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How much time it will take for a ball of ice at 0 °C and radius 3 cm submerged in 1 l of water at 100 °C to completely melt? (Assume the ice melts completely uniformly and treat it as a shrinking sphere.)

What type of theoretical calculations would I use to do this? I was thinking about using Newton's Law of Cooling and maybe related rates but I don't know where to start.

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    $\begingroup$ There is too many variables. $\endgroup$
    – Poutnik
    Nov 14, 2023 at 4:07
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    $\begingroup$ Newton's law would only apply if the water were flowing past the ice. A full calculation would involve using Ficks law of diffusion for ice and water, so not easy as you will need to know about Fourier methods to solve the equations or do so numerically (and allow for enthalpy changes). You could start with a thin rod melting at the end with ice held at 0 degrees C. There are some examples in chapter 10 of chemistry-maths-book.com $\endgroup$
    – porphyrin
    Nov 14, 2023 at 8:32
  • $\begingroup$ @porphyrin It would need to solve water convection as well, that would be changing in time and the ice could get irregular shape. It would be complex system with non trivial simulation. $\endgroup$
    – Poutnik
    Nov 14, 2023 at 8:57
  • $\begingroup$ @Poutnik, I agree, its very very complicated but appears not to be. The temperature of the ice is unknown anyway no matter any other complications. This simple question illustrates why we invariably have to build rather simple models to explain things by ignoring this and that to try to reach an answer. $\endgroup$
    – porphyrin
    Nov 14, 2023 at 10:34
  • $\begingroup$ This seems like a rather standard engineering problem you can approach with semi-empirical formulas and in a similar fashion as you would derive Planck's equation of freezing, however the obtained differential equation will become incredibly complex as all these formulas (find the heat transfer coefficient from the Nusselt number, using free convection relations which will need Rayleigh and Prandtl numbers...) have a dependence on the characteristic dimension which is changing over time. This is of course neglecting the bulk temperature changes significantly which can be checked as well. $\endgroup$
    – Noah
    Nov 16, 2023 at 9:47

3 Answers 3

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As a very, very crude, 'ball-park' or 'back of envelope' etc. estimate we can use the thermal diffusion coefficients of ice and water. The values for ice and water are $\approx 1$ and $0.14$ mm$^2$/s respectively (Wikipedia, Thermal diffusivity). If we assume that heat transfer is limited by thermal diffusion in water into the ice (as its so much faster when in the ice) and that the temperature at edge of ice is always the same (100 deg) it takes $L=\sqrt{6Dt}$ to diffuse average length $L$ in time $t$, thus taking $L=15$ mm, the time is $t\approx 4.5$ minutes. This should be the shortest time, as the ice cannot melt faster than the thermal energy can reach it.

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  • $\begingroup$ Yet another aspect, aside of conduction and convection, is heat transfer caused by thermal radiation of condensed phases. As it is proportional to the difference of the 4th power of temperatures, it is progressively raising. I would not , of course, dare to try to involve it in any simple model. $\endgroup$
    – Poutnik
    Nov 14, 2023 at 14:40
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The simplest model I know is to define a diffusion layer (or boundary layer) around the ice sphere. The temperature of the ice is set to 0 °C, the temperature of the bulk water is set to 100 °C, and there is linear diffusion in the boundary layer to transfer the heat from water to ice.

A similar approach is used for the dissolution of crystals in water. Depending how vigorous you stir, the boundary layer is larger or smaller (we did an experiment on the dissolution of gipsum in physical chemistry lab, and here is an example research paper). The boundary layer is on the order of hundreds of microns for aqueous room temperature systems. In the case of ice melting, having a temperature gradient within the boundary layer makes estimation of its magnitude more complex as the viscosity depends on temperature.

In any case, you could measure radius vs time for a melting ice sphere, and check whether a single magnitude of the boundary layer describes the kinetics well (or not). From my real-world experience, the 4.5 minute estimate in Porphyrin's answer seems much too long (here is a video to get the order of magnitude). Because I don't have a clue what the size of the boundary layer would be, I can't make an estimate with the model I am proposing.

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It is simple. Drop an ice cube into a pot of boiling or just hot water and time it. The temperature gradient is from 0 to 100 Celsius because the ice was equilibrated at zero and the water boils at 100.

This could be a calorimetry experiment. Run the process in a calorimeter measuring the energy needed to maintain the liquid T at 100 [or other temperature] as a function of time. [Or measure the liquid temperature drop.] There are many variables to consider; that is the work of the experimenter, to identify and address the variables.

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    $\begingroup$ The question is how to model the process i.e. on a computer or if sufficiently simple solving the relevant equations analytically, not how to do an experiment $\endgroup$
    – Ian Bush
    Dec 16, 2023 at 13:52

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