Final answer:
The power from the wind turbine at a height of 300 m for a rotor diameter of 90 m is approximately 190.46 kW. The wind velocity at 10 m is 2 m/s and the shear ratio is 0.15. Using the given information and the power formula, we can calculate the power output of the wind turbine.
Step-by-step explanation:
To find the power from the wind turbine at a height of 300 m with a rotor diameter of 90 m, we can use the formula P = 0.5 * Cp * A * rho * V^3, where P is the power, Cp is the efficiency factor (Betz limit), A is the area of the rotor (pi * r^2), rho is the air density, and V is the velocity of the wind. First, we need to find the velocity of the wind at a height of 300 m using the shear ratio. The wind velocity at 10 m is given as 2 m/s, and the shear ratio is given as 0.15. So, the wind velocity at 300 m can be calculated as V1 = V0 * (H1 / H0) ^ alpha, where V0 is the velocity at 10 m, H0 is the height at 10 m, H1 is the height at 300 m, and alpha is the shear exponent. Substituting the known values, we get V1 = 2 * (300 / 10) ^ 0.15. Now, we can calculate the area of the rotor as pi * (diameter/2)^2. Substituting the known values, we get A = 3.14 * (90/2)^2. Finally, we can substitute all the values into the power formula to calculate the power from the wind turbine.
Here is the step-by-step calculation:
P = 0.5 * 0.45 * (3.14 * (90/2)^2) * 1.23 * (2 * (300 / 10)^0.15)^3
Simplifying the equation, we get P = 0.5 * 0.45 * 3.14 * 45^2 * 1.23 * (2 * 30^0.15)^3
Calculating further, we get P ≈ 0.5 * 0.45 * 3.14 * 45^2 * 1.23 * (2 * 2.8)^3
Simplifying the equation, we get P ≈ 0.5 * 0.45 * 3.14 * 45^2 * 1.23 * 19.36^3
Calculating further, we get P ≈ 0.5 * 0.45 * 3.14 * 45^2 * 1.23 * 7186.98
Finally, calculating the equation, we get P ≈ 190,456.27 W or 190.46 kW. Therefore, the power from the wind turbine at a height of 300 m for a rotor diameter of 90 m is approximately 190.46 kW.