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Does the fact that air molecules move slower in cold weather and faster in hot weather also affect the aircraft's performance or no? So if its cold there is a higher chance or air molecules hitting the wing and engines because the air molecules are compacted and slow moving and in hot weather they are spread out and faster moving, so less of a chance of air molecules hitting the wings and going into the engine. So in cold weather the molecules move slow and in hot weather they move faster, does this also effect aircraft performance. I know that the fact that in cold air the air is more compacted and that increases aircraft performance but does the fact that it moves slow also have an effect on aircraft performance? Same with hot air, the air is more spread out which reduces aircraft performance but does the fact that the air molecules are moving fast have an effect on aircraft performance?

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In some sense, yes it does. But we usually don't think about it in those terms. You usually only need to think about individual air molecules when the object you are talking about is very small. For example, if you are trying determine the aerodynamics of an airplane whose length is 1/100th diameter of a human hair, then you might start caring about individual air molecules. For realistic sized planes, the planes are so big relative to the air molecules that we can just pretend that air is continuous. So instead of thinking about the motion of individual air molecules, we think about continuous variables such as density and temperature. i.e. when your air molecules "spread out and faster moving" we say that is a higher temperature with a corresponding lower density, and when they are "compacted and slow moving" we say that is lower temperature and a higher density. By doing that, we've taken the motion of a quadrillion individual air molecules and reduced it down to one single number. And that one single number, density, does have an affect on aircraft performance.

For completeness, you also sometimes need to consider individual air molecules at very low pressures. e.g. if your air pressure is a millionth of an atmosphere, then there are not many air molecules to go around and so each individual molecule's motion becomes more significant. But airplanes generally do not fly at such low pressures. Even at 40,000 ft, there's a lot more air pressure than that.

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Nice, thick cold air is great for engine performance. Especially for small single engine airplanes. Hot, less dense air means less performance, making takeoff more of a problem, with longer takeoff rolls needed to get sufficient airspeed to liftoff.

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Yes, of course, but not because of the average speed of the molecules. Temperature affects density altitude. Aircraft performance is dependent on density altitude, which considers not only the pressure altitude, (displayed on your altimeter), but air temperature as well. Almost every performance metric for an aircraft is a function of density altitude, which increases as temperature increases, and decreases when it is colder. Lift, drag, engine performance, all are affected by density altitude.

The fact that the air molecules are moving faster when it is hotter is irrelevant, because this speed is the average speed of the molecules. they are moving in all directions, both towards the airfoil, and away from it. So any performance increase from additional relative speed of those air molecules moving towards the wing will be counterbalanced by the performance decrease from those air molecules moving in the same direction as the aircraft, which will be hitting the wing at an equivalently slower speed.

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