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Propeller Aircraft Performance and The Bootstrap Approach

The Bootstrap Approach: Background

The Bootstrap Approach

Of the four forces acting on the airplane – thrust, drag, lift, and weight – thrust is the most difficult to measure or predict. That is why most books about aircraft performance simply assume that propeller efficiency h is some constant. Commonly cited values are h = 80% and h = 85%. Then thrust T = h P, where P is the engine power. Unfortunately, propeller efficiency is in fact not constant; it varies with air speed and RPM or, more precisely, with the dimensionless ratio of those two variables:                                                                             

The basic Bootstrap Approach, strangely enough, makes no assumption about propeller efficiency.  It has an alternate way, which we now explain, for coming up with thrust.  Because a section of propeller blade at distance r from the hub moves (in a sense) in two directions at once – longitudinally with velocity V and to the side with speed 2pnr – there are two distinct propeller "coefficients," one (C_T) having to do with thrust and the other (C_P) having to do with absorbed power. (In the third direction, along the length of the propeller blade, we assume the propeller is rigid enough that it doesn’t move at all.) The propeller thrust coefficient is                                                        

1. Reference wing area S = 174 ft²;

2. Wing aspect ratio A = B²/S = 7.38 (B = wing span = 35.83 ft);

3. Mean sea level (MSL) full–throttle rated torque M₀ = P₀/2pn₀ (P₀ rated power, n₀ rated propeller revolutions per second).  For this Cessna 172, P₀ = 160 HP = 88,000 ft–lbf/sec and n₀ = RPM/60 = 2700/60 = 45 rps. Hence M₀ = 311.2 ft–lbf.  But in most of our formulas, though it makes them a little longer, we’ll retain P₀ and n_0;

4. The proportional mechanical power loss independent of altitude, C, which can almost always be taken as 0.12. This governs full-throttle torque at altitude through the power drop-off factor (Greek capital ‘Phi’):
   

Relative atmospheric density (Greek small 'sigma')  s = r/r₀ where r is atmospheric density and standard density r₀ = 0.002377 slug/ft³.The time-honored form (Gagg and Farrar, 1934) for this drop off factor is

5.     Propeller diameter d = 6.25 ft.

To simplify later calculations, it’s convenient to assume a “standard weight” for the airplane. For our sample Cessna 172 we choose W0 = 2400 lbf, maximum certified gross weight. Standard relative air density is taken to be unity.

Glide test for Drag Parameters

Of the four remaining "harder-to-get" BPD items, two typify drag and two characterize thrust. The drag numbers are the usual:

6. Parasite drag coefficient, C_D0; and

7. Airplane efficiency factor, e.

Getting C_D0 and e by the usual method, linear regression analysis of many glides, is overkill. Instead, simply find, by trial and error, the speed for best glide V_bg and its corresponding glide angle g_bg (Greek small ‘gamma’) at one known aircraft weight W in an atmosphere of known relative density. Let us take W = 2200 lbf and h_r = 5000 ft. That latter makes s = 0.86167 and F(s) = 0.84281. (For convenience of the checking reader, we carry more decimal places than makes strict sense).

Consider that we time glides from 5100 ft to 4900 ft; Dh = 200 ft. Glide angle (in calm wind) is shallowest when product V_T×Dt, true air speed times elapsed time, is greatest.  To find that maximizing V, one can just as well use calibrated air speed Vc.   Best glide angle is later calculated from

The relation between true and calibrated air speeds is:

For our sample Cessna, take V_Cbg = 68.9 KCAS = 116.29 ft/sec and Dt = 16.96 sec From Eq. (9), V_Tbg = 74.3 KTAS = 125.3 ft/sec. From Eq. (8), g_bg = 5.40 deg.

The two required drag parameters are obtained from:

and

Substituting our numbers into Eqs. (10) and (11) gives us C_D0 = 0.0370 and e = 0.720. Those numbers (especially C_D0) would have been different if we had run the glide tests with some flaps extended.

Climb and Level Flight Tests for Thrust Parameters

Our last two BDP items are:

8. Slope of the linear propeller polar, m;

9. Intercept of the linear propeller polar, b.

Of several alternative flight test regimens for evaluating m and b, we choose: trial-and-error climbs to find speed for best angle of climb, V_x, and subsequently b, followed by a test for maximum level flight speed, V_M, and then m. 

V_x is the full–throttle partner of V_bg.   The latter is the most nearly positive (smallest negative) glide angle you can achieve. Accordingly, when product V×Dt is smallest one has found V_x.  For our sample Cessna 172, assume V_Cx = 60.5 KCAS = 102.1 ft/sec.  The true value is then V_Tx = V_x = 65.2 KTAS = 110.0 ft/sec. The Bootstrap formula which kinds polar intercept b is :

Substituting our sample values into Eq. (12) gives b = –0.0564.

We conclude our flight tests with a full-speed level run (still at 5000 ft, still at 2200 lbf) and find V_CM = 104.8  KCAS = 176.9 ft/sec.  In the true terms needed in our formulas, V_TM = V_M = 112.9 KTAS = 190.6 ft/sec.  The Bootstrap formula for polar slope m is:

Substituting our values into Eq. (13) gives us m = 1.70. The Bootstrap Data Plate of Table 4 is complete.

Go to next section-Boostrap Approach: Formulas and Graphs

The ALLSTAR network would like to thank Dr. John T. Lowry, of Flight Physics, for providing this section of material and giving ALLSTAR permission to use it.  Dr. Lowry is the 1999 AIAA Flight Research Project Award winner.  Though the ALLSTAR network edited the material for clarity, and maintains the copyright over the format of the material presentation, the material is wholly Dr. Lowry's and is copyrighted to him (© April 1999).  Any questions about this material should be directed to Dr. Lowry.

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Updated: January 18, 2011