The Bootstrap Approach to Aircraft Performance
(Part Two – Constant-Speed Propeller Airplanes)

Calculating performance numbers for an airplane equipped with a constant-speed propeller requires a different bootstrap model than was used with fixed-pitch props. In this promised follow-up to his earlier article on performance of fixed-pitch propeller aircraft, aviation physicist John T. Lowry offers a downloadable spreadsheet plus some guidance on how to use it to calculate key V-speeds and performance numbers at any given combination of loading and atmospheric conditions.

0

AirmanshipThe constant-speed version of the bootstrap approach isin some ways less complicated than the fixed-pitch version. Preliminary flight tests forconstant-speed airplanes (to obtain some of the nine “Bootstrap Data Plate”numbers) require only glides, no climbs. And partial-throttle operations forconstant-speed airplanes are automatically included from the get-go.

On the other hand, the constant-speed propeller thrust model is much more complex,making use of a general aviation version of Boeing’s old General Propeller Chart developedback during World War II. Not to worry, though. Almost all the complexity is dealt withautomatically in the Excel ’97 spreadsheet(bootstp2.xls, 194K) that accompanies this article. I suggest you download it now. [If your spreadsheet programcan’t read Excel ’97 files, you can try the SymbolicLink (SLYK) version (bootstp2.slk, 324K).]

See the appendix below.

Also, if you haven’t already done so, I recommend you read my earlier article “The Bootstrap Approach to AircraftPerformance (Part One)” that discusses fixed-pitch prop airplanes. And possiblythe four linked AllStar (Aviation Learning Laboratory for Science, Technology andResearch) bootstrap articles accessible from the author’s single-page Web site athttp://www.mcn.net/~jlowry. Bootstrap basics largelyoverlap for the two types of propellers.

The Constant-Speed Bootstrap Data Plate

Here’s the Bootstrap Data Plate for constant-speed-prop airplanes:

Symbol and Meaning Value Units
S, reference wing area 174 ft2
B, wing span 36.0 ft
P0, rated MSL power at rated RPM 235 hp
N0, rated RPM 2400 RPM
d, propeller diameter (2 blades) 6.83 ft
CD0, parasite drag coefficient 0.02874
e, airplane efficiency factor 0.720
TAF, propeller Total Activity Factor 195.9
Z, fuselage/propeller diameter ratio 0.688

Table 1. Bootstrap Data Plate for Cessna 182, a constant-speed propeller airplane.

If you compare it with the fixed-pitch-prop BDP, you will see that the constant-speedversion deletes two data items (m and b, the slope and intercept of thelinearized propeller polar), and adds two new parameters in their place: the propeller’sTotal Activity Factor (TAF) and fuselage/propeller diameter ratio (Z). Why?Constant-speed propeller blades change pitch automatically, and have a differentpropeller polar for each different blade setting angle. So we need a new way to deal withconstant-speed propeller action in our performance model.

Total Activity Factor (TAF)

The propeller thrust model we’re using – a “General Aviation General PropellerChart” (GAGPC) adapted for GA use from a model created by Boeing back in the 1940s -requires Total Activity Factor (TAF), which can be calculated simply by making aseries of measurements of your propeller blades and plugging them into the spreadsheet.(When you get ready to do this, read the Appendix to this article for detailed propmeasuring instructions.)

The propeller’s TAF is defined as the number of blades (called BB so asnot to confuse it with wing span B) times one single blade’s Blade Activity Factor.BAF measures power absorbed by the propeller blade. Since power goes as force timesspeed, and (aerodynamic) force on the propeller blade section goes as the square of(tangential) speed, absorbed power goes as the cube of the speed. And therefore, for agiven RPM, as the cube of the distance the blade section is from the hub. Absorbed powerat any propeller station also increases directly with blade width at that section. Add upthe power absorption along the length of the blade and you get Blade Activity Factor. Theprecise formulas for BAF and TAF are in Part A of the downloadablespreadsheet that accompanies this article. Practical details for making the measurementsneeded to get your propeller’s BAF and TAF are in the Appendix.

Our example Cessna 182 propeller has BAF = 97.94 and BB = 2; therefore TAF= 195.9.

Fuselage/Propeller Diameter Ratio (Z)

The second new item, Z helps fine-tune our performance predictions for thesefaster high-performance aircraft by accounting for how much the fuselage interferes withpropeller efficiency by blocking the airflow. It’s calculated simply by comparing the propdiameter (which you can find in the POH or measure directly) with the diameter of fuselagewhen measured one propeller diameter behind the plane of the propeller.

Measuring for Z

Figure 1. Measuring fuselage diameter dFone propeller diameter d behind the propeller plane.

There are several acceptable methods for determining the fuselage diameter. One is tomeasure the perimeter with a long tape measure, then divide by pto calculate the effective diameter. Another method is to measure the fuselage width andheight, and to use the average of those two measurements. It rarely much matters which youchoose.

I get Z for the Cessna 182 about 0.688.

The General Aviation General Propeller Chart

That’s really all you need to know about our propeller thrust model in order to use thespreadsheet and start calculating performance figures for your constant-speed-propairplane. Gluttons for punishment, however, may be interested in a little more backgroundon where the model came from and how it works. If so, read on. (If not, feel free to skipahead without guilt.)

The Boeing Airplane Company developed the B-17 bomber in the late 1930’s. I was raisedin Alamogordo, New Mexico, during the war (“the big one,” as they say). B-17swere a frequent and powerful sight skirting that little railroad town. We kids gawked,pausing in our games of marbles or red rover, as they thundered slowly over ourplayground, low against the skein of mountains, headed back to their base a few milessouth. I digress. But they were beautiful.

The B-17 was powered by four giant radials turning four 10-foot-diameter constant-speedprops. To get the drag polar (values for CD0 and e), Boeing took theirprototype B-17 to 20,000 feet, shut down all four engines, and ran the time-honored glidetests. (Or so I’m told.) However, these engine-out glides made the Boeing management teamunderstandably nervous, so they began looking for a better mousetrap. This led todevelopment of the Boeing General Propeller Chart. In level flight, thrust T equalsdrag D, but to get drag characteristics from thrust one needs propeller efficiency (Greek eta), engine shaft power P,and air speed V. The basic relation is T = EtaP/V = D. Piece of cake.

Boeing engineer R. Uddenberg went to work analyzing reams of data, for 10-foot-diameterpropellers, previously collected by NACA (NASA’s predecessor). Uddenberg figured out a wayto make a correlation between the propeller’s geometry, engine power, air speed, andresulting propulsive efficiency. End of problem.

Well, not quite … at least for our general aviation airplanes. Unfortunately, thedifference in scale between (say) a 285-horsepower Bonanza and some big bomber powered byfour Pratt & Whitney Wasp Majors producing 2,685 horsepower each – or even by WrightCyclone 1,000 horsepower engines on the later B-17E – is just too great. Alas, the BoeingGeneral Propeller Chart doesn’t work for general aviation airplanes.

So we took general aviation propeller data and constructed a new chart, a GeneralAviation General Propeller Chart. The chart is fairly complicated and requiresconsiderable knowledge of propeller theory to use manually, but the accompanyingspreadsheet sidesteps those technicalities and does all the work for you. (For thoseinterested in the gory details of the GAGPC, it’s all available in my book Performanceof Light Aircraft available either from AIAA or from Amazon.com.)

For small airplanes (particularly single-engine airplanes), we also need one additionalfinagle factor, a Slowdown Efficiency Factor (SDEF). This can be motivated by therealization that presidential politics, though only peeking over the far horizon, willsoon be upon us. Inspired by that august (and November) event, we propeller heads havelearned to lie a little. Our propeller thrust model starts out by pretending thatpropeller efficiency depends only on thepropeller “advance ratio” J, defined at V/nd, where Vis the true air speed, n the propeller revolutions per second (RPM/60), and dthe propeller diameter. (Think of the advance ratio J as being the ratio of how far theprop tips move forward compared to how far they move circumferentially.)

But the truth is that the fuselage (and the wings, and the propeller boss) interferewith airflow through the propeller. Blade stations closer to the prop hub are exposed toslower air than stations farther out along the length of the propeller blade. J istherefore not the same all along the propeller blade. Since this reality is inconvenient,we account for it with an approximation (i.e., we lie about it). We adjust for thisfuselage interference, in a somewhat ad hoc fashion, by tacking on a SlowdownEfficiency Factor (SDEF) that depends on how much the fuselage gets in the way.Some fairly antiquated British and American data give us the SDEF graphs shown inFigure 2 for both tractor and pusher aircraft. “Free stream” values of , given directly by the GAGPC, are then multiplied bythe appropriate Slowdown Efficiency Factor.

Slowdown Efficiency Factor graph

Figure 2. Slowdown Efficiency Factors for tractor (SDEFT) and pusher (SDEFP) propeller airplanes.

You don’t need to worry about any of this, of course, because the companion spreadsheetdoes the work for you.

Constant-Speed-Prop Airplane Performance

Okay, at this point we assume you’ve downloaded the spreadsheet, measured yourpropeller and your fuselage diameter, done your glide tests, written down your airplane’sBootstrap Data Plate parameters, and entered the BDP data into Part B of the spreadsheet.

Now it’s time for the good stuff: using the spreadsheet to calculate performancenumbers for your airplane. To do this, your next step is to choose values for the four”operational” variables:

  • gross weight W
  • density altitude h
  • propeller RPM
  • engine power P (as a percentage of rated sea-level full-throttle power)

You get that last figure from your airplane’s POH or the engine operator’s manual byhaving chosen an appropriate manifold pressure MAP. For my sample airplane, aCessna R182, I chose 3100 pounds, 8000 feet, 2300 RPM, and 65% power. Put your choicesinto Part B, recalculate the spreadsheet, and you’re ready to get some performancefigures.

The upper portion of Part C of the spreadsheet gives you some important speeds, such asthe speed for best rate of climb under the power setting you’ve chosen, and thecorresponding rate of climb itself. Our sample airplane could achieve 372 ft/min at 77KCAS. Now that’s not Vy and the airplane’s best ROC; for those you’d need tohave entered your full-throttle and highest RPM values into Part B. But the 372 ft/min at77 KCAS is what this airplane will achieve at this 65% power setting.

But let’s consider the V-speeds. When the throttle is full open (or, in some cases,gliding), the various bootstrap V-speeds also come from knowledge of thrust and/or drag.Here’s the cast of V-speed characters:

  • Maximum level flight speed VM, and minimum level flight speed Vm (when and only when it is greater than stall speed VS) are where thrust and drag are equal, T = D.
  • Speed for best angle of climb Vx is where the so-called “excess thrust,” Txs = TD, is a maximum. The corresponding best angle of climb is found from Equation (2).
  • Speed for best rate of climb Vy is where so-called “excess power,” Pxs = (T-D)V , is a maximum. The corresponding best rate of climb is found from Equation (1).
  • Speed for best glide Vbg is where drag D is a minimum.
  • Speed for minimum (gliding) descent rate Vmd is where power required Pr = DV is a minimum.

Since thrust plays no role in those last two (gliding) V-speeds, there’s no differencein this regard between constant-speed and fixed-pitch propeller airplanes. Theaccompanying spreadsheet includes all these relationships.

The main performance section of the companion spreadsheet is organized according toairspeeds V, but if you’re handy at “what-if” analysis you can arrange toget out performance data organized by density altitude or gross weight. Figures 3 and 4for instance show (for a similar Cessna 182, flaps up and weighing 3000 pounds) V-speedsand maximum rate of climb plotted against density altitude:

V-speeds for Cessna 182

Figure 3. How some important V-speeds (incalibrated terms) vary with density altitude.
This is for a Cessna 182, flaps up, weighing 3000 pounds.

Recall that we’re only talking quasi-static (non-aerobatic) performance here. Thatmeans we take air speed velocity vector V to be “almost” constantduring any one maneuver. Exceptions due to very slow effects of altitude change (whileclimbing or descending) or very slow effects of fuel burn are considered negligible, soour “point performance” calculations ignore those very slow changes in trueairspeed or gross weight. We also assume, (until the next article) that the airplane keepsits wings level. This and the “small flight path angle approximation” imply thatlift L always equals weight W. That in turn means that theairplane’s motion is only due to the other two forces acting on the airframe,thrust T and drag D. There’s nothing new here on drag; it’sthe same old D = GV2 + H/V2 (i.e., the sum of parasiteand induced drag) that it was in the fixed-pitch case. But our method for calculatingthrust T has completely changed.

The lower portion of spreadsheet Part C gives performance numbers for a range ofairspeeds. I’ve chosen (see column B) from 60 KCAS to 180 KCAS in half-knot intervals, butyou can change them if you have a much slower or much faster airplane. Column C has thepropulsive efficiencies, calculated behind-the-scenes using formulas and lookup tablesover to the right side of the spreadsheet (don’t look right after a heavy meal); thosevalues of include Slowdown Efficiency Factors.Column D then gets the corresponding thrust, and column E adds together parasite andinduced drag calculated off to the right (where non-gluttons-for-punishment won’t see it).

Cessna 182 rate-of-climb

Figure 4. How best rate of climb varies (almostlinearly) with density altitude.
This again is for a Cessna 182, flaps up, weighing 3000 pounds.

Once we know thrust and drag (and knowing air speed V and gross weight W),it’s easy to calculate rate of climb ROC (column F) and flight path angle (column G):

Equation 1:
and
Equation 2:

Using my sample data, for say V = 70 KCAS, let’s clarify Equations 1 and 2 withnumerical examples. Thrust T is 418.3 lb and drag D is 280.6 lb; theirdifference is 137.7 lb. Divided by W = 3100 lb gives sin= 0.0444 (by Equation 2), which makes angle-of-climb = 2.55 degrees. When we move to Equation 1 andmultiply this 0.0444 figure by V we have to be careful to use true airspeed inft/sec; that is given over in column L as V = 133.3 ft/sec. That makes ROC =5.92 ft/sec. Multiply by 60 sec/min to get a cockpit-friendly figure, ROC = 355ft/min. For the gliding rate of sink (column H) and glide angle (column I), simply repeatthe above calculations with T = 0, reversing signs to switch climb to sink andflight path to glide path.

Or ignore all this and let the spreadsheet do the work.

Aircraft performance figures ultimately come from (what else?) knowledge of the forceson the airframe. The advantages of the bootstrap approach comes from its fairly simple butstill quite accurate ways of dealing with propeller thrust, the sticking point, for bothfixed-pitch and constant-speed propeller aircraft.

What’s Next?

So far, we’ve kept the airplane’s wings level. Next article we relax a bit and let itmaneuver, banking and turning. This will turn out to be easy, within the bootstrap scheme,and it will bring some surprises. Especially at higher altitudes. Stay tuned!


Appendix.
Using the Spreadsheet (bootstp2.xls)

Before you can get anything out of the spreadsheet you must first perform three piecesof practical work:

  1. Run glide tests (to get parasite drag coefficient CD0 and airplane efficiency factor e) as described in the first bootstrap article. During the glide tests, pull your propeller pitch control back to coarse pitch in order to reduce windmilling drag.
  2. Measure your propeller (to get Total Activity Factor TAF).
  3. Measure the fuselage perimeter, or its height and width, one propeller diameter behind the plane of the propeller (to get fuselage/propeller ratio Z).

You will need your airplane’s POH for additional Bootstrap Data Plate (BDP) informationitems. Recording the results of your mini-research project in the appropriate cells inbootstp2.xls will quickly get you the performanceinformation you’re looking for.

Because constant-speed propeller airplanes have an additional “degree offreedom,” partial-throttle performance is accessible right from the start. Howeverthis does assume POH cruise performance tables (except for the normally”optimistic” air speeds said to result from given values of altitude, RPM, andMP) are correct. There is little reason for the manufacturer to fudge cruise table “%BHP” columns. Those cruise tables are normally only for maximum gross weight, butthat makes no difference to the power setting. If your airplane is lighter it will go abit faster, though probably still not as fast as that elusive Teflon-coated beauty onwhich the cruise tables are based.

It’s a good idea to make yourself a backup copy of bootstp2.xls immediately; there are some complicated table lookups offon the right side. If you start rearranging the sheet, you may run afoul of those lookups.Here are some details on the three input areas (Parts A, B, and C) of bootstp2.xls.

Part A. First, record propeller blade radius R and the number of blades BBin cells B8 and B9. Then empty my sample blade width data out of columns F and G from rows18 through 50. Print out cell block A13:H50. Take the printout out to the hangar, run astrip of masking tape the length of one propeller blade, and mark off the stations (x= 0.20, 0.25, 0.30, … , 0.95, 1.00) mentioned on your printout. See Figure A1.

Since our usual non-Metric measuring tapes use inches and sixteenths of inches, thesheet turns calculated decimal inches into whole inches plus so many sixteenths. If yourspinner is so large it obscures some early stations just treat those blade widths as zero.At each station mark, measure and record the corresponding blade width. Write down thosewidths as so many whole inches (e.g., 5) plus so many (including fractions, if you’d like)sixteenths of an inch (e.g., 3.4). Return to your spreadsheet, enter the data into blockF18:G50, and recalculate [F9] the spreadsheet. Voila! You have your propeller’s TAFin cell F53. Record this figure somewhere in your POH and also in cell B73. Generalaviation values of BAF are generally between 70 and 140. (I didn’t make that TAFtransfer automatic because you may want to use bootstp2.xls for more than one airplane;you wouldn’t want to have to re-enter propeller blade width information each time youshifted from one airplane to the other.)

Propeller measurements

Figure A1. Sheet for collecting data to find yourpropeller’s Total Activity Factor.

Recall what I said earlier about performance following the forces. Level cruise (foryour chosen power and propeller speed settings) is the speed at which the powered flightpath angle g Power is zero; that means when thrust Tequals drag D. With my sample data that was at 112 KCAS = 126.3 KTAS. The POHclaims 148 KTAS. Obviously our old dented-up 182 is somewhat slower. Actually the upperperformance section gave level cruise air speed 111.5 KCAS; that half-knot discrepancy isdue to rounding error. In this cruise condition, with these power parameters, bootstp2.xlssays the propeller efficiency is about 73.5% and that thrust and drag are each about 290pounds.

I hope you enjoy using bootstp2.xls. Formore details on how the bootstrap approach works and how to figure other aspects of yourairplane’s performance (including takeoff and landing) go to my Web site http://www.mcn.net/~jlowry and click on the book Performanceof Light Aircraft available either from AIAA or from Amazon.com. And don’t neglect toshare your experiences using bootstp2.xls with your fellow AVweb pilots in thefeedback section (buttons below). If you run into a problem or have a question, eitherpost it to feedback, which I’ll check periodically, or e-mail me directly: jlowry@mcn.net.