Performance of a thermos bottle relative to contents

Question

I'm not a physicist but I majored it at high school (a long time ago) and I study university math.

Me and my roommate discussed whether the performance of a Thermos bottle is influenced by how full it is. So if it is not full do the contents cool down faster, slower or equally?

Thanks.

Answer 1

I don't have a thermos flask to hand, otherwise I'd do the experiment (the only sure way to answer :-). In the absence of experimental data I'd guess that the half full flask will cool faster.

The heat flow will be roughly proportional to the temperature difference between the inside of the flask and the ambient temperature outside. The constant of proportionality is the heat transfer coefficient.

Whether the flask is full or half full won't make a lot of difference to the internal temperature because heat circulation inside the flask will be fast. This is because liquids have a high thermal conductivity, plus you get convection in both the liquid and the gas above it. Evaporation/condensation at the gas-liquid interface will also keep the liquid and gas at similar temperatures.

So the full flask and half full flask will lose heat at the same rate because the interior of the flask is at the same temperature. However the half full flask has only half the specific heat, so for a given heat flow it will reduce temperature twice as fast.

This argument is quite general and would apply to any container as long as the heat flow was slow enough that the interior liquid/gas temperature remained even. At high heat flows the gas above the liquid will cool faster than the liquid because heat transfer is slower in the gas. This complicates the analysis, though I think the half full flask would still cool faster.

Answer 2

I am not a physicist either.

As I understand it, heat can be lost by conduction, by convection and by radiation, The purpose of the bottle is to reduce all three.

If you half the amount of liquid, the question is whether you also half the loss of heat, or do more or less.

Analysis is difficult because the weak part of the bottle is the cork. If it is full, there is hot liquid near the cork that looses heat faster, and then gets conduction and convection heat fron the rest. There is also a lesser problem with the bottom, since it is an additional surface where heat can be lost.

When the bottle is half full, the liquid is further away from the cork. But the air inside will conduct some of the heat (conduction, and convection) to the empty part of the bottle, and radiation may internally add some. If the empty part became as hot as the liquid, the heat loss would be the same as before, for a lesser mass of liquid. hence it would cool faster,

If it does not get as hot, it means that some heat is lost to keep it cooler. If the bottle were homogenous (no cork effect, no bottom effect), that would mean that, in addition to its normal heat loss through the side, the remaining liquid has to provide for the heat loss in the empty space above it. Hence it cools down faster.

The bottom is a disadvantage for the half full bottle, since its loss is the same in both case, and thus contributes comparatively more to cooling when the liquid mass is lower.

Now, I would need more data and/ or knowledge to analyse the effect of the cork. With a very conducting cork, the full bottle would loose heat quickly (assuming the liquid touches it) through convection and conduction in the liquid. With a totally insulating cork, it would at worse balance the effect of the bottom of the bottle, so that the analysis without cork or bottom would be valid.

So with a reasonnably good cork, my conclusion is that a half full bottle will cool faster.