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I was having a discussion with a friend who's going for a record speed over distance attempt. We were discussing which altitude will have the fastest TAS, and I suggested it would be the highest altitude the plane can fly at.

I now realize this is incorrect, as shown by many charts such as this one:

enter image description here

The above chart shows for a give power output TAS increase linearly with altitude, but power decreases exponentially with altitude. What I don't understand is why.

As air density goes down, there is simultaneously less oxygen and less friction. I would have expected those to to act linearly and oppositely, i.e. TAS increases by as much as the percentage of oxygen-- and thus power output-- decreases.

However, clearly some other factor is at play, whereby the available power decreases faster than the density.

What is it? I'm thinking it might have something to do with inefficiencies in spinning blades-- propellers and turbines alike-- and pumping engines, but I don't have a convincing hypothesis which ties together observation and expectation.

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  • $\begingroup$ Is it a turbojet, a turboprop or a propeller airplane? $\endgroup$
    – sophit
    Commented Jan 9, 2023 at 18:05

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The power that can be produced by a normally aspirated piston engine reduces with altitude -- as you note, the air gets thinner. This is typically modeled as:

$\frac{HP}{HP_0}=1.132 \sigma - 0.132$

Where $\sigma=\frac{\rho}{\rho_{SL}}$ is the ratio of the density at altitude to the density at sea level. Similarly, $HP_0$ is the power available at sea level.

When aircraft use forced induction, we usually use it to compensate for this lapse (not to boost sea level power). This is called turbo normalization. A turbo normalized aircraft will maintain sea level power up to some critical altitude and then the available power will begin to lapse.

The aerodynamic forces on the aircraft also vary with altitude. Take lift for example...

$L=C_L q S$

Where $C_L$ is the lift coefficient, $q$ is the dynamic pressure, and $S$ is a reference area.

The dynamic pressure $q=0.5 \rho V^2$. So, at fixed lift coefficient and velocity, the lift would drop off with altitude as $\rho$ drops.

Of course, lift must equal weight, so as altitude increases, we must either increase $C_L$ or increase speed $V$.

To fly at a constant $C_L$, we increase true air speed (TAS). We call this constant equivalent airspeed (EAS). Equal EAS will also be equal dynamic pressure ($q$).

Other aerodynamic forces behave similarly with altitude -- if we fly at constant $C_L$, then we also fly at approximately constant drag coefficient $C_D$ (ignoring Mach and Reynolds number effects). So $D=C_D q S$ describes the drag force.

Since we need $L=W$ and if we maintain constant $q$ and constant $C_L$ with altitude, then we will also maintain constant drag coefficient $C_D$, drag force $D$, and even lift-to-drag ratio $L/D$ with altitude. We will fly at constant EAS, but increasing TAS.

However, the power absorbed by the air due to drag is the product of drag and velocity (true) $P=D V$. So, it takes increasing power to maintain flight at constant $C_L$ and constant EAS with altitude.

Unfortunately, as discussed previously, our normally aspirated piston engine's ability to produce power decreases with altitude.

Depending on how these factors balance, an aircraft's maximum speed may first increase with altitude and then decrease -- or it may be maximum at sea level and decrease with altitude.

Of course, we usually don't fly at maximum speed for extended periods of time. Instead, we fly at a speed for best range (or slightly faster than that). The speed for best range does not care about the lapse in available power (until we reach the altitude where we have insufficient power).

Because of this, the speed (TAS) that a piston prop aircraft operates at will usually increase with altitude.

Some of the above details are affected by changes in propeller efficiency -- and how that varies depends on whether you are operating a fixed pitch or constant speed propeller. In either case, the above factors will determine a lot of what is going on.

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  • $\begingroup$ Perfectly sensible. For a given power output, TAS goes up with the cubic root of density. And, IIUC, the subtle nuance at 100% throttle is that frequently as altitude goes up, necessitating an increasing AoA because EAS is slowing down, $Cl$ goes up sufficiently more quickly than $Cd$ such that the drag falls faster than the power output decreases. (Whew, that's a mouthful!) Did I get that right? $\endgroup$ Commented Jan 9, 2023 at 23:51

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