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Originally Posted by C4 Diesel
It would be an issue of thermal mass, not temperature gradient. Assuming the bottle is a tall cylinder, even if they were both closed, the temperature gradients would be very similar. This is because the smaller dimension is the radius, not the height, and since decreasing the amount of water in the bottle does not affect the radius and only affects the height (assuming it was standing relatively upright), the direction containing the more drastic temperature gradient would not change. The bottle would only cool faster because there is actually less to cool.
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C4 - lacking sufficient information about the position of the bottles, etc - I honestly can not make a better statment about this portion, save what I'll say below.
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The temperature gradients would look something like this... Imagine taking a rainbow and shaping it into a two-dimentional rectangle (with the longer dimension being height, such that it somewhat resembles a vertical bottle) with rounded corners. The colors would represent the temperature profile. The full bottle would be that rainbow, the other would be one that was 1/4 shorter.
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I believe I understand your analogy, and thus I have this question: if you have a rectangular rainbow that is x units long, and another rectangular rainbow which is x/4 units long, with the same "non-dimensionalized" spectrum across them, I do not understand how this would not affect the color gradient, as I would think it would have to be "compressed" on the shorter rectangular rainbow. Would that compression not essentially represent a more severe temperature gradient, in the longitudinal direction? I wholeheartedly agree with your radial statements.
While I appreciate the thermal conductivity of plastic, I am guessing that the air in a plastic bottle with a thickness of ~ 1/16 " or less will reach thermal equilibrium with the air on the outside of the bottle if left overnight. I have never conducted the experiment, nor have I performed the calculations, thus I can not state as fact. I do think that a zero-flux gradient at the center of your radial coordinate system and a low-temperature temperature condition on the edge of the circle is less severe than two low-temperature conditions on the end of the longitudinal axis. Of course, there will result a zero-flux condition in the middle of the cylinder as well; either way I agree that the volume (which I believe is related to the thermal mass?) differences will directly affect the rate of cooling and phase change.