If wind speed doubles, the Wind Power Density increases by a factor of ________.

Wind Power Density and Its Dependence on Wind Speed:

Definition: Wind Power Density (WPD) is a measure of the power available in the wind per unit area perpendicular to the wind direction. It is an important parameter for evaluating the potential of wind energy at a particular site and is given by the formula:

WPD = (1/2) × ρ × V³

Here:

• ρ is the air density (typically around 1.225 kg/m³ at sea level and standard atmospheric conditions).
• V is the wind speed in meters per second (m/s).
• The factor of 1/2 arises from the kinetic energy equation.

From the formula, it is evident that wind power density is directly proportional to the cube of the wind speed (V³), meaning that even a small increase in wind speed results in a significant increase in the power density.

Correct Option Analysis:

The question asks what happens to the wind power density when the wind speed doubles. Let us analyze this step-by-step:

WPD_initial = (1/2) × ρ × V³

WPD_new = (1/2) × ρ × (2V)³

WPD_new = (1/2) × ρ × 8V³

WPD_new = 8 × [(1/2) × ρ × V³]

• Let the initial wind speed be V.
• Substitute the initial wind speed into the formula for WPD:
• Now, if the wind speed doubles, the new wind speed becomes 2V.
• Substitute the doubled wind speed into the formula for WPD:
• Simplify the expression:
• From this, it is clear that the new wind power density is 8 times the initial wind power density.

Therefore, when the wind speed doubles, the wind power density increases by a factor of 8.

The correct answer is: Option 2 (8).

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: The wind power density increases by a factor of 6.

This option is incorrect because the relationship between wind power density and wind speed is cubic. A doubling of wind speed results in an 8-fold increase, not a 6-fold increase.

Option 3: The wind power density increases by a factor of 4.

This option is also incorrect. A 4-fold increase would imply that the wind power density is directly proportional to the square of the wind speed (V²), but in reality, it is proportional to the cube of the wind speed (V³).

Option 4: The wind power density increases by a factor of 2.

This option is incorrect because a doubling of wind speed does not result in a doubling of wind power density. Instead, the cubic relationship means the power density increases by a factor of 8.

Option 5: No option provided.

This option is not relevant to the question, as the correct answer has already been determined.

Conclusion:

Understanding the cubic relationship between wind speed and wind power density is crucial for evaluating the potential of wind energy. As demonstrated, doubling the wind speed leads to an 8-fold increase in wind power density, highlighting the significant impact of wind speed variations on energy generation. This principle is central to the design and optimization of wind energy systems, emphasizing the importance of selecting sites with consistently high wind speeds to maximize energy output.