Centrifugal maths question

[AnotherUsername]

A wind turbine with 3m radius and a blade total weight of about 12kg rotating at 250rpm has a weight of around 3 tonnes.

Now, the blades mass isn’t spread evenly it’s 75% in the first meter and rest spread along the other 2m.

My question is, with a tip speed of around 100m/s but the bulk of the mass moving at a much slower speed but same rpm, if the blade become detached at what speed would it travel and how far (down wind!)

I just can’t imagine if it’s the tip speed of the root speed or something in between!

[walamai]

AnotherUsername said:

..... a blade total weight of about 12kg rotating at 250rpm has a weight of around 3 tonnes.

What's the 3 tonnes figure, are you talking about the pylon, machinery etc? I may be misunderstanding but can't see how that relates to the question.

AnotherUsername said:

if the blade become detached at what speed would it travel and how far (down wind!)

I was going to make a joking response that the distance downwind would be negligible, since a detached blade would fly off perpendicular to the wind direction.
However a blade might presumably behave like an airplane prop and could 'screw' itself into the wind. So its conceivable that it could travel *up* wind if it became detached. (This effect might be negligible if a single blade came off since it wouldn't be held in the correct orientation to be a 'propeller'.)

[thebraketester]

AnotherUsername said:

My question is, with a tip speed of around 100m/s but the bulk of the mass moving at a much slower speed but same rpm, if the blade become detached at what speed would it travel and how far (down wind!)

Well at “t=broken blade” the tip would have a speed of 100ms, and the root would be Travelling much slower, and it would start to spin around an axis. What happens in the few seconds after that is I imagine extremely complex.

Edited by thebraketester on Tuesday 23 August 16:01

[AnotherUsername]

Yes, complex! The root is where the mass is but centrifugal force of around 3 tonnes and the long length rather complicates things!

[ktcanuck]

Given how close to the ground it is I find it hard to imagine but its characteristics and shape remind of a Sycamore seed and we know how that behaves in free space.

[AnotherUsername]

ktcanuck said:

Given how close to the ground it is I find it hard to imagine but its characteristics and shape remind of a Sycamore seed and we know how that behaves in free space.

Very good analogy!!!! I guess it depends when it got ejected too - anything below the horizontal would have it crash pretty quickly i suppose.

[SpeckledJim]

This is sort-of like throwing a golf club. Not that I do such things, but I know people who do...

It wangs off into the cabbage in a wobbly end-over-end kind of way.

[AnotherUsername]

SpeckledJim said:

This is sort-of like throwing a golf club. Not that I do such things, but I know people who do...

It wangs off into the cabbage in a wobbly end-over-end kind of way.

Throwing it from the heavy end too

[Frik]

Where is the blade centre of gravity and what speed is that travelling at? Ignoring aero effects (not realistic, but useful at the point of detachment) the blade will continue in a perpendicular direction to a line drawn between the turbine axis and blade CoG. The blade may well spin, so the tip is moving relatively faster through the air than the CoG but the blade overall will be travelling slower.

[bucksmanuk]

The tip bit leaves the assembly at its velocity – in this case 100 m/sec
The root of the blade leaves at its velocity – in this case- much less -but directly related to its radius. I can’t do symbols on here – but it’s an omega- R relationship
Omega = rotational speed usually in rads/sec
R = radius at which that bit of material exists, in m/sec.

The blade will want to spin around its centre of rotational mass, much like the sycamore seed so beautifully alluded to above. I’m pinching that analogy….

The direction of travel is determined by the blade angle of attack, and it will travel forward a bit - as per what a propellor would do.

The tip would lose speed rapidly until the system would rotate as per the golf club thrown through the air. Dependent upon where the centre of mass was considered to be in the blade.


As we all like a bit of education…..

Those of us sad enough to do rotor dynamics have to study stuff like this and write reports- called a containment study. We go on the principle that if a rotor should explode (assumed to be a fatigue crack) a third of the shaft mass falls off as the worst possibly shaped chunk - then hits the casing at its worst/weakest point on its trajectory. Does it come through?

As you can imagine the impact dynamics/physics and attendant maths, with weird shaped bits and strange shaped distorting casings, behind what’s going on here are horrendous, and simulations TEND to be more “this is what we are guessing at” more than anything.

I am not aware of it being any more scientific than that in the way of assumptions – but I’m happy to be proven wrong. The “worst third at the weakest point on the trajectory” is an assumption that would stand up in court in front of an expert witness. According to an ex technical director.

I would imagine the blisk/blade designers at Rolls Royce Derby could fill us in with more details.

I'll have a go at the maths later

[Allegro_Snapon]

I wouldn't bother with the math for a Electric windmill.....

https://youtu.be/7nSB1SdVHqQ

........as the environment absorbs it.

There are a few rules of thumb which will be as accurate as any maths.....................

The math for a Trent turbine engine on a planes wing to ensure it doesn't slice the fuselage in half...now that is interesting and pretty confidential as it is key to the economics of plane engines. Needs lots of high computing power Computational Fluid Dynamics (Navier-Stokes equations included without saying). Clever plane people, still get the odd plane downed by thrown turbine blades cutting fuselage of more often hydraullic control lines / fuel lines (Air crash investigation has plenty of examples).

Math for power station turbines.....we didn't bother as too many variables......empirically we worked on a thrown turbine blade that can get out the turbine casing, through the roof / wall of the turbine building was about 1/2 mile landing zone (Calder Hall late 1950s). Iron Bridge had a nice fire on one on their units..https://www.youtube.com/watch?v=q-OLw8KzEdc.

Now thrown blades on a hydro-powered station again, forget the maths, if you are nearby = death. https://en.wikipedia.org/wiki/Sayano-Shushenskaya_...

Sometimes Man just cannot outmath the fluke occurrences that occur........

[Talksteer]

Allegro_Snapon said:

The math for a Trent turbine engine on a planes wing to ensure it doesn't slice the fuselage in half...now that is interesting and pretty confidential as it is key to the economics of plane engines. Needs lots of high computing power Computational Fluid Dynamics (Navier-Stokes equations included without saying). Clever plane people, still get the odd plane downed by thrown turbine blades cutting fuselage of more often hydraullic control lines / fuel lines (Air crash investigation has plenty of examples).

Disc failure isn't modelled itself other than that a disc if it fails the crack proceeds rapidly to the rim whereupon the disc upfolds and breaks into three equally sized parts. There is no feasible way to contain the disc and it will go through the plane like a hot knife through butter. If the plane survives its is by luck.

Containment of the fan blade is modelled with some degree of sophistication but is still proven empirically. Containment of compressor or turbine blades is checked by doing a very basic sum which relates kinetic energy of the blade to thickness of the casing. In almost all cases the casing will be thick enough to contain a single released blade due to the thickness required to hold the gas pressure at that section of the engine and the seal segment between the blade and the casing is also pretty tough.

Critical parts lifing is reasonably sophisticated but again is still highly empirical, which is actually a reason to keep it secret as empirical data is by far the hardest to generate.

[Brother D]

Talksteer said:

Allegro_Snapon said:

The math for a Trent turbine engine on a planes wing to ensure it doesn't slice the fuselage in half...now that is interesting and pretty confidential as it is key to the economics of plane engines. Needs lots of high computing power Computational Fluid Dynamics (Navier-Stokes equations included without saying). Clever plane people, still get the odd plane downed by thrown turbine blades cutting fuselage of more often hydraullic control lines / fuel lines (Air crash investigation has plenty of examples).

Disc failure isn't modelled itself other than that a disc if it fails the crack proceeds rapidly to the rim whereupon the disc upfolds and breaks into three equally sized parts. There is no feasible way to contain the disc and it will go through the plane like a hot knife through butter. If the plane survives its is by luck.

Containment of the fan blade is modelled with some degree of sophistication but is still proven empirically. Containment of compressor or turbine blades is checked by doing a very basic sum which relates kinetic energy of the blade to thickness of the casing. In almost all cases the casing will be thick enough to contain a single released blade due to the thickness required to hold the gas pressure at that section of the engine and the seal segment between the blade and the casing is also pretty tough.

Critical parts lifing is reasonably sophisticated but again is still highly empirical, which is actually a reason to keep it secret as empirical data is by far the hardest to generate.

Disks have a lot of energy... (that's part of a disk from the other engine that travelled thru the airframe to get lodged in the other engine).



http://www.iasa.com.au/folders/Safety_Issues/FAA_I...

[Sarkmeister]

Many years ago I used to work for an energy company, my role to forecast the yield from proposed wind farms. One of my colleagues actually wrote a simple bit of software to work out exactly what you are asking, I seem to remember it was called BladeThrow. I assume this is lost amongst the various other stuff we produced the time.

[V8LM]

There are several papers written on this. This one is open access - https://docs.wind-watch.org/Sarlak_et_al-2016-thro...

These aren’t:

https://journals.sagepub.com/doi/abs/10.1260/03095...
https://www.jstor.org/stable/43749983

[AnotherUsername]

All very interesting especially the last one.

In summary it depends on speed, angle of release, pitch and how what bit comes off!