Rework of prop efficiency
assuming rpm = 8000.
Summing slices method:
sum of Vsq x A x r = 0.795 m^5/s(sq) over 8 stations x2, v1 = 30 m/s,39,47,56,64,73,83,89.
where v1= rad/s x r1 = 832 x 0.036m = 30m/s
Vsq = 900,1521,2209,3136,4096,5329,6889,7921.
VsqA =.13, .26,.4,.53,.7,.8,.96,.95 where A=(x10^-4) 1.5,1.7,1.8,1.7,1.7,1.5,1.4,1.2.
check: 900 x 1.5 x 10^-4 = 0.13 - good so far.
Vsq Ar = 4.7,12.2,22.8,36,53.9,70,96,102
Check: 0.13 x 0.036 = 0.0047
so total should be multiplied by 10^-3 -OK
grand total = 397.6 x 2 = 795.2 x 10^-3 = 0.8
multiply by 2xdenxCa should give torque. (x0.072)
= 0.0576 - this seems to be out by a factor of 10
a rough calculation gives 50 x 15 x 10^-6 area = 7.5 x 10^-4 (x2) =15 x 10^4
and r=0.09m
F = 2xdenx Vsq A x 0.03
V = rad/s x r = 832 x .09 = 75m/s
F= 6 N
Tor = 6 x .09 = 0.54 N-m
So calc with slices must be a bit out somewhere.
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say F = 0.58 N
then Tor seen by motor =:
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Thrust = 28 N in theory.
Thrust = 20 N actual - attribute to tip circulation, dead area on disk, variation due to only 2 blades.
motor power needed = 476 W
torque needed = 476/832 =0.57 N
add drag = .57 + 0.58 = 1.152 N
and power needed = 1.152 x 832 = 958 watts.
so-
*** drag coeff Ca is more like 0.01 for this propeller ***
This is not unheard of-possibly due to small bernoulli number...
Largish velocity but small active length, or blade width in this case.
drag torque = 0.19 N,
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Torque needed = 0.78 N, and power needed is 632 W,
200/632 = 31.6%
Power use is 1.32 times expected. 1/1.32= 0.76
giving motor efficiency of 76%
and prop efficiency of 1/( (8/28) + 1) = 78 % (prop thrust is 8 N less than expected.)
Both those figures sound quite reasonable!!!
My guess of 8000 rpm must be close.
<edit>
Another factor to add: Transmission, or torque efficiency.
= 1/((drag proportion of theoretical) + 1) = 75%
Now, if I multiply all these together as decimals, I will get actual thrust efficiency as a proportion of ideal (frictionless).
= 0.76 x 0.78 x 0.75 = 44 %
I think I have calculated all these decimal fractions in such a way that they can be multiplied like this.
This method expresses loss amount as a proportion of the theoretical amount.
Any comments from math or statistics experts?