December 12, 1999 The Bootstrap Approach to Aircraft Performance (Part Two — Constant-Speed Propeller Airplanes)

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by	John T. Lowry, Ph.D.

About the Author ... John T. Lowry is a consulting aviation physicist and private pilot (with tailwheel and high-performance aircraft endorsements) who has piloted airplanes from the Texas Gulf Coast to the Alaska Arctic Circle. John's Ph.D. in physics was earned at the University of Texas at Austin in 1970. Before he discovered the joys and rigors of aviation ten years ago, his academic specialty was irreversible quantum statistical mechanics. In an even earlier life, John was a Naval submarine officer and Academic Director of the Navy Nuclear Power Preparatory School. Dr. Lowry, 64, lives with his wife Mary Ann, a graphics designer, in Billings, Montana. Articles in This Series The Bootstrap Approach to Aircraft Performance: Part One — Fixed-Pitch Propeller Airplanes Part Two — Constant-Speed Propeller Airplanes Part Three — Maneuvering

The constant-speed version of the bootstrap approach is in some ways less complicated than the fixed-pitch version. Preliminary flight tests for constant-speed airplanes (to obtain some of the nine "Bootstrap Data Plate" numbers) require only glides, no climbs. And partial-throttle operations for constant-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 developed back during World War II. Not to worry, though. Almost all the complexity is dealt with automatically in the Excel '97 spreadsheet (bootstp2.xls, 194K) that accompanies this article. I suggest you download it now. [If your spreadsheet program can't read Excel '97 files, you can try the Symbolic Link (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 Aircraft Performance (Part One)" that discusses fixed-pitch prop airplanes. And possibly the four linked AllStar (Aviation Learning Laboratory for Science, Technology and Research) bootstrap articles accessible from the author's single-page Web site at http://www.mcn.net/~jlowry. Bootstrap basics largely overlap for the two types of propellers.

The Constant-Speed Bootstrap Data Plate

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

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

Total Activity Factor (TAF)

The propeller thrust model we're using — a "General Aviation General Propeller Chart" (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 a series 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 prop measuring instructions.)

The propeller's TAF is defined as the number of blades (called BB so as not 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 times speed, 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 a given RPM, as the cube of the distance the blade section is from the hub. Absorbed power at any propeller station also increases directly with blade width at that section. Add up the power absorption along the length of the blade and you get Blade Activity Factor. The precise formulas for BAF and TAF are in Part A of the downloadable spreadsheet that accompanies this article. Practical details for making the measurements needed 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 these faster high-performance aircraft by accounting for how much the fuselage interferes with propeller efficiency by blocking the airflow. It's calculated simply by comparing the prop diameter (which you can find in the POH or measure directly) with the diameter of fuselage when measured one propeller diameter behind the plane of the propeller.

Figure 1. Measuring fuselage diameter d_F one propeller diameter d behind the propeller plane.

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

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 the spreadsheet and start calculating performance figures for your constant-speed-prop airplane. Gluttons for punishment, however, may be interested in a little more background on where the model came from and how it works. If so, read on. (If not, feel free to skip ahead without guilt.)

The Boeing Airplane Company developed the B-17 bomber in the late 1930's. I was raised in Alamogordo, New Mexico, during the war ("the big one," as they say). B-17s were 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 our playground, low against the skein of mountains, headed back to their base a few miles south. I digress. But they were beautiful.

The B-17 was powered by four giant radials turning four 10-foot-diameter constant-speed props. To get the drag polar (values for C_D0 and e), Boeing took their prototype B-17 to 20,000 feet, shut down all four engines, and ran the time-honored glide tests. (Or so I'm told.) However, these engine-out glides made the Boeing management team understandably nervous, so they began looking for a better mousetrap. This led to development of the Boeing General Propeller Chart. In level flight, thrust T equals drag 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 = P/V = D. Piece of cake.

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

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

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

For small airplanes (particularly single-engine airplanes), we also need one additional finagle factor, a Slowdown Efficiency Factor (SDEF). This can be motivated by the realization that presidential politics, though only peeking over the far horizon, will soon be upon us. Inspired by that august (and November) event, we propeller heads have learned to lie a little. Our propeller thrust model starts out by pretending that propeller efficiency depends only on the propeller "advance ratio" J, defined at V/nd, where V is the true air speed, n the propeller revolutions per second (RPM/60), and d the propeller diameter. (Think of the advance ratio J as being the ratio of how far the prop tips move forward compared to how far they move circumferentially.)

But the truth is that the fuselage (and the wings, and the propeller boss) interfere with airflow through the propeller. Blade stations closer to the prop hub are exposed to slower air than stations farther out along the length of the propeller blade. J is therefore 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 this fuselage interference, in a somewhat ad hoc fashion, by tacking on a Slowdown Efficiency 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 in Figure 2 for both tractor and pusher aircraft. "Free stream" values of , given directly by the GAGPC, are then multiplied by the appropriate Slowdown Efficiency Factor.

You don't need to worry about any of this, of course, because the companion spreadsheet does the work for you.

Constant-Speed-Prop Airplane Performance

Okay, at this point we assume you've downloaded the spreadsheet, measured your propeller and your fuselage diameter, done your glide tests, written down your airplane's Bootstrap 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 performance numbers 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 by having chosen an appropriate manifold pressure MAP. For my sample airplane, a Cessna R182, I chose 3100 pounds, 8000 feet, 2300 RPM, and 65% power. Put your choices into Part B, recalculate the spreadsheet, and you're ready to get some performance figures.

The upper portion of Part C of the spreadsheet gives you some important speeds, such as the speed for best rate of climb under the power setting you've chosen, and the corresponding rate of climb itself. Our sample airplane could achieve 372 ft/min at 77 KCAS. Now that's not V_y and the airplane's best ROC; for those you'd need to have entered your full-throttle and highest RPM values into Part B. But the 372 ft/min at 77 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 V_M, and minimum level flight speed V_m (when and only when it is greater than stall speed V_S) are where thrust and drag are equal, T = D.
   
• Speed for best angle of climb V_x is where the so-called "excess thrust," T_xs = T-D, is a maximum. The corresponding best angle of climb is found from Equation (2).
   
• Speed for best rate of climb V_y is where so-called "excess power," P_xs = (T-D)V , is a maximum. The corresponding best rate of climb is found from Equation (1).
   
• Speed for best glide V_bg is where drag D is a minimum.
   
• Speed for minimum (gliding) descent rate V_md is where power required P_r = DV is a minimum.

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

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

Figure 3. How some important V-speeds (in calibrated 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. That means we take air speed velocity vector V to be "almost" constant during any one maneuver. Exceptions due to very slow effects of altitude change (while climbing or descending) or very slow effects of fuel burn are considered negligible, so our "point performance" calculations ignore those very slow changes in true airspeed or gross weight. We also assume, (until the next article) that the airplane keeps its wings level. This and the "small flight path angle approximation" imply that lift L always equals weight W. That in turn means that the airplane'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's the same old D = GV² + H/V² (i.e., the sum of parasite and induced drag) that it was in the fixed-pitch case. But our method for calculating thrust T has completely changed.

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

Figure 4. How best rate of climb varies (almost linearly) 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 with numerical examples. Thrust T is 418.3 lb and drag D is 280.6 lb; their difference 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 and multiply this 0.0444 figure by V we have to be careful to use true airspeed in ft/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 = 355 ft/min. For the gliding rate of sink (column H) and glide angle (column I), simply repeat the above calculations with T = 0, reversing signs to switch climb to sink and flight 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 forces on the airframe. The advantages of the bootstrap approach comes from its fairly simple but still quite accurate ways of dealing with propeller thrust, the sticking point, for both fixed-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 it maneuver, 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 pieces of practical work:

1. Run glide tests (to get parasite drag coefficient C_D0 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) information items. Recording the results of your mini-research project in the appropriate cells in bootstp2.xls will quickly get you the performance information you're looking for.

Because constant-speed propeller airplanes have an additional "degree of freedom," partial-throttle performance is accessible right from the start. However this does assume POH cruise performance tables (except for the normally "optimistic" air speeds said to result from given values of altitude, RPM, and MP) 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, but that makes no difference to the power setting. If your airplane is lighter it will go a bit faster, though probably still not as fast as that elusive Teflon-coated beauty on which 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 off on 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 BB in cells B8 and B9. Then empty my sample blade width data out of columns F and G from rows 18 through 50. Print out cell block A13:H50. Take the printout out to the hangar, run a strip 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, the sheet turns calculated decimal inches into whole inches plus so many sixteenths. If your spinner 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 those widths 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 block F18:G50, and recalculate [F9] the spreadsheet. Voila! You have your propeller's TAF in cell F53. Record this figure somewhere in your POH and also in cell B73. General aviation values of BAF are generally between 70 and 140. (I didn't make that TAF transfer 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 you shifted from one airplane to the other.)

Figure A1. Sheet for collecting data to find your propeller's Total Activity Factor.

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

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