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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.

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I think your tags are fine (edited out that bit of your question). –  Kyle Jun 26 '13 at 12:29
    
Possibly you may find this other question interesting: Heat preserving performance of container relative to content. It is a follow-up on yours, with a different result. Comments welcome, even negative ones. –  babou Jul 10 '13 at 19:44

2 Answers 2

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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.

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Since you added the license bit (which maybe is in conflict with the usage of this site, but forget about that), I'll take the opportunity to ask you something. "You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work)." Why don't people using such a license actually specify the manner then? –  Glen The Udderboat Jun 26 '13 at 15:47
    
@Gugg Actually, I think "manner" refers to the wording of the attribution. I do not believe that constraints on the means would be in the spirit of the license (see section 4c of the full license text). But if you are really interested, I can ask for confirmation from the horse's mouth. The license bit is not a whim. - By the way, could you tell me what's wrong with my answer ? I thought it more precise and more general, without unneeded hypothesis. –  babou Jun 26 '13 at 17:04
    
1) Well, I am sort of interested, because I had to deal with it with pictures taken from Wikipedia and cartoons. And I don't think it is clear what "the manner specified" and the sentence surrounding it imply if no manner is specified. 2) I don't know enough to have an opinion on your answer. –  Glen The Udderboat Jun 26 '13 at 17:28
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That license is at best redundant, at worst wrongly asserting ownership, cf. the terms of service: "You agree that all Subscriber Content that You contribute to the Network is perpetually and irrevocably licensed to Stack Exchange..." –  Chris White Jun 26 '13 at 18:21
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Any content posted on SE 2.0 is CC-BY-SA, attributed to you (so anyone quoting it must mention your username, et cetera). However, it is licensed to SE, which means that they reserve the right to delete or modify this content (and other rights). –  Manishearth Jun 26 '13 at 18:24

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.

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