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Let's assume I have a small bottle (neglecting any insulation from the walls of the bottle) containing e.g. 150ml water at 4°C and place it in a larger pot with sufficient water at 80°C so that the water levels inside and outside the bottle align. How can I calculate the time it takes for the water inside the bottle to reach 40°C?

How does this change with greater/smaller quantities of water inside the bottle (while still adding just enough hot water align the levels)?

How does this change if the water inside the bottle is initially at room temperature 21°C?

And yes this is about heating baby food ;)

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  • $\begingroup$ what is the container made of and how thick are the walls? $\endgroup$ – costrom Feb 8 '16 at 23:59
  • $\begingroup$ The outer container is a pot used for cooking with about 2mm thick steel walls. The inner container (small bottle) has about 1mm thick walls of plastic. $\endgroup$ – Felix Lamouroux Feb 9 '16 at 0:01
  • $\begingroup$ Would this change a lot much if the bottle was made of 2-3mm glass? Or is it safe to ignore this if I am mostly interested in +/- 5 seconds accuracy? $\endgroup$ – Felix Lamouroux Feb 9 '16 at 0:03
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    $\begingroup$ There is no easy way to calculate this for liquids because the heat exchange will depend on whether there is any convection in the water or not. You can calculate the solution for the heat (conduction) equation for your geometry, but this may or may not give the right answer. The problem is a lot better defined for solids which can not convect. $\endgroup$ – CuriousOne Feb 9 '16 at 0:03
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    $\begingroup$ @costrom: Yes, I would think so. So if you are interested in the upper bound, solving the heat equation for a (cylindrical?) geometry might do the trick. $\endgroup$ – CuriousOne Feb 9 '16 at 0:05
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There is no easy way to calculate this for liquids because the heat exchange will depend on whether there is any convection in the liquid or not. You can calculate the solution for the heat (conduction) equation for your geometry, but this may or may not give the right answer. The problem is a lot better defined for solids which can not convect.

The solution to the heat equation in cylindrical coordinates can be found in many physics books and scripts. It will neglect the bottom. There is another question that you need to consider: how homogeneous does the temperature have to be in the bootle? Unless the liquid is being stirred, there will be a significant gradient. If you look at the way chemists are doing their temperature dependent reactions, there is almost always some rather strong stirring or agitation going on, otherwise things may react differently in one part of their beakers than in another. You may want to stir, too.

The good news is that stirring simplifies the physical problem and it will, at least that's my gut feeling, reduce the time for the heating and the variation in the time it takes to get to the right temperature. The heat equation can then be reduced to the boundary, which is characterized by an effective area and an effective thermal resistance.

As a final comment: I would never leave the health of my baby to theoretical calculations. The only safe method is to measure the temperature.

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