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Re: [Phys-L] Aeroplanes and air temperature



On 06/21/2017 09:50 AM, antti.j.savinainen via Phys-l wrote:

there has been recent news on aeroplanes which cannot fly because the
air temperature is too high. Here is one explanation why this is so:
https://www.wired.com/story/phoenix-flights-canceled-heat/

How do you find the explanation?

Nonsense. Layers upon layers of nonsense. It is a just-so story,
insofar as it is a ludicrous explanation of an observed fact, but
unlike Kipling it is not well-written, entertaining, or allegorical.

The diagram of how a wing produces lift is entirely wrong and almost
entirely irrelevant. The actual physics is explained here:
https://www.av8n.com/how/htm/airfoils.html

It's true that density is related to temperature, and lift is related
to density, and takeoff requirements are related to lift. However,
if that were the whole story, the dependence would be rather weak,
for reasons that can be figured out using high-school physics:

The basic equations for landing are:

P = ρ kT (ideal gas law)

L = ½ ρ v^2 C_L

W = L (you have to support the weight)

F_B = μ W (friction)

where
P = ambient atmospheric pressure
T = absolute temperature
ρ = air density

L = lift
v = true airspeed
C_L = coefficient of lift, dimensionless, typically 1.5 or less,
unaffected by density and airspeed over a wide range

W = weight of the airplane, assumed constant for present purposes

F_B = braking force
μ = coefficient of friction (quasi-static rolling friction)
assumed constant for present purposes

So, when the temperature goes up 6 °C, that's a 2% increase in absolute
temperature. That makes a 2% reduction in density. To support the
weight of the airplane requires 1% more true airspeed. This requires
the force to act for a 1% longer time (to get rid of all the momentum)
and therefore requires a 2% longer runway. Scaling laws!

Under normal conditions, a Bombardier CRJ can land on a runway that
is 6000 feet long, maybe less:
https://en.wikipedia.org/wiki/Bombardier_CRJ700_series

The long runway at Sky Harbor is 11,489 feet long. The second runway
is 10,300 feet long: http://airnav.com/airport/KPHX
So you tell me, how hot does it have to get before an airplane can't
produce enough lift to land there?

There are several steps in this calculation, but each step is trivial.
I tell students I'm not superman, and you're not either. I'm not going
to tell you how to leap tall buildings in a single bound. But I can
show you where the stairwell is.

==================================

Now, if you want to know what the real limitation is, you have to look
at the engines.

First of all, engine power is strongly dependent on density (even at
constant temperature). You have to push X moles of fluid through a
series of restrictions. That's a problem when the density is low.

What's worse, there is an additional direct dependence on temperature
(even at constant density). This is highly nonlinear. It has to do
with overheating the internal components of the engine. So you have
to de-rate the engine at high temperatures.

What's worse is that the runway length required for takeoff is a
highly nonlinear function of engine power. In fact there is a
singularity. There is a denominator that goes to zero. That's
because the engine first has to produce enough thrust to overcome
drag; then the leftover thrust, if any, can be used for acceleration.

To say the same thing another way, if you de-rate the engines enough,
the aircraft can reach what might be called terminal velocity,
rolling down the runway but not accelerating, even with the engines
producing as much power as they can. Takeoff would require more
than infinite length.

You will *not* be able to figure out the details using high-school
physics or even grad-school physics. Takeoff performance is very
very complicated. In practice it is approximated by detailed
simulations, and then determined authoritatively by experiment.

Now we can state what's probably the real problem: I haven't
looked at the CRJ Airplane Flight Manual, but it seems highly
plausible that nobody ever did the experiments to determine the
takeoff performance at high temperatures, so the tabulated data
in the AFM goes up to something like 45 °C and stops. It would
be illegal to extrapolate ... and also very tricky to extrapolate
correctly, because you are dealing with a singular function.

I reckon it would cost less than a million bucks to do the required
test flights and file the paperwork to lift the restrictions. It
wouldn't surprise me if they were working on it right now.