Understanding Wind Turbine Power Generation

The electric power generated by a wind turbine depends on several factors, including the speed of the wind, the size of the turbine's rotor blades, and the air density. For an ideal wind turbine, we can determine the theoretical relationship between power output and these parameters using physics principles.

Physics of Wind Power

Wind possesses kinetic energy due to its motion. The power available in the wind is the rate at which this kinetic energy passes through a given area.

The kinetic energy ($KE$) of a mass ($m$) moving at velocity ($v$) is given by:

$$KE = \frac{1}{2} m v^2$$

Consider the mass of air passing through the swept area ($A$) of the turbine blades per unit time. If the air density is $\rho$ and the wind velocity is $v$, the mass of air passing per second is:

$$m/t = \rho \times A \times v$$

Therefore, the power available in the wind ($P_{wind}$) is:

$$P_{wind} = \frac{1}{2} \left(\frac{m}{t}\right) v^2 = \frac{1}{2} (\rho A v) v^2 = \frac{1}{2} \rho A v^3$$

The swept area ($A$) is related to the radius ($r$) of the turbine blades by:

$$A = \pi r^2$$

Substituting this into the power equation:

$$P_{wind} = \frac{1}{2} \rho \pi r^2 v^3$$

This shows that the power available in the wind is directly proportional to the air density ($\rho$), the square of the rotor radius ($r^2$), and the cube of the wind velocity ($v^3$).

Ideal Wind Turbine Efficiency

An ideal wind turbine cannot capture 100% of the wind's power. The maximum theoretical efficiency, known as the Betz limit, is approximately 59.3%. Therefore, the actual electrical power ($P_{electric}$) generated is a fraction of the power available in the wind:

$$P_{electric} = \eta_{Betz} \times P_{wind} = \eta_{Betz} \times \frac{1}{2} \rho \pi r^2 v^3$$

Where $\eta_{Betz}$ is the maximum theoretical efficiency.

From this formula, we can see that the electric power generated is directly proportional to the cube of the wind velocity ($v^3$), assuming air density ($\rho$) and rotor radius ($r$) are constant.

Analyzing the Options

• Option 1: Proportional to square root of wind velocity - This is incorrect. The relationship is with $v^3$, not $v^{1/2}$.
• Option 2: proportional to cube of wind velocity - This is correct, as derived from the physics principles ($P_{electric} \propto v^3$).
• Option 3: proportional to square root of radius of the turbines - This is incorrect. Power is proportional to the area, which depends on the square of the radius ($r^2$), not the square root ($r^{1/2}$).
• Option 4: inversely proportional to density of air - This is incorrect. Power is directly proportional to the air density ($\rho$), not inversely proportional.