How Much Power Does a 30m Diameter Windmill Generate?

[desi01]

Homework Statement

In this era of high energy prices, there is a focus on renewable sources of energy such as wind energy. Consider a large windmill 30m in diameter. On a windy day, suppose that the windmill entrains a stream of air 25 m in diameter at a speed of 14 m/s. Downstream of the windmill, the entrained stream exits over a large diameter at a speed of 5 m/s. The pressure equals atmospheric pressure at both inlet and outlet. Density of air: ρ = 1.2 kg/m3

Find the power (in megawatts) generated by the windmill.

Homework Equations

1/2 Δ(v^2) + gΔz + (ΔPressure)/density + lost work + shaft work/mass flow rate = 0

The Attempt at a Solution

So I know that that is no change in Pressure and no lost work so those go to 0. I don't think that you would need the gΔz part either?? So basically I'm thinking plug in the velocities given and that would only leave shaft work which I'm solving for and mass flow rate. Now I don't know what velocity and area I would use for that either so that's where I'm stuck?? Am I headed in the right direction?
Thanks.

 

[Redbelly98]

Welcome to Physics Forums. You're correct that gΔz is not needed (it is zero).

You can use the initial velocity and area, and density, to get the mass flow rate.

Also, note that "shaft work" would actually be power here.

 

[tomcue]

Yes, you are on the right track. To calculate the power generated by the windmill, you will need to use the equation for mechanical energy balance. This equation takes into account the different forms of energy involved in a system, such as kinetic energy and potential energy, and balances them with the work done by or on the system.

In this case, the windmill is converting the kinetic energy of the entrained air into mechanical energy. The velocity and area of the air stream at the inlet and outlet will be important factors in calculating the mass flow rate and shaft work.

To determine the mass flow rate, you can use the equation: mass flow rate = density x velocity x area. In this case, the area will be the cross-sectional area of the air stream, so you will need to use the formula for the area of a circle (πr^2) to find the area.

Once you have the mass flow rate, you can use it to calculate the shaft work using the formula: shaft work = mass flow rate x (Δv^2)/2. This will give you the power generated by the windmill in watts. To convert to megawatts, divide by 1 million.

So to summarize, to find the power generated by the windmill, you will need to:

1. Calculate the mass flow rate using the density, velocity, and cross-sectional area of the air stream.
2. Use the mass flow rate to calculate the shaft work.
3. Divide the shaft work by 1 million to get the power in megawatts.

I hope this helps. Good luck with your calculations!