There are a couple of formulas for normally aspirated engines (the RoC formula works for all engines), to calculate (I) the excess thrust needed for RoC, and (II) HP loss from density altitude.
The HP loss is equal to DA x .03 x Sea level effective HP (delivered at the prop).
RoC in fpm is excess HP x 33,000 divided by mass in lbs.
For typical GA SEP the second formula will suggest that SL excess thrust is around 30-40% of Rated HP. The HP loss formula suggests that this excess thrust will disappear around 12,000-13,000 DA. However, typical service ceilings are around 14,000-16,000, and some NA types will have service ceilings above 18,000. e.g. Cessna 180 at 20,000’.
Is there an aerodynamic (prop or aerofoil) efficiency gain with altitude that accounts for this higher service ceiling? Best rate of climb converges on best angle of climb speed as you approach service ceiling, and obviously TAS is higher.
Getting a layman’s explanation of this service ceiling dividend would be great.
…and now in English, please? (just an overload of acronyms for this simpleton)
RoC ==>?
HP ==?
DA ==> (could only think of decision altitude but that’s an IFR term )
SL ==> ?
NA ==>?
RoC is rate of climb
HP is horse power
DA is density altitude
SL is sea level
NA is normally aspirated
GA is general aviation
SEP is single engine piston
TAS is true air speed
“Is there an aerodynamic (prop or aerofoil) efficiency gain with altitude that accounts for this higher service ceiling? "
No, the explanation is simpler. Higher ceilings can be achieved by either better aerodynamics, more power or lower weight. All three will also increase what you call excess thrust (it is really excess power) at Sea Level. So 40% excess power (available for climb) is definitely not a maximum.
The formula for HP loss is fairly accurate for all non-turbo piston engines. However, the percentage of excess power at sea level varies quite a bit between aircraft types/weights. That is the explanation for different ceilings.
Huv thank you. Agree (thrust minus drag)/weight.
If we take a Cessna 180K with a MGW of 2900 lbs (I took 50lbs off for climb fuel), and a sea level best rate of climb of 1,060 fpm – the excess thrust formula shows 93 hp which is 40% of rated power. This implies you need the remaining 60% to maintain height at Vy. Which sounds too high, as a 180 can loiter at 100mph on 35% power, or conversely the increase in drag to slow to Vy is high, which is unlikely.
The service ceiling (assume 50 fpm) is 17,700 feet. The HP density altitude formula suggests that at 17,700 a normally aspirated engine would produce around 47% of sea level power. To get 50 fpm with a mass of 2900lbs you need 4 hp of excess thrust, but according to the sea level calculation we have a power deficit (47% vs 60%).
With altitude Vx increases, and Vy reduces, converging at the service ceiling. TAS however is increasing, so there may be an aerodynamic efficiency gain as you climb higher? Climb angle reducing, but angle of attack increasing slightly with a lower drag penalty?
A normally aspirated 125 hp Super Cub, not exactly the most optimised aerodynamics, held the small aircraft altitude record at 32,000 feet. Not sure if it still stands, but this defies the formula. It was light and a woman pilot.
One thing to bear in mind is the possible difference between Service Ceiling, which I think is the altitude above which the aircraft can not maintain a 300 FPM rate of climb, and a limitation in the POH.
On my SR 22, the POH limitation is 17,500 ft but the aircraft is clearly capable of climbing higher.
It’s much less than 300fpm. I think it is 50fpm for CAA and 100fpm for FAA, hence my TB20 has an N-reg ceiling of 18k and a G-reg ceiling of 20k 
If I was doing 300fpm at 20k I would have a ceiling of probably around 25k, which would be rather nice!
Also the ceiling is rarely a legal limitation. It is in some cases (maybe the SR22?) but not in the TB20, or anything else I have been familiar with.
Turbocharged airplanes tend to have a legal ceiling (maximum operating altitude). My aircraft (TR182) was flight tested until FL240 and still climbs at around 300fpm up there but in order to not cannibalize T210 sales, Cessna added an arbitrary FL200 limit right before certification. It runs fine at FL240 and I can go there when needed. However, one would try to avoid that in a non pressurized aircraft for safety reasons.
Does the TR182 have pressurized magnetos?
No but rather big magnetos. D3000. Flight tested until FL240. I am not a fan of pressurized magnetos, they suck humidity in IMC and generally suffer from contamination. They are quite rare.
