[Retric]

Unless I am missing something obvious, lift-induced drag is largely independent of altitude. However, parasitic drag is significantly reduced by lower air pressure. Thus the advantage from high altitude flight.

[mliben]

Less dense air -> higher angle of attack required to produce required lift -> true lift vector is more offset from vertical -> horizontal component of lift is actually producing drag

Like I said in the other comment, if the plane is operating at the range-optimal speed, I think the air density does not impact the range capability (it cancels out) but it does increase the range-optimal speed, allowing for faster travel.

[Retric]

Drag required to make lift is only a subset of total drag.

A car for example doesn’t need to produce lift, but it still displaces air which causes drag. The same is true of an aircrafts fuselage, which is generally not used to generate lift but still increases total drag. https://en.wikipedia.org/wiki/Parasitic_drag

Also, an aircraft is generally designed so that at cruse speed and altitude the wing incidence angle provides appropriate lift. https://en.wikipedia.org/wiki/Angle_of_incidence_(aerodynami.... Which means at optimal curse distance the cabin would almost perfectly level independent of optimal cruse speed or altitude.

[mliben]

Yes, the range-optimal speed is where the parasitic drag is equal to the lift-induced drag.

If you go through the analysis, the air density drops out of the range equation if you assume are operating at the range-optimal speed (which is higher at lower air densities).

[gnramires]

> the range-optimal speed is where the parasitic drag is equal to the lift-induced drag

Not quite true -- range-optimal speed is where the sum of those terms is minimal. With some assumptions, this is where the derivative is 0, dDrag/dv = 0, and since derivative is linear, this means: the range-optimal speed is where the (infinitesimal) increase of parasitic drag (with speed) is equal to the decrease of lift-induced drag (in other words, opposite derivatives).

[mliben]

Using the simple drag polar approach,

D = A*v^2 + B/v^2 (D is total drag, first term is parasitic drag, second term is lift-induced drag)

dD/dv = 0 where v = (B/A)^(1/4)

Plug in v = (B/A)^(1/4)

D = sqrt(AB) + sqrt(AB), aka dD/dv = 0 exactly when parasitic drag is equal to the lift-induced drag

[gnramires]

I stand happily corrected :)

[Retric]

That’s only relevant up until you approach the speed of sound. Passenger aircraft are designed to stay subsonic for a host of very good reasons.

[mliben]

Definitely, for sure. Like I said in the other thread, supersonic is a whole other thing, and I don't think anyone is trying to electrify anything supersonic any time soon :)