A friend posed a question to me yesterday: if I have two glasses of 60 F water, would they heat at a different rate if one were in a 70 F room and one were in a 90 F room. Myself having some education in an engineering school, I replied, 'of course'. He said that it was not so obvious, and that they would heat at the same rate.

I found this website on heat transfer. I scanned the resource and replied back to him that it appears it is a function of delta Temp and the heat transfer constant, so it would seem I am right. But I am thinking that the glasses themselves buffer the transfer - that they act as something of an insulator, and so it would appear that they are heating at nearly the same rate.

So I ask - everything being the same, two rooms, one at 70 and one at 90, if the water is held in two glass bottles, where the opening is so small we might say the area is large enough only so it doesn't cause a sealed environment, would the heat transfer appear to happen at the same rate to the observer?

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    $\begingroup$ When trying to answer questions like this it is sometimes usefull to take the options to extreme - if you put two glasses of 60 F water, one in 70 F room and another in 1000 F room, would they heat at the same rate? I think the answer is pretty obvious. $\endgroup$ – Alexander Jul 3 '15 at 20:15
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    $\begingroup$ Another way to think of it is: you know the answer in the case where the bottles are perfectly conducting, and the bottles in your question are identical. So why would it change the relationship? It can only slow down the heat transfer by a constant factor. $\endgroup$ – tok3rat0r Jul 3 '15 at 20:19

Your model is correct. The speed of the transfer depends on the difference in temperature and the thermal resistance of the glass and the water, which is constant.

Interestingly the speed is not constant. The closer you get to the final temperature the more it slows down. Technically you never really get there exactly. There is no finite time where you can say "now it's exactly 70F". If you want to measure an actual time, you need to set some threshold that's below 70F, for example 69.99F.

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  • $\begingroup$ "Technically you never really get there exactly." Given that it's a dynamical system, isn't there a nonzero chance that the temperature difference will fluctuate both above and below an equilibrium? $\endgroup$ – Asher Jul 3 '15 at 23:07
  • $\begingroup$ Experimentally speaking, due to finite apparatus resolution, you will measure the desired temperature at a finite time. $\endgroup$ – Alexander Jul 3 '15 at 23:09
  • $\begingroup$ I meant that the statement described an ideal, not real, situation and that in reality the container would reach (and at times slightly exceed) the room temperature. Compare similarly the ideal case of a smooth ice/water phase transition versus the realities of sublimation, supercooling, etc. $\endgroup$ – Asher Jul 4 '15 at 2:23

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