What would the temperature limit of heating metal block?

[tomothy]

I have this problem and it's something that's been annoying me all day that I just can't get my head around.

Imagine a metal block, in direct thermal contact with a heating element and "the surroundings"; but the heater and "the surroundings" are not in direct thermal contact. (I'm supposing also that the surroundings will remain at a constant temperature). The heater is turned on with a constant potential difference and constant current running through it. The effect is to heat the block.

My question is: will the block reach the same temperature as the heating element, or will it reach a steady temperature somewhere in between the temperature of the surroundings and the heating element, due to constant heating of "the surroundings" by the metal block? Or alternately, would this depend on the conditions of the system, like the heat capacity of the metal block and its surface area?

I will happily accept a condescending response because I think I may just be over-thinking this problem and missing a blindingly obvious answer!

 

[edgepflow]

Welcome to PF.

Depending on the arrangement, the temperature of the metal block will vary between the surface temperature of the heater and the surroundings. The shape of the distribution can be determined from conduction theory. The temperature within the metal block could vary in 1, 2, or 3 dimensions depending on the layout.

 

[tomothy]

aaah thank you very much! So the average temperature of the metal block will never be as high as the temperature of the heater? Also, does the rate at thermal energy is transferred by heating from the block (in a simplified way) increase as the temperature difference between its surface and the surroundings increases (is it proportional?)?

 

[edgepflow]

The average temperature of the metal block will always be less than the heater surface unless the block's surface is insulated or the box is totally enclosed in the heater.

The heat transfer rate is proporational to the temperature difference between the hot and cold surfaces. So, as time passes and the block heats up, the rate of heat transfer from the heater to the block decreases and the rate of heat transfer from the block to the surroundings increases. After much time has passed, the block will exactly pass the heater power to the surroundings.