Wind Turbines
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.
The Power produced (Pkin) 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.
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.
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 (p) 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= 1262)/4 = 12469 m2
Wind Power = 0.5 x A x x V3
= 0.5x12469x 1.2 x (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.
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