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The Canadian Flight Training Manual stated this when talking about maximum endurance:

For reciprocating engines, maximum endurance is achieved at sea level.

The book does not explain further as to why. Air density is my first guess, when the engine gets a better mixture of air to burn with its fuel, and so we get the most power for the same amount of fuel.

But if that is true then it sort of contradicts why maximum range is at higher altitude.

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    $\begingroup$ Endurance is about fly for longer, not faster. At sea level you can also fly slower in order to stay afloat, which decreases the engine power needed already. $\endgroup$
    – user3528438
    Jan 25, 2021 at 7:30
  • $\begingroup$ At lower power settings the carburetor flap limits air intake. So at higher altitude one only need to open the flap more (with throttle) and lean to prevent too much fuel from being used. There is plenty of air at max endurance power settings even if you go up a few thousand feet. $\endgroup$
    – Robert DiGiovanni
    Jan 26, 2021 at 2:20
  • $\begingroup$ But you have to burn more fuel for the same thrust as you go up, and its a trade off between increased fuel consumption and benefits of increasing IAS TAS spread. Jets really shine at higher altitudes because their thrust does not drop off with increasing TAS as much as props do. $\endgroup$
    – Robert DiGiovanni
    Jan 26, 2021 at 2:34

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Try working it from the other end. A propeller needs to generate a certain amount of thrust to maintain a given indicated airspeed to fly.

In thinner air, it must turn faster to do so.

Turning faster is higher RPMs. Higher RPMs mean more engine friction.

More fuel must be burned per unit time to overcome engine friction at higher RPMs.

RPMs to generate a given amount of thrust are lowest at sea level (maybe lower in Death Valley (on a colder day)).

Because the horsepower requirement for maximum endurance speed is much lower than the maximum rated output, the horsepower loss due to thinner air is not an issue until one is higher up. This is why "leaning" is required at higher altitudes.

But "air density" is the correct answer.

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    $\begingroup$ What if I told you it's better for max endurance to fly low, even if there were no engine friction? $\endgroup$
    – Sanchises
    Jan 25, 2021 at 16:21
  • $\begingroup$ @Sanchises I'd be interested, although flying in ground effect may be dangerous. Aerodynamicly, IAS would produce similar drag regardless of altitude. At max endurance, throttle could be leaned or enriched as needed. It could make enough power to turn the prop. There is sufficient air to burn the fuel at say, 5000 feet, because the engine does not need to develop full power. The prop only needs to spin faster at its IAS to develop the thrust. But I am willing to hear additional info if there is more. $\endgroup$
    – Robert DiGiovanni
    Jan 25, 2021 at 23:07
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Fuel flow is largely proportional to engine power, which in turn is largely proportional to power required (there are some efficiencies involved but that depends on the exact combination of engine and prop - we're interested in the general rule here).

Theoretical power required is TAS x drag (not IAS!). Drag is determined by IAS however. So maximum endurance is reached by finding an optimum of the product IAS x drag (VBE), and then making sure TAS is as small as possible for this IAS, which is at sea level.

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  • $\begingroup$ TASxDrag(or Thrust) is the energy state of the aircraft (per unit time). Required engine output thrust at a given IAS is constant. The efficiency (fuel consumption per unit thrust/time) is affected by internal engine friction losses (and prop AOA). But your statement that TAS should be as small is possible is +. $\endgroup$
    – Robert DiGiovanni
    Jan 25, 2021 at 15:06
  • $\begingroup$ @Robert I purposefully chose not to include engine and prop efficiency, because those can be significantly reduced (or exacerbated) by designing for a particular flying regime. The things I mentioned are fundamental to flying with a reciprocating engine in Earth's atmosphere. $\endgroup$
    – Sanchises
    Jan 25, 2021 at 16:18
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The following paper gets into the details. Optimum Range and Endurance of a Piston Propeller Aircraft with Cambered Wing

Figure four skips the math and provides a graph.

More than one flight over water has taken advantage of the ground effect when lacking fuel or power, which is an analog of the maximum endurance problem.

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    $\begingroup$ The article is behind a paywall, and I cannot see the images either. Can you provide a similar image from the open domain? $\endgroup$
    – ROIMaison
    Jan 25, 2021 at 13:51
  • $\begingroup$ Oh, sorry, I thought it was publicly available at no cost. Give me a day or so and I will get something publicly assessable, or I will excerpt from the document. Once again, I did not mean for things to be obscured. $\endgroup$
    – mongo
    Jan 25, 2021 at 20:53
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    $\begingroup$ Here's one which addresses the endurance calculations, but it lacks a nice graph. The objective is to stay aloft with minimum power, which happens at low density altitudes. I will keep looking for a better reference. nptel.ac.in/content/storage2/courses/101104007/Module2/Lec8.pdf $\endgroup$
    – mongo
    Jan 25, 2021 at 21:00
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    $\begingroup$ Here is a nicer explanation, but still lacks the clear graph. But it deals with the aerodynamics pretty clearly. dept.aoe.vt.edu/~lutze/AOE3104/range&endurance.pdf $\endgroup$
    – mongo
    Jan 25, 2021 at 21:03
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    $\begingroup$ OK sorry about all the references, but none of these are perfect. However in Aerodynamics for Naval Aviators, look at Page 171, and note the bottom figure. I think the interpretation of that is pretty intuitive for most pilots. You can see the book here, for free: faa.gov/regulations_policies/handbooks_manuals/aviation/media/… $\endgroup$
    – mongo
    Jan 25, 2021 at 21:17

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