How to Calculate RPM of Wind Turbine generator

[alokkumar]

I am going to design 10 kw, Permanent Magnet, Direct Drive Wind Turbine Generator. The main problem is, how to determine the rotational speed of the Wind Turbine. This is important for me because I have to design the PMG and its rated rpm values.

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First calculation:


The Power captured by Rotor blades from Wind is given by formula :-
P = 1/2*Cp*ρ*(πr^2)*v^3

Cp = Power coefficient of turbine
r = radius = length of the blade
v = velocity of wind.


Also, Power in any rotational object is given by formula :-
P = τ * ω
τ = Torque
ω = Angular Velocity of the rotating blades.


If we equate both the equations, then still Torque and angular velocity is unknown. Then how can we calculate angular velocity ? unless we know Torque ?


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Alternate calculation is given below. I may be wrong here :-


We know that Kinetic Energy of the rotating Blade is given by formula :-

K.E. = 1/2 * I * ω^2

I = Moment of Intertia of Rotating Blades
i.e. I = 1/3 * n * M * r^2 where M = Mass of blades;
r = length of blade;
n = number of blades



Can Kinetic Energy of rotating Turbine be equated with Power in the Rotating Turbine multiplied by Time ?

i.e. K.E. = P x Time ?

Putting Power equation in the above formula :-

K.E. = 1/2*Cp*ρ*(πr^2)*v^3 x Time

Please correct if I am wrong in my assumption. Also please suggest me a better calculation to determine RPM of Wind Turbine if input Wind Power is known.

 

[mfb]

If we equate both the equations, then still Torque and angular velocity is unknown. Then how can we calculate angular velocity ? unless we know Torque ?

Angular velocity will increase until the net torque is zero (so friction and generator give the same torque as the wind, just in the opposite direction). Cp could depend on the angular velocity...

Can Kinetic Energy of rotating Turbine be equated with Power in the Rotating Turbine multiplied by Time ?

What do you expect to get?

Spoiler

Something like a typical time to start the rotor?

 

[alokkumar]

Thanks mfb for the reply !

In the first calculation, assume Cp, Wind velocity, air density, length of the blade is known then we can calculate Power captured by Wind Turbine from wind. But, how can we compute angular velocity ? You are saying angular velocity will increase until net torque is zero. Please note, this calculation can be done only by measuring the torque and velocity relationship on the actual moving Wind Turbine. But, my point is that I need to know the angular velocity not from experiment but from mathematical calculation. This will allow me to design the rated RPM of the Permanent Magnet Generator.

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In the second alternate calculation I am expecting a mathematical formula for angular velocity by computing Kinetic Energy of the Wind Turbine system.

The equation is :-

K.E. of Rotating Wind Turbine = (Power in Rotating Wind Turbine) x (Time)

we know K.E. or Rotating Wind Turbine = 1/2 * I * ω^2 --- equation a)

we also know Power in Rotating Wind Turbine = 1/2*Cp*ρ*(πr^2)*v^3 ----equation b)

If Time is 1 sec then equation a) and equation b) can be written as :-

1/2 * I * ω^2 = 1/2*Cp*ρ*(πr^2)*v^3 * 1

If Cp, ρ, r, v, I is known then ω can be determined. ω is the angular velocity.

Am I correct in my assumption above ?

 

[mfb]

But, my point is that I need to know the angular velocity not from experiment but from mathematical calculation.

Angular velocity will depend on the generator (and friction). There is no way to avoid that, you have to design both rotor and generator to work together.

If Time is 1 sec

What is special about 1 second? Why 1 second, and not 1.2 seconds, or 1 day?
No, that approach does not work at all.

 

[Baluncore]

The kinetic energy of the rotor is irrelevant. Keep turbine mass as low as is safe.

First you must consider the range of RPM that would be sensible for your turbine blades. You know blade length, = radius, so you can work out maximum speed to keep tip velocity sensibly below the speed of sound.

Then you must decide what RPM would be best for your generator. Maybe you will need a gearbox.

Then you can design your turbine blade profile for an optimum wind velocity and angle of attack.

The optimum number of blades will be inversely proportional to the RPM. That way the air volume passing the turbine will be “sliced” efficiently by the blades.
I would strongly recommend three blades because that is a very stable configuration.

 

[russ_watters]

So to put a finer point on this: while you think you need to be able to calculate the RPM, you probably don't. You probably need to select the rpm based on power and generator needs.

 

[spectastic]

So to put a finer point on this: while you think you need to be able to calculate the RPM, you probably don't. You probably need to select the rpm based on power and generator needs.

in other words, pick a rotor with correlation between rpm and power (like a curve specified by the manufacturer)? I'm not familiar with wind turbines, but how much energy loss is typically attributed to friction loss in the moving parts, and what about the conversion from kinetic energy to electricity?

 

[Baluncore]

I assume you mean the kinetic energy of the wind, as the KE of the rotor is irrelevant.
http://en.wikipedia.org/wiki/Wind_generator#Efficiency
http://en.wikipedia.org/wiki/Betz'_law

 

[Suhan]

How far I think the inlet kinetic energy will not be completely used...say if v1 be the inlet velocity then there will be outlet vel. as u can't deaccelerate the air to a vel of zero..u mig get the energy conversion by .5m(Vin^2-Vout^2) ...now here the problem rises with what will be the outlet vel and what will be the blade vel. or the rpm ..You can get the rpm if you know the power output.. but before designing ucant say the machine will run at particularly at this rpm and will give some rated power until u test it

 

[mfb]

How far I think the inlet kinetic energy will not be completely used

That is taken into account in the efficiency given in post 1.
This thread is more than a year old.

Sure you can determine it in advance. It's just not something that can be done in two lines.

 

[Suhan]

okay assuming no losses what might be the exit velocity how to calculate that ..I think that will depend upon the blade angle and the amount of energy transfer taking place from the air .. so can u suggest me some way to calculate the exit speed if I know the blade angle at inlet . say radial turbine with 1 set of rotors only no stators...will the blade speed depend upon the flow angle and blade angle at inlet??

 

[Suhan]

Okay is the blade speed say mean speed will be the difference between the tangential component of actual inlet vel of air and relative velocity ...where as the relative vel is tangential to blade and actual vel is the floe direction of air at inlet...huh I think I am getting it thanks for the support

 

[mfb]

Please start a new thread if you have your own questions on this topic. I think the initial question from 1.5 years ago has been answered.

In general, the speed will depend on many factors.

 

[David Lewis]

A wind turbine rotates at the speed where torque produced by the turbine equals torque absorbed by the generator. If you know the torque coefficients of the turbine and generator, it's a simple calculation.