Flight Mechanics - Maximum Cruise Altitude

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So, I was helping someone with some science fiction stuff and got to poking into flight mechanics.

So we've got a balance of lift and weight, and drag and thrust.

The speed at which thrust and drag cancel determines the speed of flight. The speed of flight determines lift, and lift must cancel weight for a steady cruise.

But looking up the formulas for drag and lift, then using a constant thrust and weight, and requiring a steady-state so that the forces cancel, I get this:

[itex]F_{weight} = K_{L} v^{2} \rho[/itex]
[itex]F_{thrust} = K_{d} v^{2} \rho[/itex]

Where [itex]K_{L}[/itex] is all the coefficients in the lift equation, and [itex]K_{d}[/itex] is all the coefficients in the drag equation.

I assumed that for cruise, I'd find some solutions to uniquely solve both of these simultaneously for a given speed or a given altitude. So I was expecting to solve one of these for velocity or air density (air density is 1-to-1 with altitude, so I use the terms interchangably here since increasing air density is always lower altitude), substitute it in to the other, and be able to solve for a velocity given an air density, or an air density given a velocity.

But since they have the same form, that's not possible.

So I took another approach - I solved each equation for one of the two variables (air density or velocity)

[itex]\rho = \frac{F_{weight}}{K_{L} v^{2}}[/itex]

[itex]\rho = \frac{F_{thrust}}{K_{d} v^{2}}[/itex]

It's kind of what I thought earlier, but twice - for each velocity, there's a certain air density that balances lift and weight, and for each velocity, there's a certain air density that balances drag and thrust.

So I thought perhaps cruise must be where these intersect. But, again, still not quite - these two either have no intersects or infinite intersects, depending on the values of the parameters, since they have the same form.

So what does this mean? Well, if there's no solutions (values of [itex]\rho[/itex] and [itex]v[/itex] that make both equations true), then the plane cannot cruise at any altitude. If there's a solution, the plane can cruise at that altitude. If there's infinite solutions, then the plane can cruise at any altitude.

So in some way it makes sense - to vary the altitude, you vary the thrust so that you enter that state of having infinite solutions, so no matter where you are, you can cruise there. But the big flaw here is the simplifications imply that if this plane can fly at one altitude, it can fly at any.

So now I'm wondering what determines the maximum cruising altitude.

My thought was that it just broke down with the thrust - because this assumes thrust is constant, when in reality propeller or jet engines perform differently at different mach numbers. So far, that's still my understanding of it.

But there's two issues with this, even -
1) from what I've read, jet engines produce more thrust at higher speeds. Does this cap out at a certain point? Because if it didn't, then by those flight equations, there'd be no max speed. I'm guessing this caps out at a certain point, though, limited by the design of the engine. For turbofans on airliners, they can't cruise supersonic. For jet fighters, maybe mach 2 or so. For the SR-71/A-21, this would be mach 3.3+ ?

2) I saw a news story recently where two pilots forgot to raise the landing gear. This resulted in the plane struggling to climb as high. This doesn't effect the thrust, so why did it limit the altitude? Yes, it results in more drag, of course, but why does more drag keep the aircraft from climbing as high?

I'm tempted to simply answer, "more drag, slower speed, less lift", but that doesn't quite sit with the flight equations above.

The flight equations say if I can fly at one altitude, I can fly at any altitude, just needing higher speed at higher altitudes - until I get so high that the speed necessary to maintain flight is speeds where the engine cannot operate (due to fluid dynamics at higher mach numbers in the engine).

But the aircraft couldn't climb any more and was only at 230 knots. So even that explanation I thought I finally had figured out, fails somewhere, since even with a max of 230 knots, if it can climb at all, it should be able to climb to where the air is thinner, thus less drag, thus higher speed, thus more lift and keep climbing until that higher speed reaches mach numbers the turbofan can't exceed (>0.93 or so).

But that's not what happened. Could that have happened, though, it's just the climb was very slow? Or am I missing some important details here? What determines the maximum altitude an aircraft can fly at in these terms?

Another effect worth considering is the convergence of things like speed of sound, velocity not to exceed (VNE), and stall speed. As you go up in altitude the speed of sound comes down, and the stall speed goes up to meet the speed of sound and the VNE. This a limiting factor because at the limit of an aircraft's operational altitude you cannot even turn the aircraft without stalling the inside wing.



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The speed at which thrust and drag cancel determines the speed of flight. The speed of flight determines lift, and lift must cancel weight for a steady cruise.
If you can vary the angle of attack there is a wide range of speeds at which lift = weight.


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What determines the maximum altitude a plane can reach..

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 (Mmo'>M mo Mmo ) 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.
Oh, so they have less thrust at higher speed. I guess, yeah, I'd heard jet engines have more thrust at higher speeds, but I guess maybe there's a peak and it falls off again?

Fantastic reply, though. I'm surprised, I did a lot of searching and didn't find that. Must've slipped by somehow. Thanks! This is an excellent reply!

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