Hub loss:

In all manuals hub correction exponent (see e.g. Aerodyn Man p. 6)
is described as
f = (B/2) * (rRhub)/ (r sin(phi)) .
But isn’t it
f = (B/2) * (rRhub)/ ((1r)* sin(phi)) ?
One can easy check it out if Rhub is set to zero and the value is set for .1 at the hub or
at .9 at the tip. That’s simply symmetric.

Now to the important things:
According 2D Betz blade chord formula it is not of importance
if one has e.g. 3 blades with 1m chord or 2 blades with 1.50 or 1 blade with 3 m.
when lift coefficient and everything is kept constant.
The local flow angle is the same I think.
But Prandtls tip and rootcorr has as parameter the number of blades.
How can it be? Why is the loss higher for more blades?
Kind regards
Hermann
Dear Hermann,
I’ll answer your first question and let someone more knowledge about rotor aerodynamics respond to your second question.
Actually, Eq. (13) of the AeroDyn theory manual has:
f = ( B/2 )( r  Rhub ) / ( RhubSIN(phi) ).
This coefficient goes to zero near the root and gets very large (f >> 1) at the tip. The coefficient"f" is used in Eq. (7) for the hubloss factor, “F”, as follows:
F = ( 2/pi )*COS^1( EXP( f ) ).
So, the hubloss factor “F” goes to zero near the root and goes to unity at the tip, which is appropriate.
I hope that helps.
Best regards,
Dear Jason.
Yes, we are talking about eq. 13.
But I read here (B/2) * (rRhub)/ (r sin(phi)) .
Version January 2005 is my paper here the wrong one?
Lets check symmetry for the tip and hubloss. and take r/R= 0.1 for the hubloss and r/R)=0.9 for tiploss and Rhub=0.
And assume 2 blades.
Then
f= 0.1111/sin(phi) for the tip with tip formula),
f=1.000 /sin(phi) according (my) formula above and
f=infinite/sin(phi) when I use the formula you provided in your post.
Yes I know the intention.
A plot you can find e.g. in Mendez, Greiner: Wind blade chord and twist angle optimization by using genetic algorithms.
Page 6. They wrote down the same formula which is given in aerodyn manual, but got symmetrical results.
But If I you check out Betz formula for blade shape
c(r) = const * 1/NrofBlades * 1/cl (Very simplified)
. Therfore NrofBlades * cl is constant.
(Or in other words: chord * liftcoeff * nrofblades must be constant to have the same rotor in linear theory.)
Therefore the loss must be a function of cl * NrofBlades, not alone Nrofblades.
This seems to be questionable for me.
Kind regards
Hermann
Dear Hermann,
There was a correction to the AeroDyn Theory Manual, dated December 2005. The corrected version is available from: wind.nrel.gov/designcodes/simulators/aerodyn/. The version of the AeroDyn Theory Manual that I found in the NREL publications database is the old version, dated January 2005. In the corrected version (December 2005), Eq. (13) is stated as in my post above. In the old version it is as you indicated. The corrected version (December 2005) is what is actually implemented in the source code. I’ll see if we can update the NREL publications database to have the corrected version uploaded.
Again, I’ll let someone more knowledge about rotor aerodynamics respond about the correctness of the model.
Best regards,
Dear Jason!
Thank you for your answers. Only to to clarify my opinion:
“My” formula would be symmetric to the tiploss.
I am aerodynamicist, but that is IMO math.
Kind regards
Hermann
I ordered already the reprints of Prandtl and Betz
maybe this will clarify the stuff.
Dear Hermanm,
First things first: Prandtl’s formula is used for the tiploss. The idea behind Prandlt’s correction is that one is attempting at capturing the loss going from a “solid” actuator disk to a limited number of blades. Therefore the loss decreases with the number of blades as it should. The argument of the exponential function increases in absolute value that is.
The idea behind the hub loss is indeed to emulate the tip loss. However, one could argue that it is not quite the same, given the interference aerodynamics present near the hub etc. This may be an answer as to why they do not appear “symmetric” along the span as you mention. The classic correction simply replaces the numerator in the exponential function from (Rtipr) to (rRhub); Aerodyn has also a different denominator (Rhub rather than r). This causes a slight higher loss near the hub than otherwise, but we are talking about small differences of small numbers.
The truth and the matter is that not enough research has been conducted on the hubloss, and perhaps the main reason is that power extraction near the root is limited anyways.
We will be dedicating efforts to validate these and other models with data from our test site turbines in the future.
Rick
Hello and thank you.
Here’s the original article where It’s derived  I found it (but bought it 4 month ago):
goedoc.unigoettingen.de/goescho … dle/1/6356
Prandtl, Ludwig; Betz, Albert (2010): Vier Abhandlungen zur Hydrodynamik und Aerodynamik 
In: Dillmann, Andreas (Ed.) Göttinger Klassiker der Strömungsmechanik; 3. Universitätsverlag Göttingen
It’s freely available per pdf. It’s made by conformal mapping.
What I do not understand:
Betz provides a formula which is providing constant circulation. So far, so good.
It plays no role if you have 1 Blade with 3m chord or three blades with one.
But for this one blade the circulation would be three times so strong.
Why are the losses here higher?
Kind regards to the people behind the big ditch
Hermann
Is it possible to implement in WTperf the radial beginning position of hubloss?
Usually you have a hub with a radius of 1.25m and then the same size a cylinder.
Therefore the hubloss has to begin at a radial station of 2.5m
(sorry  I am calculating in Siunits).
Or can I do it? Is there any documentation of WT_perf available? Flowchart etc.
Dear Hermann,
WT_Perf is based upon PROPPC with a few modifications. The theory for PROPPC was documents in an unpublished report. You can view the report here:
http://wind.nrel.gov/designcodes/papers/HibbsRadkey_BEM_Theory.pdf
Marshall