Steady state heat conduction into a very large solid

[Racer_Rob]

Say you have a flat resistor that is producing heat. You place the resistor against a sheet of steel and wait for equilibrium. One side of the steel is now at the same temperature as the resistor (assuming negligible contact resistance), the other free-air side of the steel is at a lower temperature than the resistor surface but not as low as ambient. So on this side convective cooling will bring the temperature gradient down to ambient.

You repeat the experiment but this time the free-air side of the steel is put up against a very very large solid object and again you wait for equilibrium, what will the temperature gradients look like now?

Because the solid object is so massive then I imagined the heat flow will cause it's temperature to rise by a negligible amount. Perhaps a small thickness of the solid that's against the steel will have an increased temperature but further into the object this will quickly be brought down to ambient. This feels intuitively right at least.

However, if you really wait for equilibrium (which may be a very long time) then the temperature profile across the large solid must be a straight line, from the temperature of the steel on one side to ambient on the other surely? So say half way through the solid, the temperature rise must be significant, is this correct?

 

[Chestermiller]

The resistor has the same area as the steel sheet and the object, so that the heat conduction is one dimensional, correct? If this is correct, then you are talking about a transient 1D heat conduction problem with convection at the far boundary. If this is correct, then your qualitative analysis of what is happening is correct. Basically, the region of the large body where the temperature is significantly affected grows with time. Eventually the temperature profile in the large body will transcend the entire body and be linear. You can find solutions to this problem (and ones like it) in Carslaw and Jaeger. You can also find transient heat conduction solutions in Heat Transmission by McAdams and Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

 

[Racer_Rob]

Hi Chet, those assumptions are correct, I was imagining 1D conduction. Although surely once the temperature gradient through the large object is a straight line then it's steady state not transient because the temperatures are not changing with time from then on.

I think I over complicated the problem in my own mind initially because the answer didn't 'feel' right. I suppose that's because in this case it's hard to appreciate what equilibrium (several hours or days later) will look like, plus convective losses out of the sides of the large object will stop the temperature rise from propagating too far into the object.

 

[Chestermiller]

I think you have a good feel for what is happening.

Chet

 

[sardar]

Yes, your intuition is correct. In the case of steady state heat conduction into a very large solid, the temperature gradient across the solid will indeed be a straight line, with the temperature at one end being equal to the temperature of the resistor and the temperature at the other end being equal to ambient temperature. This is because, in steady state conditions, the heat flow into the solid will be equal to the heat flow out of the solid, resulting in a constant temperature gradient.

As you mentioned, there will be a small increase in temperature at the surface of the solid that is in contact with the steel, but this temperature increase will quickly dissipate as you move further into the solid. This is due to the high thermal conductivity of the solid, which allows for efficient heat transfer and distribution throughout the material.

It is important to note that the time it takes for the solid to reach equilibrium will depend on various factors such as the thermal conductivity and thickness of the solid, as well as the heat generation rate of the resistor. In some cases, it may take a very long time for the solid to reach equilibrium, as you mentioned.

Overall, your understanding of the temperature profile in this scenario is correct. The temperature gradient across the solid will be a straight line in steady state conditions, with a small increase in temperature at the surface in contact with the steel.