The other day I asked a question here. The question concerns two separate, radiating spheres that have constant heat generation and a consequent steady state temperature. Since one sphere has a smaller magnitude heat generation, but is identical to the other in all other respects, it will have a lower temperature. If the initially far apart spheres are moved much closer together, then their temperatures will both rise after steady conditions are again established.
The net thermal energy exchange will be from hot sphere to cold sphere, and we call this net flow of energy heat. This makes sense because the initially colder sphere had an increase in temperature. But since the initially hotter sphere also increased its temperature, it seems strange to say that there was no heat flow that caused the initially hotter sphere to have a higher temperature.
Can we say they both experienced a heat flow that increased their temperature? Otherwise, how can we explain that the initially colder sphere had an increase in temperature due to heat flow, but the initially hotter had an increase in temperature that was not due to heat flow?
Part of my confusion is due to the fact that some sources define heat thusly:
Heat #1: The transfer of energy due to a temperature difference.
Now this says nothing about the direction of transfer, so that we must rely on the second law of thermo to infer that heat is only transferred from hot body to cold body. This makes sense given the Clausius statement of the second law of thermo: "It is impossible for heat to spontaneously flow from a cold body to a hot body." Note that Clausius refers to heat. Thus together, the definition of heat above and the second law of thermo lead us to conclude that heat is the flow of energy from hot body to cold body.
Other sources define heat as
Heat #2: the flow of energy from a hotter body to a colder body due to the temperature difference.
This definition makes the Clausius statement of the second law of thermo superfluous. Of course heat cannot spontaneously flow from a cold body to a hot body because then it wouldn't be heat by definition!
If we take the first definition, then heat could at least by the definition alone flow from a cold body to a hot body, except the second law of thermo prohibits that. With the second definition of heat, it is definitional that the flow cannot be from a cold body to a hot body and the Clausius statement is more of an implication of the definition of heat rather than a law of physics.
Still other sources define heat as
Heat #3: The net energy exchanged between bodies at different temperatures.
This definition acknowledges that energy may indeed be spontaneously exchanged in both directions between the spherical black bodies, and also allows that the Clausius statement is not superfluous. It says heat is the net of the exchange and Clausius says this is always from the hotter body to the colder body. I think I like this definition the best.