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I couldn't open the attachment myself on LML so here it
is:
Here’re some ruminations for the mathematically-inclined to look over and add
any corrections they deem necessary. I tried to solve for the power-off
rate-of-descent and altitude loss at three different bank angles, 30° , 45° , and 60° . I used the parameters of my 235 with me and 30 gallons:
W-1400 lb, S-24.9’, A-78 sq.ft., V Y-110 mph IAS, Î , Oswald efficiency factor,
0.82*. At VY, best L/D,
induced drag, DI, and
parasite drag, DP, are
equal.
V ft/s= 22mph/15, r =2.37689E-3, Q=r V2/2, CL=W/QA, AR=S2/A, D P= QAPD
CDI=CL2/p ARÎ
=W2/Q2Ap Î S2 DI=QACDI=W2/Qp Î
S2=DP APD=DP/Q
Even though we would not be flying at sea-level, using sea-level values
simplifies the calculations for this mathematical exercise.
Using my V Y and my
plane’s parameters, Q=30.93 at VY so APD=1.282
Assuming that A PD is a
constant, we can rearrange terms and solve for V2 with different
values of g-load-multiplied weight.
V2=2gW/r S(p
Î A PD)½ This may be simplified to
V=(26033g)½
In circular flight at a bank angle of Q , the g load
is 1/cosQ and the centripetal force, CF, g
units,=tanQ = V2/g 0r, where g0=32.16, so radius
r=V2/CFg0
Multiplying the obtained radius by p to get the
180° turn circumference, then dividing this by V we get
the time t for the turn. Since the total drag is twice the parasite drag or
induced drag, HP=2QA PDV.
The rate of descent to give the required HP, VS=550HP/W, ft/sec.
So for bank angles of 30° , 45° , and 60° , the following values
were obtained:
g: 1.155,1.414, 2.0; IAS mph: 118.2,130.7,155.5; radius ft.: 1618.6, 1142.6,
933.8; t sec.: 29.33, 18.73, 12.86; drag lb.: 80.43, 112.5, 159.1; HP: 25.34,
39.23, 65.99; V S ft/sec:
9.955, 15.41, 25.92; Alt. Loss: 292‘, 289‘, 333‘.
It appears there may be an optimum bank angle of 45°
that gives the minimum loss of altitude. I’ve heard that is what sailplane
pilots use. Now the altitude loss figures do not account for the loss entering
the descent nor the loss during recovery, but the higher bank angle turns result
in less distance away from the runway, which has some merit. Food for
thought!
*Î is 0.75+ for square tips with rounded edges,
0.75- for rounded-planform tips, 0.80 for Hoerner tips with sharp edges, 0.81
for square tips with sharp edges (Mooney), and 0.82 for raked tips with sharp
edges (mine).
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