Thermal conduction of heat by a solid body

[A Dhingra]

Hello..

The query is regarding thermal conduction of heat from a solid body:
Suppose we have a long metallic rod which is insulated on the curved surface and one of the bases. If we supply Q amount of Heat for a small time to the rod from the non-insulated end, initially the temperature of the end in touch with the sources rises and then due to temperature difference in the rod the heat flows. Since the amount of heat provided is fixed and is cut-off after a while, then the rod tends to attain the uniform spread of heat all over the space, tending towards the spreading of energy (hence rise in entropy.)
Now if I supply heat continuously to the rod at one end, then since the rod doesn’t get the opportunity to tend towards distributing the energy uniformly, there should be a temperature difference between the end in touch with source and the one farthest away from it. (Please correct me if I have made any conceptual error). So the rod as a system is unable to tend towards spreading the energy maximum, the entropy is not rising to its maximum ever in this situation. That means, as long as the rod is supplied with heat the temperature should rise, but in practise a steady state is achieved. How does that happen?

One more thing,
I have read at a few places that conduction in metals is significantly because of the free electrons, and very less due to the vibration in the lattice structure. Reading this I thought, may be the situation I am talking about is because of this fact. The steady state might be reached because the energy of vibration the particular solid can bear (being in the same phase) has a upper limit, after this is reached all the energy is transferred.
If now the other base of the rod is brought in contact with say flowing water, then after the steady state is reached all the heat supplied to the rod is given away by it to the water (the rate of flow of water should be less than that of Heat).
Is this correct?
I would wish to know details of how the conduction actually occurs in the three situations at the atomic or sub atomic level.

 

[Chestermiller]

The solution to the heat conduction problem you describe can be found in Heat Conduction in Solids by Carslaw and Jaeger. For the situation you describe, the temperature throughout the rod will continue increasing forever, and will never reach a steady state. At very long times, the temperature distribution will approach a function of position plus a constant times time. The constant will be equal to the heat supply rate divided by the mass of the rod times the heat capacity.

 

[haruspex]

You need to clarify what you mean by 'supplying heat'. In practical situations, the heat supply will be at or limited by a temperature. As the temperature of the rod rises (the heat apparently has no way to leave the rod) it will approach that of the source and heat flow will tend to zero. Or are you supposing that the source gets hotter at whatever rate is needed to keep pumping the heat in steadily?

 

[Studiot]

Good point huruspex and shows why you have to carefully examine the premises of thought experiments.

 

[A Dhingra]

You need to clarify what you mean by 'supplying heat'. In practical situations, the heat supply will be at or limited by a temperature.

The heat is supplied through the steam, i.e., water in a closed metallic container is continuously heated and the steam is allowed to get in touch with the rod.

So how does this affect the flow of heat from the source, and Will the temperature be same throughout the rod (when heat is supplied continuously, and the other end is insulated.) or not?

 

[Studiot]

You don't need a fancy Fourier thermal analysis to know what haruspex has already stated.

Heat will be input to the rod until it has reached the temperature of the steam throughout the rod at which point a steady state will have been reached and no further heat input or temperature rise occurs to the rod.

Since you have specified a 'long rod' this could take a very long time and will only happen after infinite time for an infinite rod.

 

[Chestermiller]

You need to clarify what you mean by 'supplying heat'. In practical situations, the heat supply will be at or limited by a temperature. As the temperature of the rod rises (the heat apparently has no way to leave the rod) it will approach that of the source and heat flow will tend to zero. Or are you supposing that the source gets hotter at whatever rate is needed to keep pumping the heat in steadily?

I am supposing the latter. That was my interpretation of what you were referring to in the original post. Obtaining the long-time behavior for this problem is much easier than for the case of constant temperature at the boundary, because it involves solving an ODE rather than a PDE (unless you are only considering the final state, where the final temperature in the constant boundary temperature case is just equal to the boundary temperature).