Understanding Wind Turbine Power Output and Wind Velocity

Wind turbines convert the kinetic energy of wind into electrical energy. The amount of power a wind turbine can generate depends significantly on the speed or velocity of the wind.

How Wind Velocity Affects Power Output

The theoretical power available in the wind is directly proportional to the cube of the wind velocity. This relationship is described by the following formula:

$$ P = \frac{1}{2} \rho A v^3 $$

Where:

• $P$ is the power available in the wind
• $\rho$ (rho) is the density of the air
• $A$ is the swept area of the turbine blades (the area through which the wind passes)
• $v$ is the velocity of the wind

As you can see from the formula, the power is proportional to $v^3$. This means small changes in wind speed can lead to large changes in the available wind power.

Analyzing the Effect of Doubling Wind Velocity

The question asks what happens to the power output if the velocity of wind is doubled, within the operating range of the wind turbine. The operating range refers to the wind speeds between the cut-in speed (when the turbine starts producing power) and the rated speed (when the turbine reaches its maximum design power output). Within this range, the power output generally follows the cubic relationship with wind speed.

Let the initial wind velocity be $v_1$ and the initial power output be $P_1$. According to the formula, $P_1 \propto v_1^3$.

Now, if the velocity is doubled, the new velocity $v_2$ is $2v_1$. The new power output $P_2$ will be proportional to $v_2^3$.

$$ P_2 \propto v_2^3 $$

Substituting $v_2 = 2v_1$ into the relationship:

$$ P_2 \propto (2v_1)^3 $$

$$ P_2 \propto 2^3 \times v_1^3 $$

$$ P_2 \propto 8 \times v_1^3 $$

Since $P_1 \propto v_1^3$, we can see that $P_2$ is 8 times $P_1$.

This means that doubling the wind velocity within the operating range where the power output is proportional to the cube of the speed will result in the power output becoming eight times the original output.

Conclusion

Based on the cubic relationship between wind power and wind velocity ($P \propto v^3$), if the wind velocity is doubled from $v$ to $2v$, the theoretical power available increases by a factor of $(2)^3 = 8$. Assuming the wind turbine is operating within its normal range where the output scales with the available power, the power output will also increase by approximately the same factor.

Therefore, if the velocity of wind is doubled within the operating range, the power output will be eight times.