This question has been addressed in the case of a thermos bottle: Performance of a thermos bottle relative to contents
I am asking the question again without the hypothesis that it is a thermos bottle.
Given a container with a warm liquid inside ("warm" meaning warmer than the medium surrounding the container), will it cool faster, slower or equally, when it is half-full than when it is full.
To simplify the analysis, it is assumed that the opening and cap of the container have the same properties with respect to heat as the rest of the container, so that they may be ignored in the analysis.
Answer
I started asking myself this question because I was somehow unsatisfied with the answers to the previous question "Performance of a thermos bottle relative to contents" concerning a very specific kind of container. There were assumptions made in the answers. Even though these assumptions were fair, given that a thermos bottle is a rather precise and well known object, the reasonning in the answers did not make them explicit. I tried to avoid it, by reasonning explicitly about the cork of the bottle, but I still used properties of the bottle shape without saying so explicitly (I became aware of it later). And even though my assumption was close to the actual facts, a thermos bottle is not a cylinder. The bottom is often somewhat spherical, which could have called for an extra line of justifications (how high should the bottle be compared to the radius of the bottom half sphere, unless the top is also considered a half-sphere ?).
Sometimes, we also make (explicit or implicit) assumptions that are not needed.
The other thing that bothered me is that people will often vote for simple answers they understand quickly (not necessarily the best answer or even a correct one). At least that is the feeling I get. If warranted, this would justify making lots of unstated assumptions when answering. Not to mention the fact that fast answers get a better chance at upvotes, when acceptable.
Then, considering the thermos question, I started wondering about what could make our statements wrong, and what could be the assumptions that are often made implicitly, just for that kind of problem (though I actually made one or two explicit in stating this new problem). Here are some such assumptions, probably an incomplete list (other ideas are welcome):
role of the cork: can it be ignored as not significant ?
homogeneity of the bottle sides: is it the same kind of material all over ?
shape of the container: is it just a bottle, which we tend to assume ?
heat conductivity of the bottle side: is it isotropic ?
uniformity of liquid cooling: well, that is always wrong, but liquid conductivity is so efficient that it seems a good approximation. Is it?
Then I wondered whether falsifying these assumptions could also falsify the conclusion.
Once I had satisfied myself that it was the case, I asked the question, carefully stated so as not to induce any assumption (for example by always using the word "container" instead of "bottle"). I was not trying to trap anyone, only experimenting. I unfortunately got few reactions (thank you to those who did react). I should have started that more anonymously as some users clearly wondered what I was after (comments welcome).
So here is my answer.
The picture is a cross-section image of a container that will cool faster when it is full than when it is half-full. It consists of a large disk on top of a sphere, with the same volume so that only the sphere contains liquid when it is half-full. The opening between them is large so that heat can flow easily between the two parts when it is full.
Clearly, the disk has a very large surface to volume ratio and will act effectively as a radiator to cool the liquid it contains, while the shere has the smallest possible ratio and will not cool fast. However, when the container is full, heat will flow through the liquid from the sphere to the disk so that all the liquid content will cool rather fast, though the disk will get cooler faster than the sphere.
If the container is only half-full, only the sphere contains warm liquid, with a small surface/volume ratio. Hence it will cool more slowly.
If the container is large enough, this should be sufficient.
It can be improved by remarking that a horizontal disk shape is not very good for convection heat transfer. Replacing the hollow disk with an inverted bell shape would work better.
You can even reinforce further the effect by using for the side of the container a material that is more heat conductive transversally than laterally, so that heat goes out quickly but is not conducted efficiently from one part of the container side to another. That avoids heat being transferred to the disk when the container is half full, and still permits the disk to cool efficiently the liquid it contains when full. It can be easily produced by using a heat insulating material with copper nails piercing it at close regular intervals.
The same effect could be achieved with a bottle having a standard shape, but with isolation only in the bottom part. This is somewhat close to what I said about the effect of a conductive cork in the thermos case.
Of course, this is no major discovery in elementary physics. At best a moderately easy puzzle game. But it may be telling about our reasonning process.
Coming back to the remark about votes. I am wondering whether the initial downvote for that question (without an explicitly related explanatory comment) was motivated by such an unstated assumption. Was it really justified?
Now, it is possible that people who are very proficient in a field will vote more accurately. It is probably more the case, but I think not always (I do have one example in mind, not from physics). Proficiency is an ill-defined concept, and schools of thought are often biased, even in hard sciences.
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