Propeller Aircraft Performance and the Bootstrap Approach

Propeller Aircraft Performance and The Bootstrap Approach

The Bootstrap Approach: Background

The Bootstrap Approach

The Bootstrap Approach is a parametric performance method. You take the airplane out and fly it, for an hour or two, doing very specific climbs, glides, and a level speed run. Those routine maneuvers must be done at known weight and altitude, but it doesn’t matter what that weight or that altitude is. That gives you the data, after you get back down, to calculate the parameters making up the "Bootstrap Data Plate" (BDP). In all, the BDP consists of nine parameters. When you want to know the airplane’s performance – say angle of climb – under specific circumstances (weight and altitude), you simply take the appropriate Bootstrap formula, substitute in the BDP parameters for that airplane, and the weight and altitude figures, and do the calculations. Out comes the airplane’s performance number. Now let’s take a look at where the Bootstrap Approach comes from.

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  = 80% and  = 85%. Then thrust T = 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:

/ (1)

where J is the "propeller advance ratio." As the propeller rotates through one circle the airplane advances a distance V/n. J is then the ratio of that advance distance to the propeller’s diameter d. Figure 1 is an example of how propeller efficiency varies with advance ratio.

Figure 1. Efficiency graph for McCauley 7557 propeller on some Cessna 172s.

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 nr/2 – there are two distinct propeller "coefficients," one (CT) having to do with thrust and the other (CP) 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

/ (2)

and the propeller power coefficient is

/ (3)

Figure 2 shows, for the same propeller as in Figure 1, how these coefficients vary with advance ratio J. Propeller efficiency can be obtained, knowing the two coefficients, from

/ (4)

Figure 2. All the important information about a propeller’s function can be obtained from its thrust and power coefficient functions.

The Bootstrap Approach uses a little known but close approximate relation between these two coefficients: that the so-called "propeller polar," defined as CT/J2 plotted against CP/J2, is linear. That means that for any reasonable propeller there are two numbers m and b so that

/ (5)

The Bootstrap Approach depends upon our finding those parameters m and b, and a few others, experimentally, by means of flight tests. For the same propeller as above, Figure 3 shows the propeller polar and the best fit line approximating it.

The Bootstrap Data Plate

To predict the airplane’s performance using the Bootstrap method, a so-called "Bootstrap Data Plate" or BDP, consisting of nine numbers, must first be ascertained. Table 4 is a sample BDP for a particular Cessna 172 airplane.

Bootstrap Data Plate Item / Value / Units / Aircraft
Subsystem
Wing area, S / 174 / ft2 / Airframe
Wing aspect ratio, A / 7.38 / Airframe
Rated MSL torque, M0 / 311.2 / ft-lbf / Engine
Altitude drop-off parameter, C / 0.12 / Engine
Propeller diameter, d / 6.25 / ft / Propeller
Parasite drag coefficient, CD0 / 0.037 / Airframe
Airplane efficiency factor, e / 0.72 / Airframe
Propeller polar slope, m / 1.70 / Propeller
Propeller polar intercept, b / –0.0564 / Propeller

Table 4. Bootstrap Data Plate for a particular Cessna 172.

Figure 3. For most propellers, the best fit line to its polar diagram has a goodness-of-fit parameter R2 = 0.95 or better.

Where do these nine BDP items come from? Five come from the Pilots Operating Handbook (POH) or common knowledge. Those are:

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 = P0/2n0 (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 n0;
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’):

/ (6)

Relative atmospheric density (Greek small 'sigma')where  is atmospheric density and standard density = 0.002377 slug/ft3.The time-honored form (Gagg and Farrar, 1934) for this drop off factor is

/ (7)

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, CD0; and

7. Airplane efficiency factor, e.

Getting CD0 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 Vbg and its corresponding glide angle 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= 5000 ft. That latter makes  = 0.86167 and () = 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; h = 200 ft. Glide angle (in calm wind) is shallowest when product VT×t, true air speed times elapsed time, is greatest. To find that maximizing V, one can just as well usecalibrated air speed Vc. Best glide angle is later calculated from

/ (8)

The relation between true and calibrated air speeds is:

/ (9)

For our sample Cessna, take VCbg = 68.9 KCAS = 116.29 ft/sec and t = 16.96 sec From Eq. (9), VTbg = 74.3 KTAS = 125.3 ft/sec. From Eq. (8), bg = 5.40 deg.

The two required drag parameters are obtained from:

/ (10)

and

/ (11)

Substituting our numbers into Eqs. (10) and (11) gives us CD0 = 0.0370 and e = 0.720. Those numbers (especially CD0) 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, Vx, and subsequently b, followed by a test for maximum level flight speed, VM, and then m.

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

/ (12)

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 VCM = 104.8 KCAS = 176.9 ft/sec. In the true terms needed in our formulas, VTM = VM = 112.9 KTAS = 190.6 ft/sec. The Bootstrap formula for polar slope m is:

/ (13)

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.