Understanding the Relationship Between Power Input and Torque in Wind Turbines

[boyle007]

Hello Physics Forum,

Thanks to the Physics Forum, I learned an equation that helps me understand the relation between torque and power generated for a wind turbine. The equation is

P = m * ω

and can be rewritten as

m = P/n * 30/∏

where m = applied torque, P = power generated, n = rpm, and ω is a constant * rpm

I understand the above equation, however, I would like to know the following:

If the applied torque is of a certain amount, it will produce an rpm of a certain amount, which will in turn generate a power of a certain amount.

Is that a correct interpretation?

I'll take a specific example to help me understand better. Suppose we put the following numbers into the above equation and get:


12,738 Nm = 2,000,000/1,500 * 30/3.14

If the power input produces 12,738 Nm of applied torque, then that will produce an rpm of 1,500, and that in turn will produce 2,000,000 MW of power generated. So, regardless of the source of the power input (e.g. wind or water power), so long as it produces enough power to generate 12,738 Nm of applied torque, then the power generated will be 2,000,000 MW.

The key, I am thinking, is to produce enough applied torque to generate the required rpm and therefore generate the power output. Is that a correct interpretation?

Thank you

 

[elegysix]

If the applied torque is of a certain amount, it will produce an rpm of a certain amount, which will in turn generate a power of a certain amount.

Is that a correct interpretation?
...
The key, I am thinking, is to produce enough applied torque to generate the required rpm and therefore generate the power output. Is that a correct interpretation?

I don't know the specifics of your setup, but in general an applied torque creates an acceleration, not a steady RPM (ignoring non-conservative forces). This may be a custom solution for your particular system.

In the real world an applied torque will accelerate the system until the
drag torque + generator torque = applied torque

Depending on the design of your generator, generator torque and RPM may not be directly related. This may cause power output to not be linear with RPM.

Otherwise, yes a certain applied torque means a certain RPM which means a certain power output.


however - I think in the real world wind generators and such are run at a constant RPM, and the power output is varied to maintain RPM. This way, in high winds they produce a lot of power but turn at the same RPM. (That's just my guess though)

 

[CWatters]

If the applied torque is of a certain amount, it will produce an rpm of a certain amount, which will in turn generate a power of a certain amount.

In general no.

It's interesting to compare the equations for linear motion with rotation. You will be familiar with Newton's

F=m*a

where F= Force, m=mass and a = acceleration

The corresponding equation for rotation is

T= I * a

where T = Torque, I = Moment of inertia, a = angular acceleration

As elegysix said, many turbines run at a constant rpm. Under those conditions the net torque on the rotor is zero. The torque generated by the blades exactly matches that of the generator/load. If the net torque is zero then the angular acceleration is zero and that corresponds to a constant rpm. If there was a net torque the rotor would accelerate.

Don't confuse net torque with the torque in the shaft. The net torque might be zero but that doesn't mean the shaft isn't transmitting a lot of torque from one end to the other.

 

[boyle007]

Thanks for the answers.

I am still a bit confused. I would like to know the general before the details; first the chain of causality: the prime force (wind, water) begets torque on the shaft, which begets rotation, which begets power generation. Is that a correct "chain" of causality?

If I understand elegysix correctly, then in general and ignoring the details for a moment, "a certain applied torque means a certain RPM which means a certain power output."

Secondly, if I know the numbers shown above (applied torque, rotation speed, power being generated); can one extrapolate the amount of "power input", i.e. the amount of force needed to produce the applied torque, rotation speed, and power generated? I know the power input has to be at least slightly greater than the power generated. Also, how is that "power input" measured? I know power generated is measured in watts.

Thanks once again

 

[CWatters]

I am still a bit confused. I would like to know the general before the details; first the chain of causality: the prime force (wind, water) begets torque on the shaft, which begets rotation, which begets power generation. Is that a correct "chain" of causality?

That's the basic sequence but rotation does not automatically imply power generation. There might be nothing connected to the generator.

Perhaps look at it this way..

A turbine starts from rest with no load connected to the generator.
The wind blows causing a net torque

T_net = T_rotor-T_generator

However since T_generator=0 (no load yet) that means T_net >0 and the rotor accelerates up to it's design speed.
If a load is not applied at this point it might keep accelerating to destruction.
So a load is normally applied and the power extracted is sent to the grid.
The load is constantly adjusted to keep the rpm within design limits despite varying wind speeds. This results in varying power output.

If it's operating at constant rpm the net torque is zero. (T_rotor-T_generator=0)

If I understand elegysix correctly, then in general and ignoring the details for a moment, "a certain applied torque means a certain RPM which means a certain power output."

No. In general a net Torque causes a certain acceleration not a certain rpm. It's the details of the turbine design that cause it to target constant rpm not the basic physics.

Crude turbines such as those used to pump water from a well tend to work differently to those designed to generate electricity. Crude water pumping turbines tend not to be designed constant rpm. If the wind blows they accelerate and pump water faster. To stop them spinning too fast some types employ flexible blades to shed (waste) wind.

Secondly, if I know the numbers shown above (applied torque, rotation speed, power being generated); can one extrapolate the amount of "power input", i.e. the amount of force needed to produce the applied torque, rotation speed, and power generated?

If you can measure the torque transmitted through the shaft and the rpm then yes you can calculate the available input power.

Input Power = transmitted torque * angular velocity

Angular velocity can be calculated from the rpm.

To calculate the output power you need to know the efficiency of the generator.