Technical definition of heat

Question

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?

EDIT

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.

Answer 1

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?

They did not both experience a heat flow that increased their temperature.

The initially hotter sphere had an increase in temperature due to its own internal heat generation, not due to the external flow. In this scenario the heat flow acts only to decrease the temperature of the hotter object. When the cooling heat flow is reduced and the heat generation maintained, the equilibrium temperature increases. This in no way implies a flow of heat from the cold object to the hot one.

Similarly, if you are filling a tub with water you have a faucet and a drain. With the drain open and the faucet open the water level stays low. When you close the drain the water level increases. This in no way implies a water flow into the tub from the drain. The water flows in only from the faucet and only flows out the drain. But the water level will still increase if the drain is closed

Answer 2

Before discussing your example, regarding the title of your post, the technical definition of heat is energy transfer due solely to temperature difference. In this respect by energy we mean net energy.

Initially, when the spheres were thermally isolated from on another, each was in thermal equilibrium. That is their temperatures were constant. What that meant was the rate at which each was radiating heat into space exactly equaled the rate at which energy was being generated by the nuclear reaction within. Since the spheres are physically identical in every other respect, the rate at which energy is being generated has to be greater in the higher temperature sphere.

If the spheres had no nuclear energy source within, the only reason the initial temperatures would be different is stored thermal energy, i.e., a difference in internal energy. Eventually each sphere will come into equilibrium with space at 0$^0$K.

When the spheres are brought near to one another, and assuming there are no other objects that can radiate or receive energy, each sphere is now capable of both transferring energy to and receiving energy from the other. The temperatures of both spheres will initially rise because because both are generating energy at the same time they are radiating and receiving energy. In other words because they are not finite energy sources, to the extent that mass is being converted to energy by nuclear reactions.

Although the temperatures of both will rise, the temperature of the sphere with the smaller energy generator will rise faster than the temperature of the higher energy generator until they reach thermal equilibrium with each other at a temperature greater than their original temperatures. Moreover, the increase in temperature of the lower energy generator will be greater than the increase in temperature of the higher energy generator. Since both spheres are physically identical, that means there is a net transfer of energy from the initially higher temperature body to the initially lower temperature body.

The idea that the temperature of BOTH spheres can increase during the heat transfer can be hard to wrap your head around because usually the final temperature is something in between the original temperatures, or will equal the higher temperature body if it is a thermal reservoir relative to the lower temperature body, But that’s only because usually we are not dealing with bodies that are generating energy internally simultaneously while radiating energy.

That is very much in line with my own analysis. Thanks. However, the question I am more interested in is why do we not call the energy absorbed from the colder sphere by the hotter sphere as "heat" when such energy was both transferred and also lead to an increased temperature?

Heat is not "divided up" by the two bodies. The energy absorbed or radiated at any give time by the colder body is not heat any more than the energy absorbed or radiated at any given time by the hotter body. It is the net energy that transfers from one body to another that is legitimately called heat.

You may recall my answer to one of your earlier posts where I pointed out that when considering heat transfer by conduction microscopically, energy is being transferred from the colder body to the hotter body due to collisions between higher energy particles in the colder body with lower energy particles of the hotter body, while simultaneously energy is being transferred from the hotter body to the colder body due to higher energy particles in the hotter body colliding with lower energy particles in the colder body. But because there are a greater number of higher energy particles in the higher temperature body than the lower temperature body the net transfer of energy is from the hotter to the colder. It is the net transfer of energy that is heat. Not the individual transfers of energy between the particles.

EDIT TO QUESTION

The first statement (Heat 1) is the definition of heat. It doesn't preclude the transfer of energy from cold to hot, otherwise we wouldn't have air conditioners.The second statement (Heat 2) is from the second law dictates the the direction that can occur spontaneously, i.e., without external work. So net energy can transfer from cold to hot, but not spontaneously, i.e., without external work. The third statement (Heat 3) is fine because it only clarifies the actual definition of heat (Heat 1) because what we are really talking about is net transfer of energy, not energy transfer occurring at the particulate level.

Hope this helps