# How quickly does the temperature rise in a water container submerged in a hot water bath?

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 ;)

• what is the container made of and how thick are the walls? Feb 8, 2016 at 23:59
• 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. Feb 9, 2016 at 0:01
• 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? Feb 9, 2016 at 0:03
• 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. Feb 9, 2016 at 0:03
• @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. Feb 9, 2016 at 0:05