# Heat transport in earth's gravitational field

Consider a Carnot machine that exploits the earth's atmospheric temperature gradient using air at lower altitude as a warm reservoir and air at higher altitude as a cold reservoir. Due to the temperature difference it will have a nonzero efficiency and thus performs some work $$W$$. It will also reduce the heat of the warm reservoir by some $$Q_H$$ and raise the heat of the cold reservoir by $$Q_C$$. In order to circumvent an explanation of parts of the machine being moved within the gravitational field, consider the Carnot machine to be small in the earth's radial direction, but to have ways of heat transport to actually make use of a non-negligible temperature difference.

At first glance this looks like a violation of the second law of thermodynamics, but I believe the solution lies within the heat transport between reservoirs and machine. Obviously, if I am using convection or conduction (momentum transfer), then the carrier will need to overcome the gravtational field and thus the required heat would not reach the Carnot machine. If I were to use radiation, however, the only possibility of losing energy from photons moving away from earth would be due to gravitational redshift, but that should be vanishingly small.

Is there an easy way to see why the above machine should not work?

• At first glance this looks like a violation of the second law of thermodynamics how so? Commented Jul 27 at 18:23
• Ah sorry, this should happen after running the machine for a while and then turning it off. The air in the earth's athmosphere should equilibrate back into its initial state (say I turn the work of the Carnot machine back into heat so I am not cooling down the atmosphere). This process transports heat from warm to cold and then back to warm, the last bit should be a violation of the second law, no? Commented Jul 27 at 18:33