Propeller Aircraft Performance and The Bootstrap Approach: Performance Basics

To Non-Java ALLSTAR Network Website

                                                                                                                                                                        JAVA-capable browser required for graphic-based menus (Exploer 3.0 or Netscape 2.0 or greater)

Please let me remind all of you--this material is copyrighted.  Though partially funded by NASA, it is still a private site.  Therefore, before using our materials in any form, electronic or otherwise, you need to ask permission.
There are two ways to browse the site: (1) use the search button above to find specific materials using keywords; or,
(2) go to specific headings like history, principles or careers at specific levels above and click on the button.
Teachers may go directly to the Teachers' Guide from the For Teachers button above or site browse as in (1) and (2).

FAQnewred.gif (906 bytes)           


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:

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

  1. 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 hr – 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.
  2.  

    Pressure

    Altitude, ft

    0 C

    dLO, ft

    0 C

    d50, ft

    20 C

    dLO, 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.

     

  3. Climb Performance. The airplane’s maximum climb rate, and the speed Vy 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, Vx, 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.
  4.  

    Pressure

    Altitude, ft

    Speed for best ROC

    Vy, 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.

     

  5. 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 Vbr or speed for best (longest) endurance Vbe 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.

  1. 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) Vbg 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?
  2. 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 (Vx 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:

    1. climb performance at full throttle;
    2. maximum speed in level flight VM;
    3. glide performance with the engine turned off; and
    4. similar maneuvers with the airplane turning, wings banked to angle f .
    5. Also calculations of thrust T and of drag D under any situation with "operational" or "pilot-chosen" values of weight W, density altitude hr , 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.


Send all comments to allstar@fiu.edu
1995-2017 ALLSTAR Network. All rights reserved worldwide.

Funded in part by

Updated: April 11, 2008