How Much Power Does a 30m Diameter Windmill Generate?

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SUMMARY

The discussion centers on calculating the power generated by a 30m diameter windmill under specific conditions. The windmill entrains a 25m diameter air stream at a velocity of 14 m/s, with an exit velocity of 5 m/s. The relevant equation for this calculation is the energy equation, which simplifies to focus on shaft work and mass flow rate due to the absence of pressure changes and lost work. The density of air is given as 1.2 kg/m³, and participants confirm that the gravitational potential energy term is not applicable in this scenario.

PREREQUISITES
  • Understanding of fluid dynamics principles
  • Familiarity with the energy equation in fluid mechanics
  • Knowledge of mass flow rate calculations
  • Basic concepts of wind energy generation
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  • Calculate mass flow rate using the initial velocity and area of the windmill
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  • Explore the impact of air density on wind energy calculations
  • Study the principles of energy conservation in fluid systems
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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.

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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.
 
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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.
 

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