What determines the maximum altitude a plane can reach?

Question

What factors determine the maximum altitude for a plane?

Is it limited by wing design, engine thrust, and so on?

Is there a formula by which one can calculate the maximum altitude a plane can reach?

Answer 1

The higher you get, the lower the density of the air becomes. This lower density results in a lower lift being generated for the same airspeed and angle of attack. Effectively, the higher you fly the higher your minimum speed becomes. So while climbing, your speed needs to increase to compensate for the lower air density. As long a you can fly faster, the lower density at altitude can be compensated for.

Basically there are two things that limit your maximum speed: thrust and speed of sound and with that your maximum altitude.

First is thrust; the higher you get, the lower the thrust your engines deliver. You might note that drag goes down with the air density as well but since you are flying faster and faster during the climb the drag doesn't decrease at all. If your maximum altitude is limited by thrust then at some point during the climb the thrust and drag are getting close to equal and that is where the climb stops. When you can no longer climb with more than 100ft per minute (for propeller aircraft) or 500ft per minute (for jet / turbofan aircraft) you have reached your service ceiling. If the aircraft maximum altitude is determined by thrust, the absolute ceiling will take very long to reach.

At high altitudes air breathing engines will get difficulties eventually. Due to the lower air density the mass flow through the engine is reduced up to a point where it causes a flame out.

The other limitation is the speed of sound, at least for subsonic aircraft. In the process of generating lift, air flowing over the top of the wing is accelerated. At one point, when the aircraft is still flying below the speed of sound, shock waves will start to form over the wing. This results in increase of drag and reduces the lift. So provided you have enough engine power at your disposal you can climb to an altitude where your minimum speed is also your maximum speed. This is called the coffin corner. In the coffin corner:

• if you fly any faster, you will exceed the maximum Mach number ($M_{mo}$) of your aircraft, resulting in high speed buffet, vibrations and possible loss of control.
• if you fly any slower, the maximum lift that the wing can provide will be insufficient to maintain altitude. Descent or the aircraft will stall.
• if you fly any higher and you will be too fast and too slow at the same time.
• if you turn, you increase the wing loading, thereby increasing the minimum speed needed to create the required lift. Also the outer wing will easily exceed the maximum speed while at the same time the inner wing is below stall speed. This can quickly develop into a spin.

Since accurate knowledge of engine performance, drag and wing characteristics of the aircraft is needed, there is not a simple formula to derive the maximum altitude for an aircraft.

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Apart from the aircraft performance related limitations above there is a certified maximum operating altitude for the pressurized cabin. This takes into account the structural properties of the hull (pressure difference between inside and outside) and the achievable emergency descent rate in case of a depressurization event.

Answer 2

The maximum altitude is limited by a number of factors, and the one which counts depends on the particular aircraft. These are:

1. Engine power output. Airbreathing engines produce less power the higher they operate due to decreasing density with altitude. In reciprocating engines, this can be overcome with turbocharging, and dedicated high-altitude piston engines use triple-stage turbochargers with intercoolers. In dedicated high-altitude designs, the engine is the smallest part of the propulsion package, most is cooling and ducting. The propeller has to be matched to the low density at high altitude, increasing in diameter for operation in low density air.

2. Combustion chamber pressure: The altitude limit of jet engines is mostly determined by the pressure ratio of the intake and the compressor. If this pressure drops below the minimum for sustained combustion, the engine will flame out. Since jet engines are in principle a big turbocharger where the piston engine has been replaced by a combustion chamber, this combustion chamber becomes the weak link.

3. Wing loading: The lower the wing loading, the lower air density can become before a wing will fail to produce enough lift. If the engines produce enough power for sustained flight (electric propulsion with solar panels, for example), the limit becomes the structural integrity of the light structure. See this answer for an applied example.

4. Maximum flight Mach number: For supersonic aircraft, the limit is given by a combination of wing loading and maximum speed. The faster the aircraft can fly, the lower air density can be. In most cases, the speed limit is given by intake efficiency, because intakes need to be optimized for their flight Mach number, and thermal limits due to airframe heating. Note that a fast aircraft with lift reserves can perform a pull-up at altitude, converting kinetic energy into potential energy (aka altitude), so the instationary maximum altitude could be several 1000 m above the stationary altitude limit.

5. Aerodynamic efficiency: This is the only factor where I can give you a simple equation, and it is determined by the aerodynamic quality of the wing and its airfoil. It applies to subsonic flight where an increase above a critical flight Mach number will reduce lift. Expressed as the minimum air density $\rho_{min}$, this is $$\rho_{min} = \frac{2\cdot m\cdot g}{(Mach^2 \cdot c_L)_{max}\cdot a^2\cdot S}$$

Here we find again wing loading $\frac{m}{S}$ as a factor, but also the maximum of the product of the square of the flight Mach number $Ma^2$ and the lift coefficient $c_L$. $a$ is the speed of sound. A good value of $Mach^2 \cdot c_L$ is 0.4, and it needs supercritical airfoils to be achieved. Use this number for modern designs and you will get a pretty accurate answer if engine thrust is sufficient. For older designs, values between 0.3 and 0.35 are a better fit. Very early designs with poor aerodynamics like the Westland Welkin would only achieve a $Mach^2 \cdot c_L$ of below 0.2.

Answer 3

In the most basic form, the aircraft's max altitude is the point where thrust required is equal to thrust available. This compares the thrust required to maintain airspeed and altitude to the thrust available from the engines. Since air-breathing engines will tend to produce less thrust as altitude increases, this means that the thrust available decreases with altitude. At some point, the aircraft will be at the lowest drag possible in level flight, and using all thrust available.

As casey points out, there will be many other factors, such as the ability of the aircraft to remain pressurized, the ability of the engine to maintain a certain thrust level, and atmospheric conditions.

However, if you are looking for the absolute max altitude it's able to reach, but not necessarily maintain, it becomes much more complicated. This would be determined by the maximum amount of energy an aircraft can attain, both in altitude and airspeed. An aircraft may be able to dive or remain at a lower altitude to gain speed, and then climb to trade that speed for altitude, reaching a higher altitude than it can maintain (see this incident for an example of an aircraft that flew to a higher altitude than it could actually maintain).

Answer 4

The absolute maximum altitude a plane can reach is only limited by the lift it can produce. This will be a function of the wing (and one of our resident engineers can explain this) and the airflow over the wing. The airflow in turn is a function of your altitude (air density) and airspeed. Airspeed in turn is a function of your thrust, drag, etc. In short, the lift you can produce is dependent on a lot of things indirectly and this will define the physical limit of max altitude.

Note that the max altitude defined by your lift is a max continuous altitude. If you have the momentum available you could use that to climb above this altitude for brief excursions, but you would not be able to maintain altitudes above this limit.

Note that this altitude is not the airplanes service ceiling, which is going to be lower due to climb rate thresholds (e.g. 100 fpm) or certification issues (e.g. 25,000 feet for pressurization/oxygen requirements).

Answer 5

An engineless aircraft is not subject to two of the five limiting factors in Peter Kämpf's comprehensive answer. The current subsonic level-flight altitude record is held by the Perlan II glider which reached 76,124 feet in September 2018, exceeding the U2's record of 73,737 feet. If Perlan II reaches its design altitude limit of 90,000 feet it will exceed the SR-71's (supersonic) level-flight altitude record of 85,068 feet.

Perlan II, while highly specialised and having a pressurised fuselage, is not dramatically different in appearance from an open-class glider. The most significant difference is in the airfoil, which is optimised for flight at 60,000 feet. This also results in a significantly wider range of airspeed at extreme altitudes (the 'coffin corner' referred to in other answers) than the U-2, which at operational altitude had only a 5-knot flyable airspeed range.