Subsequent to last
months tips on transonic airfoil sections suitable for F2A
and F3D propellers, Charlie Stone was kind enough to fax me
details of Luis Paramon's 1996 World champion F2A model and
its propeller. Regrettably the translation is awful, so that
details of the airfoil sections remain largely hidden. However,
it appears that the low Mach sections are of the Clark Y type
(t/c = 11.72%), which is your typical flat bottom section,
while the tips are thinned to 7%. The tip airfoil appears
to progress from the Clark Y to a supersonic diamond section
with the highpoint at 50%: the leading and trailing edges
are sharp.
This is interesting, as the tip Mach number is 0.87 at the
74.5mm radius. Diamond sections require fully supersonic flow
to work properly, so the use at M = .87 is novel. My guess
is that a 6% ARAD section would be better, but the diamond
shape is easy to carve into the prop blank, while the ARA
section is difficult to reproduce. There is really no resemblance
between ARAD and the diamond section, so it may be worth
my while machining up Luis' prop with the ARAD section and
giving that a whirl.
The radial distribution of chord is nearly parallel for most
of the prop, being 13.5mm at the 50% station, 12.8 at 75%
and 8.4 at the tip. This at 38000 RPM and 300 KPH gives Reynolds
numbers of 170000, 208000 and 173000 respectively; these are
good, the bad stuff starts at 150000 and less. The tip is
squared off in the computer display and the photograph, but
is claimed to be of the scimitar shape in the text. All very
interesting.
Now this month I promised you something on power absorption
and thrust. There are 2 really neat formulae for propellers
which have been known since at least 1889 and may be attributable
to the Frenchman Renard. With P for power absorption, T for
thrust, n for RPM and D for diameter, they are:
P = Cp * n^3 * D^5
and
T = Ct * n^2 * D^4
The units don't matter much, as they are swallowed up in the
coefficients Cp and Ct. If you don't know what a coefficient
is, don't lose any sleep over it. It's just a constant number,
chosen to make sure the equation gives the right answer! Even
in the above equations, the coefficients are pretty flakey:
if you look at all closely at the theory, you find that they
are slowly varying functions of Mach and Reynolds numbers,
so they are not even constants at all.
Despite this, they retain some interest.
Suppose you "hot" up your engine and gain 500 RPM. Well, this
must
be a power gain, but how much? We'll cheat a little and say
that the above equations work for static measurements on your
bench, which won't be too far wrong.
Before the rework your RPM was 22000, and after it was 22500.
Then we can write:
Pi = Cp * 22000^3 * D^5
where Pi means initial power before rework.
Pf = Cp * 22500^3 * D^5
where Pf means power after rework.
Then
Pf / Pi = (Cp * 22500^3 * D^5) / (Cp * 22000^3 * D^5)
Things the same top and bottom cancel out leaving
Pf /Pi = 22500^3 / 22000^3
= 1.07
which is a 7% gain in power. Using the second equation, we
can
write
Tf / Ti = (Ct * 22500^2 * D^4) / (Ct * 22000^2 * D^4)
= 1.046
which is a 4.6% gain in thrust. Since speed varies with the
square root of the thrust, this could mean an increase in
speed of 2.27%.
I'll leave you to try the same calculation, but varying the
diameter instead. You'll find that extra diameter requires
heaps of extra power to get the same RPM.
Time to sign off, I hope the editor can fit in the ARAD sections
left over from last month.
