Appendix 2 Appendix 2. Climb Data Reduction
1. Drag Polar Method. This is one method to develop the airplane’s drag polar equation directly from climb flight test data. It is a simplified method that assumes climb speeds where the compressibility drag is negligible (usually Mach numbers below 0.6), climb angles of less than 15 degrees, and no propeller slipstream effects on the wing lift and drag characteristics.
a. Cautions. Propeller airplanes are susceptible to slipstream drag, and all airplanes are susceptible to trim drag. This is most noticeable on airplanes with wing-mounted engines and when one engine is inoperative. Care should be given so that drag results are not extended from one flight condition to another. Examples of this are:
(1) Drag obtained in level cruise configuration cannot be extended to a climb configuration.
(2) Two-engine climb data cannot be extended to the one-engine-inoperative case.
In summary, the power and trim conditions must remain very close to those existing for the actual test conditions. Drag results are only as accurate as the available power information and propeller efficiency information. The cooling airflow through the engine is also a factor.
b. Calculation of CD and CL. Flight test data for various climb airspeeds, weights, and altitudes should be used to calculate CD and CL. The equations are as follows:
TAT AF R / CO WT 96209CD = BHP T P 3 TAS 33,000 V S e 2 T AFR / C AT O295WT 1 101.27VC TAS CL = Ve 2 S
Where:
BHPT = test day horsepower (see Appendix 1) P = propeller efficiency (obtain from propeller manufacturer or may be estimated) TAT = test air temperature - ºKelvin TAS = standard air temperature - ºKelvin A2-1
Appendix 2 R/CO = observed rate of climb - feet/minute WT = airplane test weight - pounds Ve = equivalent airspeed - knots S = wing area - square feet = atmospheric density ratio (see Appendix 7, figure 1) AF = acceleration factor (may be insignificant at lower speeds) 2 3.5 1 0.2M 1 2AF = 0.133M 1 2 2.5 1 0.2M Where: M = Mach number VC is constant, altitude below 36,089 feet
c. Data Plotting. Once CD and CL2 are calculated from various climb tests at many altitudes, weights, and airspeeds, a plot is made of CD versus CL2. This choice of parameters reduces the parabolic drag polar (CL vs. CD) to a straight line relationship. These procedures should be used to establish CDP and e for each configuration where climb data is obtained.
Figure A2-1 - Coefficient of Drag Versus Coefficient of Lift .05 .06 .07 .08 CD CDP 0 .1 . 2 . 3 . 4 . 5 . 6 2CL From this plot, the profile drag coefficient (CDP) can be determined graphically and Oswald’s efficiency factor (e) can be calculated.
A2-2
Appendix 2 CL2 CL2 / CDe = or e =2 2 CD CDP 3.1416 b 3.1416 b S S Where: b = wing span - feet S = wing area - square feet
d. Standard Day Correction. Since the CL 2 versus CD data was developed from test day conditions of weight, altitude, temperature, and power, calculations will be required to determine standard day conditions.
THPA THPR 33,000 R/C = WC (AF) Where: THPA = thrust horsepower available THPR = thrust horsepower required WC = aircraft weight to which correction is to be made (pounds) AF = acceleration factor (see paragraph b) THPA = BHPc ( )P Where: BHPc = chart brake horsepower at test day density altitude (see Appendix 1) = propeller efficiency P VT 3 CDPS 0.2883WC 2 THP R = 296209 eb V T Where: = atmospheric density ratio VT = true airspeed - knots CDP = profile drag coefficient S = Wing area - square feet e = efficiency factor b = wing span - feet WC = aircraft weight to which correction is to be made - pounds
e. Expansion to Non-standard Conditions. The methods in paragraph “d” can be used to expand the climb data by choosing weight, altitude, temperature, and the corresponding power available.
A2-3
0 D 0 0 11/16/2011 AC 23-8C Appendix 2
f. References. The following references may be of assistance in cases where compressibility drag is a factor, climb angles are greater than 15 degrees, or if the reader wishes to review the basic derivations of the drag polar method:
(1) “Airplane Aerodynamics and Performance” by C. Edward Lan and Jan Roskam. Published and sold by:
Roskam Aviation and Engineering Corporation Route 4, Box 274 Ottawa, Kansas 66067
(2) Air Force Technical Report No. 6273, “Flight Test Engineering Handbook,” by Russell Herrington, et al., dated May 1951. Corrected and revised June 1964-January 1966. Refer to NTIS No. AD 636.392. Available from:
National Technical Information Service (NTIS) 5285 Port Royal Road Springfield, Virginia 22161
2. Density Altitude Method. This method is an alternate to the Drag Polar Method. The Density Altitude Method is subject to the same cautions as the Drag Polar Method. Item numbers 1, 2, 6, 9, 12, 17, 18, and 19 are observed during flight tests and the remaining items are calculated.
Item No. Item 1 Pressure Altitude (Hp) – feet 2 Outside Air Temperature – °F 3 Atmospheric Density Ratio – .4699 4. Density Altitude (Hd) – feet. Hd = 145539 1
5. Std. Temp. @ Hp (TS) – °F + 460
6. IAS – knots
7. CAS – knots
8. TAS = 7 3
9. Observed rate of climb – ft/min
10. T/T S = 2 + 460 ____ 5 A2-4
oO O 0 0- 00 0000 ) _ _________ 11/16/2011 AC 23-8C Appendix 2
11.
12.
13.
Wc Where WC = aircraft weight to which correction is to be made
14. q πeb2 = 7 2 πeb2 295 Where: b = wing span in feet e = Oswald's efficiency factor (0.8 may be used if a more exact value cannot be determined)
215. ΔD 8 = Wc 12 _______________
16. Δ(R/C) ΔD i WS
17. Calibrated RPM (reciprocating engine)
18. Calibrated MP (reciprocating engine)
19. Inlet air temperature
20. Test day BHP corrected for temperature from Appendix 1 at Hd
21. P -- propeller efficiency (obtain from propeller manufacturer or may be estimated)
22. ΔTHP = (
23. Δ(R/C) ΔP = WC
24. R/CSTD = 11 - 13 - 16 + 24 Actual R/C = * Test Weight, 2- lbs.
∆R/C∆W = (I - 9 10 11 12 = 101.27 _____________ ____ 14 15 8
- X 33,000 ________ 22 21 20 23 Items 4, 7, and 24 are used to plot figure 25-2.
A2-5
I Appendix 3 Appendix 3. Static Minimum Control Speed Extrapolation to Sea Level
1. General. The purpose of this appendix is to identify one method of extrapolating minimum control speeds (VMC) observed during flight tests, to sea level, standard temperature conditions. There is a geometric relationship between the yawing moment about the c.g. caused by the operating engine, and the rudder deflection necessary to offset this tendency and cause an equilibrium.
2. Calculation Method. This method involves calculating a geometric constant (C2) for each observed test value, averaging the results, and calculating a sea level VMC. The equations are as follows:
1 VMC = [(C2 )( )(THP)] 3 or;
3VMCC2 ( )(THP) Where: C2 = a geometric constant = the square root of the density ratio THP = thrust horsepower (test shaft horsepower or brake horsepower times propeller efficiency)
3. Cautions and Assumptions. This method has the following associated cautions and assumptions:
a. This method is limited to airplanes with a VMC due to lack of directional control. Each test value of VMC must be observed with full rudder deflection. If, for example, the test conditions result in reaching the force limit (150 pounds rudder force) prior to achieving full rudder deflection, then observed VMC values would require special consideration.
b. The effects of wing lift in the 5 degree bank angle are ignored.
c. Do not use this method for fixed-pitch or windmilling propellers.
d. Any altitude effects that may result from drag on a rotating feathered propeller on the inoperative engine are ignored.
e. Computing a VMC value at sea level involves raising to the power of 1/3 (use 0.33333333). The number of significant digits used affects the resulting computations. For this reason, use at least 8 significant digits.
A3-1
r Appendix 3
f. Propeller efficiencies should be reasonable. They may be obtained from propeller efficiency charts provided by the propeller manufacturer, or from other acceptable sources.
4. Sample Calculations. Test data from two-engine turbopropeller airplanes have been used for illustration. Observations for one ta keoff flap setting are presented. The procedures should be repeated for each additional approved takeoff flap setting. Table A3-1 presents five data points that were collected at various altitude and temperature conditions, and the resulting C2 values that were calculated. For these tests, the inoperative propeller was feathered (auto-feather available).
Table A3-1 – Flight Test Data RUN
PRESSURE
ALTITUDE (FEET) O.A.T.
(°F) TORQUE (FT-LB)
PROPELLER RPM VMC (KCAS) SHAFT HORSE- POWER p
(2) C2 1 3500 86.3 3219 1700 91.2 .914243 1041.95 .590 1349.657 2 4200 88.3 3219 1700 91.2 .900795 1041.95 .585 1381.516 3 4800 87.3 3219 1700 90.7 .891588 1041.95 .580 1384.786 4 5500 85.2 3219 1700 90.7 .881668 1041.95 .575 1412.544 5 6300 83.2 3219 1700 90.7 .870083 1041.95 .570 1443.907
(1) Calculated from observed torque and propeller rpm
(2) Obtained from propeller manufacturer The propeller efficiencies were obtained from power coefficient versus advance ratio map, which was obtained from the propeller manufacturer. The 4-blade propellers were assumed for these calculations to have an activity factor = 80; and an integrated lift coefficient = 0.700.
The five C2 values from Table A3-1 were averaged as 1394.482. The sea level, standard temperature maximum shaft horsepower was 1050. At low speeds, the propeller efficiency changes fairly significantly with speed. For this reason, it is appropriate to determine propeller efficiencies at several speeds near the estimated sea level VMC value. Table A3-2 presents the thrust horsepower values determined for calibrated airspeeds of 90, 95, 100, and 105 knots and the VMC values calculated using these thrust horsepower values and the average C2 (1394.482).
Figure A3-1 illustrates the plot of airspeed versus thrust horsepower. One curve is of thrust horsepower available versus airspeed. The other represents the calculated VMC values versus thrust horsepower available at sea level. The intersection of the two curves represents the VMC value associated with sea level, standard temperature conditions. These calculations resulted in a final VMC value of 98.8 knots calibrated airspeed.
A3-2
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-,/ " I.)" \ / V 11/16/2011 AC 23-8C Appendix 3 Table A3-2 - Tabulated Thrust Horsepower Available and Calculated VMC VC (KCAS) SHAFT HORSEPOWER P THRUST
HORSEPOWER
AVAILABLE AT
SEA LEVEL
CALCULATED VMC C2 = 1394.482 90 1050 .610 640.5 96.3 95 1050 .640 672.0 97.9 100 1050 .665 698.3 99.1 105 1050 .688 722.4 100.2 Figure A3-1 - Thrust Horsepower at Sea Level 85 90 95 100 105 VCAS KNOTS
SEA LEVEL STD. DAY VMC
CALCULATED
VMC VALUES
THRUST HORSEPOWER
AVAILABLE AT SEA LEVEL 620 640 660 680 700 720 740
THRUST HORSEPOWER AT SEA LEVEL A3-3
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