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M_Gunz
03-26-2010, 07:17 AM
This is continuation of earlier chart monkey activities using 1942 P-47B data. There is now a spreadsheet
with graphs showing lift limit, sustainable Gs, turn radius in km and deg/sec for 4 different altitudes.

Not making claims as to accuracy here, just seeing how using the equations turns out.

Error checking is not complete. Example: the spreadsheet 5,000 ft Cd0 is 0.02037 instead of 0.021 gotten
using rounded hand-calculator values. Perhaps it is right. On the rough look the sheet seems okay.

TROLLS WILL BE IGNORED. Be as much of a fool as you want, anyone interested in the thread can also put your
name on ignore and anyone out to read goofy \$#!+ can watch you throw your feces for entertainment. I don't
care about prop blade stress risers or the effect of the position of the moon either.

So far comments about the charts on PM are good. There is discussion about modifying the equation to use a
more accurate form (from aerodynamics textbook) of some terms that accounts for AOA and thrust vectors made
by nose pitch into the turning path. It was decided to post this rather than wait at least another month.

Just the charts in PDF (http://www.mediafire.com/file/iqyjojqzdub/monkey chart.pdf)

The whole sheet as PDF (http://www.mediafire.com/file/ehe2mmnfdrz/turn turn turn.pdf)

Posts that led up to this thread:
================================================== ================================================== =======================

Let's walk through this first from power side. If power is:

W = F * v

Then the force is:

F = W / v

And if we add prop efficiency, n, we got:

F = W * n / v

This works with SI units without any conversions, example is same values as

W =2300hp = 1715110 W
v = 246.6 KTAS = 126,862 m/s
n = 0.85

F = W*n/v = 11491,57 N

--------------------------------------------------------------------------------------------------

I think we all now agree that in the power and thrust side, the relation between power and thrust is TAS and propeller efficiency dependant only so rewriting the power formula gives (Force is replaced with Thrust):

T = W*n/v

As Nexion noted above, at steady flight drag equals thrust so:

D = T

or

D = W*n/v

where

D = drag
T = thrust
W = power
v = true speed
n = propeller efficiency

Now let's take a look to the formula na85 used for to calculate required power in the original thread:

Power required (W) = q*v*S*[Cd0 + Cl^2/(pi*AR*e)]

where
q = dynamic pressure
S = wing area
Cd0 = zero lift drag coefficient
Cl = lift coefficient
e = efficiency factor
AR = aspect ratio

Dynamic pressure q can be rewritten as:

q = 0.5*p*v^2

where
p = density

And drag coefficient part can be rewritten:

Cd0 + Cl^2/(pi*AR*e) = Cd

So showing the density and simplifying the drag coefficient the formula becomes:

W = 0.5*p*v^2*v*S*Cd

As noted earlier na85 forgot the propeller efficiency so rewrite gives:

W = 0.5*p*v^2*v*S*Cd/n

And to see the relation to the thrust and drag balance, it can be rewritten:

0.5*p*V^2*S*Cd = W*n/v

Now the drag side is the basic drag formula and thrust side is the basic propeller thrust formula. Notable thing here is that the density affects only in the drag side when the true speed is used. And this where na85 and the other side made the error; they used EAS instead TAS at the thrust side without conversion. With proper conversion they would have got exactly same results as the others.

The basic assumption is again that at top speed and at given altitude the thrust equals drag ie:

D = T

And we assume following things known:

W = engine power
S = wing area
Sp = wing span
AR = Aspect ratio, calculated from span and area
p = density at this given altitude
M = mass of the plane
n = propeller efficiency
e = efficiency factor
Vs = stall speed
g = load factor, 1 in level flight, limited to certain value.

We can now expand the thrust and drag balance further:

0.5*p*V^2*S*Cd = W*n/v

can be written:

0.5*p*v^2*S*[Cd0+Cl^2/(pi*AR*e)] = W*n/v

And as lift coefficient is:

Cl = L /(0.5*S*v^2*p)

Where:
L = lift force = M*g*9.81

we can finaly write whole balance formula:

0.5*p*v^2*S*[Cd0+([M*g*9.81]/[0.5*p*v^2*S])^2/(pi*AR*e)] = W*n/v

And as an example I picked a P-47B at 15k from WW2aircraftperformance site:

V = 386 mph = 172,52 m/s
W = 2000 hp = 1491400 W
M = 12560 lbs = 5697 kg
Sp = 41 ft = 12.5 m
S = 300 sqft = 27.87 m^2
AR = 5.6
p = 0.7708 kg/m^3
n = 0.85 estimated
e = 0.8 estimated
Vs = 105 mph = 46,93 m/s
g = 6g limit

The only missing value in the balance formula is now Cd0 and using the other values it is found to be 0.0208. Now we can check what kind of load factor can be sustained as an example at 400km/h at this same altitude and putting values to the formula gives a bit over 2.25g. Note that there might be typos Smile

The balance formula might look complicated but after all it's just combination of four basic formulas:

Thrust, T = W*n/v
Drag, D = 0.5*p*V^2*S*Cd
Lift coefficient, Cl = L / (0.5*S*v^2*p)
Drag coefficient, Cd = Cd0 + Cl^2/(pi*AR*e)

================================================== ==================================================

Turn performance estimation -- working out details using data from the same plane at different altitudes.
With many thanks to community members for help in not only making this but in correcting errors on the way.

WAR DEPARTMENT
AIR CORP, MATERIAL DIVISION
Wright Field, Dayton, Ohio
June 18, 1942

P-47B Airplane, A.C. No. 41-5902
Acceptance Performance Tests

High speed in level flight with oil cooler flaps and intercooler flaps flush and throttle wide open with turbo on to give military rated power or 18,250 limiting turbo r.p.m.

Altitude Feet___________________ 5,000_____15,000_____25,000_____27,800
True Speed m.p.h._________________352________386_______420___ _____429

At these altitudes the B.H.P. is 2000HP, engine P.R.M. is 2700 and MP is 51" to 52" Hg.

I will use some of the values from Wurkeri's post (Thanks W!) since this is more about method than hair-counting.

================================================== =
W = 2000 hp = 1491400 W -------- it's from the same data, same plane AFAIK
M = 12560 lbs = 5697 kg ----------- actually 12565 lbs but I'm sure less before actual takeoff. ;^)
Sp = 41 ft = 12.5 m------------------- wingspan
S = 300 sqft = 27.87 m^2------------ wing area
AR = 5.6 ------------------------------- aspect ratio being wingspan^2 / wing area
n = 0.85 estimated
e = 0.8 estimated
Vs = 105 mph = 46,93 m/s---------- clean 1G stall IAS
================================================== =

1 foot = 0.3048 meters
1 mile = 1.609344 kilometers

Altitude Feet____________________ 5,000______15,000_____25,000_____27,800
True Speed m.p.h._________________352________386________420__ ______429
Altitude Meters___________________1,524______4,572______7,6 20______8,473
True Speed m/s____________________157.4______172.5______187.8__ ____191.8
Air density kg/m^3________________1.059______0.774______0.551____ __0.499 -- values interpolated from table below

--------------------------------------------------------------------------------------------------
air density taken from "A Sample Atmosphere Table (SI units)" at http://www.pdas.com/m1.htm
km alt____ kg/cu.meter
0________1.225
2________1.007
4________0.8193
6________0.6601
8________0.5258
10_______0.4135
---------------------------------------------------------------------------------------------------

Symbols used -- thanks Wurkeri!
===========================
W = engine power
S = wing area
Sp = wing span
AR = Aspect ratio, calculated from span and area
p = density at this given altitude
M = mass of the plane
n = propeller efficiency
e = efficiency factor
Vs = stall speed
g = load factor, 1 in level flight, limited to certain value.
===========================

Also from Wurkeri:
===========================
whole balance formula ------- without checking units and cancels, where's my cracker?

0.5*p*v^2*S*[Cd0+([M*g*9.81]/[0.5*p*v^2*S])^2/(pi*AR*e)] = W*n/v
===========================

I look at thrust alone first = W * n / v
W = 1491400 Watts
n = 0.85 estimated

Altitude Meters___________________1,524______4,572______7,6 20______8,473
True Speed m/s____________________157.4______172.5______187.8__ ____191.8
Thrust_N__________________________8,054______7,349 ______6,750______6,609

e = 0.8 estimated, Ar = 5.6; (pi * AR * e) = 14.07
S = 27.87 m^2 -------- wing area
M = 5697 kg ---------- mass of the plane
g = 1 ------------------- data values are from level flight

.5 * p * v^2 * 27.87 * [Cd0 + ( [ 5697 * 9.81 ] / [ .5 * p * v^2 * 27.87 ] )^2 / 14.07 ] = Thrust

Altitude 1524m given all our assumtions from above:
.5 * 1.059 * 24775 * 27.87 * [Cd0 + ( 55888 / [ .5 * 1.059 * 24775 * 27.87 ] )^2 / 14.07 ] = 8054
365609 * [Cd0 + ( 55888 / 365609 )^2 / 14.07 ] = 8054
Cd0 + ( 55888 / 365609 )^2 / 14.07 = 8054 / 365609
Cd0 = 8054 / 365609 - ( 55888 / 365609 )^2 / 14.07
Cd0 = 0.022 - 0.153^2 / 14.07
Cd0 = 0.022 - 0.0233 / 14.07
Cd0 = 0.022 - 0.0017
Cd0 = 0.021 -------------------- at 5,000 ft with these assumptions and this method

Altitude 4572m given all our assumtions from above:
.5 * 0.774 * 29756 * 27.87 * [Cd0 + ( 55888 / [ .5 * 0.774 * 29756 * 27.87 ] )^2 / 14.07 ] = 7349
320939 * [Cd0 + ( 55888 / 320939 )^2 / 14.07 ] = 7349
Cd0 + ( 55888 / 320939 )^2 / 14.07 = 7349 / 320939
Cd0 = 7349 / 320939 - ( 55888 / 320939 )^2 / 14.07
Cd0 = 0.023 - 0.174^2 / 14.07
Cd0 = 0.023 - 0.0303 / 14.07
Cd0 = 0.023 - 0.0022
Cd0 = 0.021 -------------------- at 15,000 ft with these assumptions and this method

Altitude 7620m given all our assumtions from above:
.5 * 0.551 * 35269 * 27.87 * [Cd0 + ( 55888 / [ .5 * 0.551 * 35269 * 27.87 ] )^2 / 14.07 ] = 6750
270802 * [Cd0 + ( 55888 / 270802 )^2 / 14.07 ] = 6750
Cd0 + ( 55888 / 270802 )^2 / 14.07 = 6750 / 270802
Cd0 = 6750 / 270802 - ( 55888 / 270802 )^2 / 14.07
Cd0 = 0.025- 0.206^2 / 14.07 ------ the 0.206 is rounded, when I square the unrounded I get .0426
Cd0 = 0.025- 0.0426 / 14.07 ------- when I square the rounded I get .0424
Cd0 = 0.025- 0.003 ------------------ but either one results in about .00301 anyway
Cd0 = 0.022 -------------------- at 25,000 ft with these assumptions and this method

Altitude 8473m given all our assumtions from above:
.5 * 0.499 * 36787 * 27.87 * [Cd0 + ( 55888 / [ .5 * 0.499 * 36787 * 27.87 ] )^2 / 14.07 ] = 6609
255801 * [Cd0 + ( 55888 / 255801 )^2 / 14.07 ] = 6609
Cd0 + ( 55888 / 255801 )^2 / 14.07 = 6609 / 255801
Cd0 = 6609 / 255801 - ( 55888 / 255801 )^2 / 14.07
Cd0 = 0.026 - 0.2185^2 / 14.07
Cd0 = 0.026 - 0.0477 / 14.07
Cd0 = 0.026 - 0.0034
Cd0 = 0.0226 -------------------- at 27,800 ft with these assumptions and this method

At this point I think there has already been debate about Cd0 *actually* increasing and whether or not our assumptions
concerning fixed efficiencies of wing and prop are correct or not. With this method and these assumptions Cd0 is not
increasing enough to bother me at all.

The equation is never the reality. I only investigate how the equation works with some fixed assumed (and reasonable)
values along with historic data.

Now I turn to working this out for turns at speed without resorting to anything but TAS, probably get this wrong the first
few times as well:

Given:
0.5*p*v^2*S*[Cd0+([M*g*9.81]/[0.5*p*v^2*S])^2/(pi*AR*e)] = W*n/v

Given all our assumtions from above and new speeds, the TAS of the IAS at altitude for about a 3G turn.
Vs = 105 mph = 46.93 m/s---------- x 1.732 = 81.29 m/s IAS

And really it's not IAS but CAS except on Tuesdays between 7AM and 9AM when they're cleaning this side of the street.
I'm not going through OAT and all that since AFAIK the historic data was corrected for Standard Atmosphere in 1942
which leaves room for correction right there that I'm not going into anyway. I'm using a table and saying close enough.

From that IAS/TAS table I get factors at altitudes to interpolate:
Altitude Meters ____________1,500____2,000____4,000____4,500____7, 500____8,000____8,500
IAS x value = TAS__________1.099____1.131____1.263____1.295____1 .493____1.525____1.558

Altitude Meters___________________1,524______4,572______7,6 20______8,473
TAS/IAS factor___________________1.101______1.268______1.5 01______1.556
Turn TAS m/s 81.29 x factor above___89.50_____103.08_____122.02______126.49
Thrust_N________________________14164_____12212___ ___10389______10022

Thrust as W * n / v = 1491400 *.85 / v
These are only the TAS that is sqrt(3) x Vs, it is not saying a 3G turn must result. It's just nice to have something to
compare with later.

.5 * 1.059 * 8010 * 27.87 * [ 0.021 + ( [ 55888 * g-load ] / [ .5 * 1.059 * 8010 * 27.87 ] )^2 / 14.07 ] = 14164
118205 * [ 0.021 + ( [ 55888 * g-load ] / 118205 )^2 / 14.07 ] = 14164
0.021 + ( [ 55888 * g-load ] / 118205 )^2 / 14.07 = 14164 / 118205 = 0.1198
( [ 55888 * g-load ] / 118205 )^2 = ( 0.1198 - 0.021 ) * 14.07 = 1.39
[ 55888 * g-load ] / 118205 = sqrt( 1.39 ) = 1.179
g-load = 1.179 * 118205 / 55888 = 2.49 G's -------------------- at 5,000 ft **with these assumptions and this method**.

.5 * 0.774 * 103.8^2 * 27.87 * [ 0.021 + ( [ 55888 * g-load ] / [ .5 * 0.774 * 103.8^2 * 27.87 ] )^2 / 14.07 ] = 12212
116210 * [ 0.021 + ( [ 55888 * g-load ] / 116210 )^2 / 14.07 ] = 12212
0.021 + ( [ 55888 * g-load ] / 116210 )^2 / 14.07 = 12212 / 116210 = 0.1051
( [ 55888 * g-load ] / 116210 )^2 = ( 0.1051 - 0.021 ) * 14.07 = 1.183
[ 55888 * g-load ] / 116210 = sqrt( 1.183 ) = 1.088
g-load = 1.088 * 116210 / 55888 = 2.26 G's -------------------- at 15,000 ft **with these assumptions and this method**.

.5 * 0.551 * 122.02^2 * 27.87 * [ 0.022 + ( [ 55888 * g-load ] / [ .5 * 0.551 * 122.02^2 * 27.87 ] )^2 / 14.07 ] = 10389
114320 * [ 0.022 + ( [ 55888 * g-load ] / 114320 )^2 / 14.07 ] = 10389
0.022 + ( [ 55888 * g-load ] / 114320 )^2 / 14.07 = 10389 / 114320 = 0.0909
( [ 55888 * g-load ] / 114320 )^2 = ( 0.0909 - 0.022 ) * 14.07 = 0.969
[ 55888 * g-load ] / 114320 = sqrt( 0.969 ) = 0.984
g-load = 0.984 * 114320 / 55888 = 2.01 G's -------------------- at 25,000 ft **with these assumptions and this method**.

.5 * 0.499 * 126.49^2 * 27.87 * [ 0.0226 + ( [ 55888 * g-load ] / [ .5 * 0.499 * 126.49^2 * 27.87 ] )^2 / 14.07 ] = 10022
111255 * [ 0.0226 + ( [ 55888 * g-load ] / 111255 )^2 / 14.07 ] = 10022
0.022 + ( [ 55888 * g-load ] / 111255 )^2 / 14.07 = 10022 / 111255 = 0.0901
( [ 55888 * g-load ] / 111255 )^2 = ( 0.0901 - 0.0226 ) * 14.07 = 0.949
[ 55888 * g-load ] / 111255 = sqrt( 0.949 ) = 0.974
g-load = 0.974 * 111255 / 55888 = 1.94 G's -------------------- at 27,800 ft **with these assumptions and this method**.

================================================== =========================
================================================== =========================

Altitude Feet_____________________5,000_____15,000_____25,0 00_____27,800
Altitude Meters___________________1,524______4,572______7,6 20______8,473
Turn TAS m/s 81.29 x factor above___89.50_____103.08_____122.02______126.49 ------------------- for IAS/CAS = 182 mph at sea level.
Max sustainable g-load at speed______2.49______2.26_______2.01_______1.94----------------------- using this method and assumptions

-----------------------------------------------------------------------------------------------------------------------------------------------------

http://www.aerospaceweb.org/qu...formance/q0146.shtml (http://www.aerospaceweb.org/question/performance/q0146.shtml)

radius = v^2 / [ g * sqrt( g-load^2 - 1 ) ] ------ since I have g-load already, I don't need to calculate bank angle
turn rate radians/sec = [ g * sqrt( g-load - 1 ) ] / v

per the turn estimate post, all in metric:
Alt Feet _____Alt Meters _______ TAS m/s _____ g-load _____ Radius Meters _____ turn-rate deg/sec
5,000 ________ 1,524 __________ 89.5 ________ 2.49 __________ 358 ______________ 9.36
15,000 _______ 4,572 _________ 103.08 _______ 2.26 __________ 534 ______________ 6.12
25,000 _______ 7,620 _________ 122.02 _______ 2.01 __________ 870 ______________ 4.63
27,800 _______ 8,473 _________ 126.49 _______ 1.94 __________ 981 ______________ 4.31

I am not up to playing with angular momentum, how the same mass at higher speed and radius works out
but I get this idea that the same engine power won't sustain the same g-load as those increase.

Okay, let's see what I did wrong this time as far as the method and the math.

Still not done, more later, please anyone check and if you find a problem say where and what.