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I'm familiar with problems of "how much ice can you melt given some amount of energy", but I'm writing to get some clarification on the time constant of this event. This question might be somewhat related to How long does it take an iceberg to melt in the ocean? but I will try to ask it in a simpler form.

I have $1 kg$ of water at $0.5^{\circ}$ that is in contact with ice. $0.5^{\circ}$ water has $2 \cdot 10^3 J$ relative to $0^{\circ}$ water. If the latent heat of ice is $3.34 \cdot 10^5 J/kg$, the water can melt $0.006 kg$ of ice.

What I'm trying to figure out now is how long that takes. If the water is in contact with the ice for 10 seconds, is 100% of the energy used up in the heat transfer process, or is some of the energy in the water advected away, unused, after 10 seconds.

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Hi phisiksquestioner. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. –  Qmechanic Apr 23 '13 at 22:10

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There isn't an answer to your question because it depends on the geometry of whatever container the water is in plus the mixing (if any) in the water.

For example, suppose the water is in a cubic container. The water at the bottom of the container will rapidly transfer heat to the ice and cool to 0ºC. The heat from the water immediately above the bottom layer now has to flow through the bottom layer to reach the ice, and rate at which this happens depends on the thermal conductivity of water and the thermal gradient. Plus since water at 0ºC is less dense than water at 0.5ºC the bottom layer of water will start rising and you'll get some form of mixing of the original and the cooled water in your cube, and this will affect heat transfer to the ice.

For any particular geometry you could do a finite element calculation, or take some approximation (e.g. ignore mixing) and solve the heat flow equation. However there isn't going to be a simple equation that describes the process. Speaking as an (ex) experimentalist, the fastest way to answer your question will be to do the experiment!

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There are some 1D text-book examples for large layers or spherical items that do have a solution. –  Bernhard Jun 23 '13 at 11:16

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