Question

Show that the power produced by a wind turbine is proportional to the cube of the wind velocity and to the square of the blade span diameter.

Short answer

Question: Show that the power produced by a wind turbine is proportional to the cube of the wind velocity and the square of the blade span diameter. Answer: The power produced by a wind turbine can be calculated using the equation P = 0.5 * ρ * (π * (D/2)^2) * V^3, where P is the power produced, ρ is the air density, D is the blade span diameter, and V is the wind velocity. From this equation, we can see that the power produced is directly proportional to V^3 and D^2.

Step-by-step solution

Understanding the problem

We are given that the power produced by a wind turbine is proportional to the cube of the wind velocity (V) and the square of the blade span diameter (D). Our task is to derive the expression for power (P) and show that it's proportional to V^3 and D^2.

Calculating Kinetic Energy

The power produced by a wind turbine is derived from the kinetic energy of the moving air. To find the kinetic energy of the air moving through the turbine, we first need to find the mass flow rate of the air and the velocity of the air. The kinetic energy (KE) can be written as: KE = 0.5 * m * v^2 Where m is the mass of the air and v is the air velocity.

Calculating Mass Flow Rate

The mass flow rate of the air through the turbine can be found using the equation: Mass Flow Rate = ρ * A * V Where ρ is the air density, A is the swept area of the turbine blades (in square meters), and V is the wind velocity. The swept area (A) can be calculated as: A = π * (D/2)^2 Where D is the blade span diameter.

Calculating Power Produced

Now we can find the power produced by the wind turbine. Power is the change in kinetic energy per unit time. Therefore, the power (P) can be calculated as: P = Change in KE per unit time Since the kinetic energy is given by KE = 0.5 * m * V^2, substituting the mass flow rate equation and the swept area equation into the kinetic energy equation, we get: P = 0.5 * (ρ * A * V) * V^2 P = 0.5 * ρ * (π * (D/2)^2) * V^3 From the above equation, we can see that the power produced (P) is directly proportional to V^3 and D^2. So, we have shown that the power produced by a wind turbine is proportional to the cube of the wind velocity and the square of the blade span diameter.