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If I have a bottle or container with X fluid ounces of water at temperature Y (say, fridge temperature or room temperature), and I put ice or ice sticks (composed of water) in it that make up Z fluid ounces, about how long should I wait before I can expect the drink to be optimally cold? The colder, the better (assuming I have ample time to drink it before it stops being cold!).

Is it coldest right when the ice is about melted, or just after a certain period of time of exposure to the ice?

I'd also be interested in whether having a vacuum sealed/insulated bottle makes any difference here, whether outside/room temperature makes any difference, and how the scenario changes if at all when placing the bottle in a freezer.

Believe it or not, I had a hard time finding a reliable answer to any of this from searching the web.

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    $\begingroup$ If this is a practical question rather than curiosity: stirring the ice+liquid makes a big difference in how long you have to wait. If it's a scientific question, one of the things it depends on is if the convective flow has good pathways to set up a current. If you just let ice cubes float at the top then the liquid lower down won't move into the ice zone. $\endgroup$
    – DMPalmer
    Commented Jun 26, 2019 at 17:50
  • $\begingroup$ True, stirring it would ensure that the heat/cold is evenly spread around. I would also think that the concept that heat rises would contribute towards the effectiveness of the ice floating at the top, though. $\endgroup$
    – Andrew
    Commented Jun 26, 2019 at 18:33

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You probably had a hard time finding an answer because it isn't straightforward. There are many factors which can influence the best approach, and the times.

For example, the temperature of the surroundings and the insulation of your container will have significant effects on the temperature over time. A highly insulated container will allow you to reach a lower minimum temperature, and cooler surroundings will do the same. Placing the bottle in the freezer will also lower the minimum possible temperature (even if you take it out of the freezer to do this experiment, a cold bottle is better than a warm one).

The initial temperature of the drink compared to the initial temperature of the ice are also very important.

To determine when it will be the coldest, you need to compare the amount of heat transferred from the ice to the drink, and from the surroundings to the drink. As long as the rate of heat transferred to the ice is greater than the rate of heat transferred from the surroundings to the drink, the system will be cooling. I would expect to find the lowest temperature near the point when the ice is melting. This is because melting ice requires the latent heat of fusion, and ice retains it's temperature during this process; so the ice can absorb heat while still remaining at the same temperature. This phase transition represents a fairly large portion of the cooling ability of ice.

The specifics of when this will happen can vary a lot depending on your setup though. Factors such as the geometry of the container and the geometry of the ice can be significant in determining the heat transfer between those objects and their surroundings (for example, surface area effects the rate of heat transfer, and geometry can effect convection currents).

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  • $\begingroup$ Is there some way that I can do a rough approximation though, some formula? I'm currently using a vacuum insulated bottle, if that helps. $\endgroup$
    – Andrew
    Commented Jun 26, 2019 at 14:52
  • $\begingroup$ @Andrew If you want to make really rough assumptions you could assume no heat transfer to the surroundings (or just guesstimate an amount of heat transfer I guess), and then look at the heat capacity, latent heat of fusion, temperature differences, and mass; and determine it that way. It wouldn't give great results, but could give a rough idea about where things would end up in equilibrium under some assumptions. $\endgroup$
    – JMac
    Commented Jun 26, 2019 at 15:04
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about how long should I wait before I can expect the drink to be optimally cold?

That depends on the rate of heat transfer between the ice and the fluid. And that depends (to a large extent) on the exposed surface area of the ice. If you put 20g of crushed ice in a fluid vs a single 20g ice cube, the crushed ice fluid will cool down significantly faster.

Is it coldest right when the ice is about melted, or just after a certain period of time of exposure to the ice?

That's a complex question. As the fluid cools, the heat transfer to the ice slows down, and the heat transfer from the environment speeds up. At some temperature (based on the insulation of the container, the temperature of the environment, and the surface area of the ice), the heat transfer between each is identical. That's the coldest the fluid will get. A large amount of crushed ice might take the fluid to almost the freezing point quickly and stay there for a long time, then climb higher as the amount of ice is reduced. A single cube might start cooling the drink and never get it very cold before the cube is completely melted.

Even measuring (or estimating) the surface area of a quantity of ice is difficult, and understanding how it evolves as the ice melts is even harder. The heat transfer rate can depend on convection currents, which themselves depend on the shape of the vessel. All told, this is a difficult thing to calculate directly. This problem would more likely be attacked by measuring the temperature vs time of a few common configurations and interpolating between them.

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    $\begingroup$ My important takeaways: vacuum insulated containers will make the cooling of water more effective due to keeping room temperature heat from entering in, the surface area of the ice and the temperature of the water over time should be factored in, and the more the ice melts, the lower its surface area and thus the weaker the cooling effect it will have on the water (though the cooler the water may already be). Since the ice and room temperature heat are competing with each other, the optimal temperature of the water would be right at the point where they balance each other out. $\endgroup$
    – Andrew
    Commented Jun 26, 2019 at 18:29

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