Calculate turbine thrust for furling

[XXLRay]

According to http://www.thebackshed.com/windmill/Docs/Furling.asp the thrust of a wind turbine is calculated by

Turbine Thrust = diameter² * wind speed² / 24

Isn't that just a rough estimation? As far as I know the power a wind turbine produces is calculated by:

0,5 * air density * wind area * wind speed³ * efficiency factor

Following this formula shouldn't the calculation of the turbine thrust look like

0,5 x 1,2 * 5,2 * 216 x 0,3

I have to claim that my formula is more complex but isn't it more precise on the other hand?

[Flux]

Yes your formula is more precise but as far as I am concerned even your formula gets you only in the ball park.

The issue is not the thrust on the prop, even if you calculate that exactly you are left with a factor not included in the equations at all.

There is a tendency for the prop to seek the wind and I don't know of any equation that even approximately addresses this issue. The seeking force seems to depend on many factors and something that works on one mill on one site may not work out on another on a different site.

My experience has been that the calculated furling point can be in error by a very large margin and if you choose an offset that is too small it will never furl at all.

Probably the larger the offset the nearer you get to the mathematical solutions given by those formulae as the seeking force seems to get swamped with larger offsets.

It's nice to be able to reduce these things to formulae but unless you can account for all the factors the answer will not be very exact. Fluid dynamics is a tricky subject and seems to contain more constants and fiddle factors than most other branches of science. Computer simulation has come a long way in the last few years but even for commercial wind power with servo yaw there is probably little incentive to look into this seeking force thing. Start with a tail half the calculated weight and you stand a chance of being safe.

Flux

[XXLRay]

Thanks a lot. That's all I needed to know.

[bobshau]

I agree that the furling equations are complicated. Most of the equations do not adequately cover how much the center of pressure on the rotor moves away from the center of rotation as a function of yaw angle. For example, we know that the center of pressure on a ship's rudder moves to the quarter chord point as a function of the angle of attack. This is why the rudder shaft is located near the quarter chord point in order to reduce the resulting torque requirements. With regard to a wind turbine rotor, I expect the center of pressure moves horizontally away from the center of rotation toward the windward side of the rotor as yaw increases. My limited search of data has led me to include the following term in my furling equation:

Rotor offset corrected for yaw=(1-.4)x actual rotor offset.

This has the effect of setting the tail weight to (1-.4)=.6 of the weight normally calculated without accounting for the movement of the center of pressure. This is close to Flux's recommendation of using 50% of the calculated weight. It has worked for my 20 foot diameter rotor with a 12" offset using a 26 pound tail assembly measured at 17 feet with 32 sq ft of sail area. This arrangement begins to furl around 20 mph.

I am still searching for good data. Any suggestions? Restoring torque vs. yaw angle would be a good place to start if anyone knows of any.

[Ungrounded Lightning Rod]

I am still searching for good data. Any suggestions? Restoring torque vs. yaw angle would be a good place to start if anyone knows of any.

I don't think there will be a simple formula like that.

I think restoring force is at least partly a gyroscopic 90-degree rotation of differences in force on the top and bottom of the blades (which are experiencing different apparent wind speeds and angles of attack).

If this is correct there are compounding factors.  Some that come to mind are:

 - Airfoil response to various apparent winds.

 - Gyroscopic effect varying with RPM.

A particular problem:  The RPM lags the wind speed, so the nodding force is a response to the current angle and an RPM based on history, while the gyroscopic effect is based on just the RPM which is based on history.  Furling happens when the wind CHANGES.  So even if things canceled out in the steady-state to produce a simple formula, it wouldn't be applicable to the situation when furling.  B-(

Aerodynamics is non-intuitive and doesn't have many good simplifications.  It takes a supercomputer to do it right.  (Gyroscopic effects are tough to calculate, too.)  So this is a job for a rule-of-thumb derived from experiments, followed by a little post-construction tuning if the machine is not an identical copy of one with know characteristics.

[SparWeb]

Maybe you would find this table of comparisons useful.  

http://www.sparweb.ca/Wind_Turbine_Comparison.htm

I also have an "interactive" version if you want to "plug-and-chug", but it needs Excel plug-ins etc.

http://www.sparweb.ca/Wind_Turbine_Comparison-i.htm

Like the others said, there are more factors at play than we know about.  Use these tables with a LOT of caution.  I cannot make any solid predictions with the information - only report what is measured, and compare the numbers.  I used equations from the Backshed, too, so you will recognize some of the math, at least.

[SparWeb]

It turns out the "interactive version" doesn't work universally.

Try this:

http://www.sparweb.ca/Forum/Wind_Turbine_Comparison.htm