# Energy and speed of a wind turbine

In summary: The ideal way to work a problem is one step at a time ...The ideal way to work a problem is one step at a time ...I think he means efficiency losses. There are friction losses in turbines, so the speed of the wind decreases.

## 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.

#### Attachments

• Thermo-HW2.4.pdf
480.8 KB · Views: 272
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?

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.

UltrafastPED said:
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 ...

The ideal way to work a problem is one step at a time ...

UltrafastPED said:
The ideal way to work a problem is one step at a time ...

Yeh, but he already did the first part ...

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.

CWatters said:
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 ...

## 1. How does a wind turbine produce energy?

A wind turbine works by harnessing the power of the wind to spin its blades, which are connected to a rotor that turns a generator. The rotational energy from the generator is then converted into electricity, which can be used to power homes and buildings.

## 2. What factors affect the speed of a wind turbine?

The speed of a wind turbine is affected by several factors, including the wind speed, the size and shape of the turbine's blades, and the efficiency of the turbine's design. Additionally, the location and surrounding terrain can also impact the wind speed and thus affect the turbine's speed.

## 3. How is the speed of a wind turbine measured?

The speed of a wind turbine is typically measured in revolutions per minute (RPM) or meters per second (m/s). This can be done using sensors and instruments that are installed on the turbine itself, or by remote monitoring systems.

## 4. Can the speed of a wind turbine be controlled?

Yes, the speed of a wind turbine can be controlled through the use of a control system. This system adjusts the pitch of the turbine's blades to optimize its performance based on the current wind conditions. Additionally, some turbines also have the ability to shut down or brake in high wind speeds to protect the turbine from damage.

## 5. What is the ideal speed for a wind turbine?

The ideal speed for a wind turbine depends on the specific design and capabilities of the turbine. Generally, a wind turbine will begin producing power at a wind speed of around 6-9 mph and will reach its maximum output at wind speeds of 25-35 mph. However, some turbines are designed to operate at lower or higher wind speeds, so it is important to consult the manufacturer's specifications for the ideal range for a specific turbine.