Conceptual question regarding the conservation of energy

[FissionChips]

Hi PF,

I've got a very trivial conceptual question regarding the conservation of energy with respect to thermodynamics and heat transfer that I can't seem to figure out.

Suppose I have an electric heating element with a 240 volt, 30 amp supply, in contact with a solid volume of an arbitrary material. P = I*V, so the element will supply a total of 7200 W. Since electric elements are 100% efficient, I can safely say that 7200 W of heat energy will be emitted from the element.

It is my understanding, (perhaps incorrect) that while the element will emit 7200 W indefinitely, and at a constant rate (assuming the voltage and current of the supply does not change), the rate at which this heat energy is received by the solid body is not constant, and is a function of the temperature differential between it and the element surface. So, if the element surface is at 800 K, the temperature of the solid body will approach 800 K logarithmically, and the heat transfer rate between the element and the solid body will become infinitely small as time progresses.

This is where I'm hung up. Assuming I'm still running my element, I'm emitting a constant 7200 W, but as time progresses, only an infinitely small fraction of this is being picked up by the solid body. If this was not the case, the solid body would continue to heat up indefinitely as 7200 Joules of heat energy were pumped into it every second.

I understand that in reality, there are losses of all kinds that account for the difference in energy emitted versus received. However, if we are to assume theoretically ideal conditions (heat transfer is only occurring between the element and the solid body, and there are no losses), where is the balance of energy going?I apologize for having to ask something so simple; I obviously misunderstand the fundamentals of heat transfer.

Thanks!

 

[Drakkith]

I understand that in reality, there are losses of all kinds that account for this. However, if we are to assume theoretically ideal conditions (heat transfer is only occurring between the element and the solid body, and no losses occur), where is the balance of energy going?

In reality the solid body will also transfer heat into its surroundings. So your heating element heats up to something close to 800k, and the solid body heats up to less than the heating element, with the portion of the body in direct contact with the heating element having the largest temperature and the portion of the body in contact with the outside environment having the smallest temperature.

If you were to assume that the solid object is perfectly insulated from the outside environment, then both the heating element and the solid body would heat up well beyond 800k, up to the point that they melt or the heating element fails from the increased temperature. Simply put, the energy would go into heating up the solid body and the heating element even further.

 

[andrewkirk]

The element would have to be encased inside the solid body, otherwise there will be losses from the element by radiation in whatever direction is not blocked by the solid body.
What I think will happen is the following.
The element's temperature of 800K assumes that the element can dissipate heat, which it usually can. But if it is enclosed, its ability to dissipate heat is related to the temperature differential between it and the solid body. So as that temp diff reduces, the element will start to increase its temp. Hence the temperature of both the element and the solid body will continue to increase, with the former always remaining above the latter, until the element burns out.

 

[FissionChips]

Thank you! For some reason, I was fixated on the element surface being at a constant temperature.

 

[Dale]

Assuming I'm still running my element, I'm emitting a constant 7200 W, but as time progresses, only an infinitely small fraction of this is being picked up by the solid body.

What happens to an object that receives more thermal energy than it loses?

Edit: never mind, I was a little bit late.

 

[rootone]

There is no ideal situation in which the heated object does not lose any heat.
If that would be the case then the object would apparently not increase in temperature as far as any external measurement could be made.
Well maybe not until the internally contained heat transformed the object into a plasma.

 

[sophiecentaur]

All of which explains why they tell you not to hang washing over electrical heaters.
It is even relevant when writing regulations for power distribution around the home. Low resistance cables can still get too hot for safety if they are routed through thick fibreglass loft insulation or coiled around a cable drum.

 

[Drakkith]

It is even relevant when writing regulations for power distribution around the home. Low resistance cables can still get too hot for safety if they are routed through thick fibreglass loft insulation or coiled around a cable drum.

Does that also apply to extension cords?

 

[sophiecentaur]

Does that also apply to extension cords?

The gauge of wire is chosen to minimise heating of cords. Feel how warm the lead on an electric kettle gets, even with heavy wire.

 

[Drakkith]

The gauge of wire is chosen to minimise heating of cords. Feel how warm the lead on an electric kettle gets, even with heavy wire.

Yeah, I've felt the cord after using my microwave before. It was pretty warm.