The maximum theoretical power of a wind turbine with a 70 m blade diameter, air density of 1.225 kg/m³, and wind velocity of 9 m/s is calculated using the Betz Law and is approximately 1.71 MW, which is closest to option (d) 1.72 MW.
The question asks about estimating the maximum theoretical power of a wind turbine based on its physical dimensions and operative conditions. To calculate the power, we will use the Betz Law, which states that the maximum theoretical power that can be harnessed from the wind is given by P = (16/27) x (1/2) x ρ x A x v^3, where ρ (rho) represents the air density, A is the sweep area of the turbine blades, and v is the wind velocity.
First, calculate the sweep area (A) using the formula for the area of a circle: A = π x r^2, where r is the radius of the circular area swept by the turbine's blades (half of the diameter). Here, the radius r = 70 m / 2 = 35 m. So, A = π x (35 m)^2 = 3,848.78 m².
Using the given air density of ρ = 1.225 kg/m³ and the wind velocity v = 9 m/s, we then substitute these into Betz's equation:
P = (16/27) x (1/2) x 1.225 kg/m³ x 3,848.78 m² x (9 m/s)^3
P = 1,710,936.45 W or approximately 1.71 MW.
Therefore, the maximum theoretical power output of the wind turbine is closest to the provided option (d) 1.72 MW.