Say there is an infinitely-tall tube extending from the ground upwards into space. Inside the tube is a heater capable of heating the air and then turning off.
My question is, at a steady-state condition, where in the tube would it be the warmest? It’s common knowledge that “hot air rises” because it is less dense, but at the same time, the higher you go in the tube the air thins out and cools down due to the diminishing atmosphere so it seems like there would be some point in the “middle*” of the tube where the warmest air would settle.
I know “middle” makes no sense in an infinite tube but just say the tube stops where earth’s atmosphere stops for all intensive purposes.
What you are describing is similar to what we on Earth call an “Atmosphere.” Generally, the warmest air is closest to the ground, although something called a “thermal inversion” can result in warmer air trapped above cooler air.
BTW, it’s hard to define what would be considered “steady state.” Are there external influences?
Assuming your heater is at the bottom of the tube, then, by definition, the air can only cool down as it moves up the tube. It’s not so much that gravity is overwhelming anything, but moreso that the closer you are to the heat source, the warmer your fluid is going to be.
Are we assuming that the tube is adiabatic (perfectly insulated) in order to achieve a steady state condition? In this case, once the air inside the tube achieves thermodynamic equilibrium it will all be the same temperature (by definition) but the thicker air in the bottom of the tube will contain more thermal energy and will feel warmer to an observer.
If we assume that the system is free to radiate heat to space or conduct to the walls then Santo Rugger is correct; it initially will be warmest near the heat source and cool down from there. At thermodynamic equilibrium it will display the same temperature gradients as the outside atmosphere.