Propeller Aircraft Performance and The Bootstrap Approach: Performance Basics

Propeller Aircraft Performance and The Bootstrap Approach

Performance Basics

What is aircraft performance?

The first question you may have is: "Just what is aircraft performance?" Performance questions include:

How fast can the airplane go?
How high can it go?
How much (weight or perhaps volume) can it carry?
How far can it go without refueling?
How steep (or perhaps how quickly) can the airplane climb?

Aircraft performance is a big subject. To make good headway, we need to pare it down. And it’s a complicated subject. For example, answers to all the above questions depend on how heavy the airplane is. Answers to most of them also depend on how high the airplane is. If one includes in "performance" acrobatic performance (usually called ‘aerobatic’), matters really get out of hand. Add to the confusion that there are land airplanes and seaplanes, propeller-driven airplanes and jets and turbo-props, monoplanes and biplanes, light airplanes and heavies, as well as multi-engine and single engine airplanes! Lots of variety even if we exclude helicopters and rockets.

Even the light single-engine land propeller-driven airplanes we consider in this article – Pipers and Cessnas and similar trainers or recreational airplanes – can have either fixed-pitch propellers or constant-speed propellers. Fixed-pitch propellers are one solid chunk of wood or metal or plastic formed to look like a propeller. Constant-speed propellers maintain the RPM value set by the pilot; that type has blades which automatically turn in their geared hubs. There is an RPM governor driven by engine oil pressure that maintains or "governs" the propeller speed. In this article, we will stick to the fixed-pitch propellers. And mostly we’ll discuss optimal, or "best" performance, for the airplane either gliding (zero throttle) or at full throttle. As you can see, aggressive pruning of the subject has cut the performance problem to manageable proportions.

Using the author’s Bootstrap Approach, we will be able to pick an airplane and demonstrate precisely how the various optimum-performance air speeds (V speeds) depend on the airplane’s weight and altitude. We will be able to learn just how fast and/or steeply a given airplane climbs (or descends) at a given weight, altitude, and air speed. We will do all this with formulas which are at the level of high school algebra.

First, we’ll delve into the performance question with concrete examples from a Pilots Operating Handbook (POH). These Handbooks (or Approved Flight Manuals) come with each new airplane. Using a Cessna 172 as our example, here are extracts of that performance information.

Performance Numbers from the Cessna 172 POH

Takeoff Performance. Minimum distance needed for the airplane to lift off the runway, and distance needed to get to a fifty foot altitude, is given for Cessna 172s with flaps ten degrees (the recommended setting) taking off from a flat dry paved runway in calm wind. Three different aircraft weights (2400 lbf, 2200 lbf, 2000 lbf) are featured and performance numbers for a large variety of so-called "density altitudes" are given. (Aircraft performance depends much more on air density than on air pressure or air temperature, so density altitude h_r– the altitude in the International Standard Atmosphere with the here-and-now value of air density – is the correct atmospheric variable to use). Some guidance on adjustments for wind and for dry grass runways is given in the POH, but what about sloped runways? Or what about various combinations? When is it best to take off uphill into the wind rather than downhill with the wind? The POH is silent on all those matters. Mr. Olson’s article very well shows you how to figure takeoff performance. It’s a detailed study because so many different forces are at work.

Pressure Altitude, ft	0° C d_LO, ft	0° C d₅₀, ft	20° C d_LO, ft	20° C d₅₀, ft
0	795	1460	925	1685
2000	960	1770	1115	2060
4000	1165	2185	1355	2570
6000	1425	2755	1665	3300
8000	1755	3615	2060	4480

Table 1. Selected POH take-off performance entries for the Cessna 172 at 2400 pounds.

Climb Performance. The airplane’s maximum climb rate, and the speed V_y needed to achieve that best rate of climb, are given for an airplane weighing 2400 pounds at various density altitudes. There are a few scattered references to speed for best angle of climb, V_x, but nothing on what that angle is. And what about for other aircraft weights W? Once we hit our stride we will be able to do much better than the POH in providing more complete performance information.

Pressure Altitude, ft	Speed for best ROC V_y, KIAS	0° C ROC, ft/min	20° C ROC, ft/min
0	76	745	685
2000	75	640	580
4000	74	535	480
6000	73	430	375
8000	72	330	275
10,000	71	225	175
12,000	70	125	NA

Table 2. Selected POH best climb performance entries for the Cessna 172 at 2400 pounds.

Cruise Performance. For an airplane weighing 2400 pounds, a table cites true air speed V (somewhat optimistically, in most cases), proportion of "rated" full-throttle horsepower (160 HP in this case), and gallons per hour (GPH) fuel consumption rate, for various RPMs. What about for weights other than 2400 lbf? No mention is found in the POH. And no indication of what the speed for best (longest) range V_br or speed for best (longest) endurance V_be might possibly be. Since the logic behind the Bootstrap theory of partial-throttle performance is so tortuous, we won’t be discussing cruise performance here. But it can be done.

Pressure Altitude, ft	RPM	% BHP	KTAS	GPH
2000	2500	76	114	8.5
	2400	69	109	7.7
	2300	72	103	6.9
	2200	55	97	6.3
	2100	50	91	5.8
4000	2550	76	117	8.5
	2500	73	114	8.1
	2400	65	108	7.3
	2300	59	102	6.6
	2200	54	96	6.1
	2100	48	89	5.7

Table 3. Selected POH cruise performance entries for the Cessna 172 at 2400 pounds.

Glide Performance. In the POH section called Amplified Emergency Procedures a diagram is given which shows that the speed for best (longest) glide (in calm air) V_bg is 65 KIAS (knots indicated air speed, before correcting for air speed indicator calibration errors) when the airplane weighs 2400 lbf, has flaps up, with no wind. The diagram also implies that the best glide angle is at a ratio of about 9.1 horizontal : 1 vertical. But what about best glide at other weights or with other flaps settings? And what about effects of headwinds or tailwinds?
Landing Performance. Minimum landing roll, and total distance needed to go from altitude 50 feet to a full stop, is given for a level dry paved runway for an airplane weighing 2400 lbf landing with full flaps. There is some guidance given for landing flaps up, with wind, or on dry grass runways. Again, as for takeoff performance, no comprehensive data. And again Mr. Olson’s essay is a good source for leading you through detailed calculations. We won’t go over that same ground.

So the pilot who cares about safe operation or about "getting the best possible performance out of his (or her) airplane," say, in climbing out of a mountain strip towards a rocky ridge (V_x needed!) or who cares about stretching fuel, must learn to figure these performance numbers for himself (or herself). We won’t have time to tackle every aspect of the performance problem, but we will go through the basics:

climb performance at full throttle;

maximum speed in level flight V_M;

glide performance with the engine turned off; and

similar maneuvers with the airplane turning, wings banked to angle f .

Also calculations of thrust T and of drag D under any situation with "operational" or "pilot-chosen" values of weight W, density altitude h_r , and bank angle f .

Go to next section- Bootstrap Approach: Background

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.

Funded in part by

Updated: April 11, 2008


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