Wind Turbine Power Calculation Equations

[recreated]

Glad it's Friday? I will be if you can help me with this :D :

I'm trying to understand an equation about calculating the power extracted by the blades of a wind turbine. I have spent a couple of hours going over it but still not making much more sense.

I'm trying to find out why P = 1/2*m*(v² - v²_d). Why the 1/2? Why the '-' sign?? And why the ² ? I assume it is to do with F=ma, so the v (velocity) is squared to show acceleration but still I can't figure it all out.

Any tips on how they got to the last equation (18.5) shown above would be amazing!

Thank you! Cannot wait for this to make sense

 

[CWatters]

https://www.raeng.org.uk/education/diploma/maths/pdf/exemplars_advanced/23_Wind_Turbine.pdf

 

[RTW69]

Does K.E.= 1/2 X M X V² look familiar?

 

[recreated]

It does now, RTW69

 

[rezeew]

:)

I can certainly help you with understanding the equation for calculating the power extracted by wind turbine blades. First, let's break down each component of the equation and its significance.

P represents power, which is the rate at which energy is transferred or converted. In this case, it is the power extracted by the wind turbine blades.

The 1/2 in the equation is a constant that represents the conversion of kinetic energy to electrical energy. It comes from the fact that the kinetic energy of a moving object is equal to 1/2 times its mass multiplied by the square of its velocity (KE = 1/2*m*v^2).

The 'm' in the equation represents the mass of the air passing through the blades per unit time. This is essentially the amount of air that the turbine is capturing and converting into energy.

The first 'v' in the equation represents the wind speed, while the second 'v' represents the wind speed behind the turbine, also known as the downwind velocity. The '-' sign between them indicates that the downwind velocity is subtracted from the wind speed.

The reason for this subtraction is due to the concept of conservation of momentum. As the wind passes through the turbine, it loses some of its energy to the blades, resulting in a decrease in wind speed behind the turbine. This decrease in wind speed is reflected in the equation by subtracting the downwind velocity from the wind speed.

Finally, the 2 in the equation is simply there to account for the fact that the kinetic energy of the wind is proportional to the square of its velocity.

As for the last equation (18.5) shown above, it is likely the result of plugging in specific values for the variables in the original equation. Without knowing the specific context or values used, it is difficult to provide an explanation for it.

I hope this helps to clarify the equation for you. If you have any further questions, don't hesitate to ask. Happy Friday!