How Does Windmill Power Depend on Blade Diameter and Wind Speed?

[tandoorichicken]

I had trouble with this, probably because there are no numbers in it:

"The power P delivered by a windmill whose blades sweep in a circle of diameter D by a wind speed v is proportional to the square of the diameter and the cube of the wind speed. Show that this dependence on diameter and wind speed is what you would expect in the case of 100% efficient energy transfer by showing that the energy transferred per unit time is equal to

P = π/8 * ρ * D^2 * v^3. (ρ is the density of air, approx. 1.25 kg/m^3)

I have no clue how or where to start.

 

[HallsofIvy]

Imagine a column of air passing through the circle "swept out" by the blades of the windmill at speed v. In unit time, that will be a cylinder with diameter D and length v. Calculate the volume of that cylinder and multiply by the density to get its mass. Now, what is the kinetic energy of that mass of air?

 

[M98Ranger]

It seems like you may have missed the key information in this problem, which is that the power delivered by a windmill is proportional to the square of the diameter and the cube of the wind speed. Let's break this down and see how it relates to the given equation.

First, let's consider the diameter of the blades, D. The power is proportional to the square of the diameter, which means that if we double the diameter, the power will increase by a factor of 4 (2 squared). This makes sense because a larger diameter will allow for more air to be captured and converted into energy.

Next, we have the wind speed, v. The power is proportional to the cube of the wind speed, which means that if we double the wind speed, the power will increase by a factor of 8 (2 cubed). This also makes sense because a higher wind speed will result in more force being applied to the blades, resulting in more energy being produced.

Now, let's look at the given equation: P = π/8 * ρ * D^2 * v^3. We can see that it includes both the diameter and wind speed, and it also includes the density of air, ρ. This density factor is important because it represents the amount of air that is available for the windmill to capture and convert into energy. The higher the density, the more air is available, and the more energy can be produced.

To show that this equation represents 100% efficiency, we can compare it to the general equation for energy transfer, which is given by E = P * t, where E is energy, P is power, and t is time. If we plug in the given equation for power, we get E = (π/8 * ρ * D^2 * v^3) * t. Now, we know that energy is also equal to force times distance, so we can rewrite this as E = F * d. And since force is equal to mass times acceleration (F = ma), we can further rewrite it as E = m * a * d.

Now, let's think about what each component in this equation represents. The mass (m) is the amount of air that is being captured by the windmill, the acceleration (a) is the wind speed, and the distance (d) is the diameter of the blades. So, we can see that this equation is essentially saying that the energy transferred