# Energy and speed of a wind turbine

## Homework Statement

Modern wind turbines can convert up to 30% of the kinetic energy of the incoming wind blowing through their blades to electricity. Assuming that the wind at a proposed offshore location blows steadily at 20 knots, how much electrical power can be generated from a turbine with rotor diameter d=80 m? By how much is the wind slowed when going through the turbine? Temperature of the air is 300K and its pressure is 1 bar. Air can be considered an ideal gas with Cv=5R/2.

Given: u=20 knots ~10.2889 m/s d=80 m T= 300k P= 1 bar Cv=5R/2

## Homework Equations

With out being given any information of the output stream how do I determine output velocity?
Am I right by making certain assumptions, such as, T and V are constant, and that there is no internal energy and if so wouldn't Cv change to Cp? If T is constant would the system be adiabatic?

## The Attempt at a Solution

I have attached a pdf of the work I have done solving for mass flow rate, Kinetic Energy of input, general equation trying to solve for velocity decrease of output and an attempt at creating an overall general energy equation.

## The Attempt at a Solution

#### Attachments

• 480.8 KB Views: 193

## Answers and Replies

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rude man
Homework Helper
Gold Member
Your method for determining electrical power looks fine. Your number for air density was higher than mine (mine = 1.16 kg/m^3) so you answer was correspondingly higher by about 11%.

Seems to me the speed of the outgoing wind depends on the friction in the turbine. If there was no friction you could say k.e.(out) = 0.7k.e.(in) but I see no justification for that. So sorry, I'm not of much help there. We need for the expert thermodynamicists to join in. Chet, you there? UltrafastPED
Science Advisor
Gold Member
From the wind speed and geometry of the turbine you can determine the volume of air passing the blades per second; you also know that the turbine harvests 30% of the kinetic energy available.

I don't think it cools the air, so the molecular kinetic energy isn't important - only the mass flow matters.

With the pressure and temperature given you should be able to calculate the mass per unit volume of air - and then put the parts together.

rude man
Homework Helper
Gold Member
From the wind speed and geometry of the turbine you can determine the volume of air passing the blades per second; you also know that the turbine harvests 30% of the kinetic energy available.

I don't think it cools the air, so the molecular kinetic energy isn't important - only the mass flow matters.

With the pressure and temperature given you should be able to calculate the mass per unit volume of air - and then put the parts together.
That answers his first part but not the second, seems like ....

UltrafastPED
Science Advisor
Gold Member
The ideal way to work a problem is one step at a time ...

rude man
Homework Helper
Gold Member
The ideal way to work a problem is one step at a time ...
Yeh, but he already did the first part ...

CWatters
Science Advisor
Homework Helper
Gold Member
I think UltrafastPED meant consider the energy in a small mass (m) of air... 0.5 mV^2

You know how much the energy has changed, the mass is constant so the velocity change can be calculated.

rude man
Homework Helper
Gold Member
I think UltrafastPED meant consider the energy in a small mass (m) of air... 0.5 mV^2

You know how much the energy has changed, the mass is constant so the velocity change can be calculated.
Not if there are friction losses ...