How to make the maximum amount of ice melt?

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Madge
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How to make the maximum amount of ice melt?

Postby Madge » Thu Apr 09, 2009 12:17 am UTC

First off, hello xkcd! I see the intro thread is now optional so hopefully you guys won't mistake me for a spambot!(I originally created this account to watch threads, but not I have a scientific question so I'm going to post it)

Anyway, last night I was enjoying a mixed drink (premixed cowboys + chocolate milk + ice), and I'd accidentally put too much of the alcohol into it, making it too strong to enjoy properly. So I added more ice, so that way there would be more ice which could melt and dilute my drink. (Adding milk would have been smarter, but let's pretend we're out of milk)

It got me thinking, though, that having more ice would make the drink colder, meaning the ice would melt more slowly, even though there was more of it.

So, if you have 300ml of a liquid that is at say 277 K, what is the optimum amount of ice you would put in it so that way you would have a maximum amount of water melt into your drink after a given time, t.

You can make any assumptions you wish with respect to the number and size of ice cubes.

It just seems an interesting problem, and probably one that will not have a practical solution. And does the size/shape/material of the cup have much of an effect?

Extra points for generalising it to a volume V of liquid at temperature T with ice cube side length of D.
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Re: How to make the maximum amount of ice melt?

Postby Rachel! » Thu Apr 09, 2009 12:23 am UTC

Madge wrote:generalising it to a volume V of liquid at temperature T with ice cube side length of D.

This is going to vary based on the strength of the drink, no? The amount of alcohol in the drink changes the freezing point of the liquid. And the perception of "coldness" varies based on the lipidity of the drink. A milk- or cream-based alcoholic beverage (say a white Russian) will taste less cold than a water-based beverage (long island iced tea) even at the same temperature.

For a more testable example, put 2 jugs of cold water and cold milk in the fridge. Let em chill for as long as you want (at least a couple hours so you know they've had time to cool down). The milk will not taste as cold as the water.

Very unscientific test, but it works :P

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Re: How to make the maximum amount of ice melt?

Postby gmalivuk » Thu Apr 09, 2009 12:24 am UTC

I think it depends entirely on what temperature the ice is to begin with. If it starts out at freezing, then basically all the heat taken from the drink would go toward melting the ice, and the only difference between more ice and less would be the speed at which this process cools the liquid to the freezing point.

But if the ice is colder than freezing, then the correct amount of ice would be exactly that amount that can be warmed up and melted by the difference in heat between the beverage at room temperature and the beverage at freezing. Any more than that and the liquid will be lowered below freezing before the optimal amount of ice has melted. Any less and obviously you won't have melted as much ice, and also your drink will still be warmer than freezing. The maximum amount of meltage would still be with the at-freezing ice, though.

(All this is assuming the air isn't having any significant effect, of course. What good's a rough physics estimate if you don't assume everything's taking place in a vacuum?)
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Re: How to make the maximum amount of ice melt?

Postby quintopia » Thu Apr 09, 2009 4:53 pm UTC

Another factor is the surface area of the ice. The greater the surface area of the ice the greater the contact it has with the liquid, and since we seem to be assuming that all the heat is being conducted away from the liquid, having this happen more efficiently will ensure a faster rate of melting.

Also, you could melt more ice by not adding all the ice at once, since the ice provides an obstruction to diffusion of liquid and diffusion of heat in liquid. Constantly stirring the beverage will help for the same reason.

The best way to allow for all of the above is to slowly pour crushed ice into the glass while stirring.

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Re: How to make the maximum amount of ice melt?

Postby danpilon54 » Sat Apr 11, 2009 5:34 am UTC

quintopia wrote:The best way to allow for all of the above is to slowly pour crushed ice into the glass while stirring.


I don't know about slowly. Sure a single ice cube might melt slower, but you have more ice cubes. It is really down to total surface area touching liquid. Of course you want the ice at 32 degrees as well. Stirring will help because it will keep the temperature of the liquid even so that you don't get isolated pockets of cold liquid near the ice cubes. I would say fill the glass to the brim with crushed ice while stirring, and drink before too much ice melts.
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Re: How to make the maximum amount of ice melt?

Postby MotorToad » Sat Apr 11, 2009 3:45 pm UTC

This is an interesting conundrum: many variables, a practical but rather unproductive application, and a good thought experiment. From anecdotal evidence (my fridge won't dispense cubes for some odd reason so I've been using crushed ice for the first time ever), adding crushed ice with all its surface area makes the drink too cold to melt the ice. In an insulated cup my crushed ice just doesn't seem to ever melt. I certainly think that maintaining the temperature of the drink as high as possible is crucial to melting the ice... but that's a bit counter-productive as the ice melting takes away all that latent heat of freezing (IIRC 944 BTU/lbm).

I think the answer is to drink as god intended: whiskey, warm with a drop or three of water optional. :D
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Re: How to make the maximum amount of ice melt?

Postby danpilon54 » Sat Apr 11, 2009 4:34 pm UTC

If the liquid gets too cold though then its done its job. You want to melt the most ice, not the highest fraction of it. The liquid has a fixed amount of energy that it can give up before it is 32 degrees. Once it gives that up, no more ice will melt (excluding heating from the atmosphere). So once youve reached the liquid-too-cold scenario you have already succeeded. Come to think of it, the best way to water down a drink is to put water in it...
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Re: How to make the maximum amount of ice melt?

Postby Madge » Sat Apr 11, 2009 11:48 pm UTC

I agree the problem variables were not properly stated.

Good point on the "drink being too cold" implying an end of the experiment, though, since the ice has nowhere to gain the heat from. I guess there's something to be said for holding the cup while the ice melts?

Does this mean that having the drink as cold as possible is the same goal as having as much ice melt as possible?

For the record, I had about 75mL of 15% alcohol and the rest was milk.
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Re: How to make the maximum amount of ice melt?

Postby douglasm » Sun Apr 12, 2009 3:10 am UTC

Madge wrote:Does this mean that having the drink as cold as possible is the same goal as having as much ice melt as possible?

Yes, provided there are no significant sources of heat other than the drink and the ice is of uniform temperature. If you can somehow cool one of the ice cubes down to far below freezing and use that heat energy to melt other ice cubes, that would let you melt more ice, but that doesn't seem very plausible and would require considerably fancier machinery and techniques than simply pouring the ice in and waiting.

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Re: How to make the maximum amount of ice melt?

Postby Mr. Freeman » Sun Apr 12, 2009 9:40 am UTC

Madge wrote:Does this mean that having the drink as cold as possible is the same goal as having as much ice melt as possible?


Not if the ice is below freezing. If the ice is below freezing and you put a sufficient amount in (and let's assume that every point in the ice cube will be at the same temperature), then the ice will heat up, the drink will cool down, and no ice will melt.

Of course, you could put an end to the entire problem by buying some Liquid nitrogen, making a few calculations and dumping the proper amount into your drink. Shouldn't really be that hard. You can assume your house is at 25 deg C, LN2 will be at 77 K until it boils off. This won't water down your drink and the alcohol isn't going to screw with the boiling point of LN2 like it will with the melting point of ice.



On another note, I wish I had a barrel of rum and sugar 300 pounds.
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Re: How to make the maximum amount of ice melt?

Postby BlackSails » Sun Apr 12, 2009 8:02 pm UTC

Wouldnt you just get that the rate of ice melting is equal to the rate at which heat enters the glass?

So assuming that the surface area of the ice is not rate limiting, you get that for every 80 calories that enters the glass from the environment, you will melt 1 g of ice.

Yeah, this assumes that the ice is at 0 degrees, but the energy involved in heating 1 g of water is only 1 calories, while the energy involved in melting 1g of ice is 80 times as much. So unless your ice is really damn cold, it wont matter very much.

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Re: How to make the maximum amount of ice melt?

Postby Zamfir » Tue Apr 14, 2009 11:42 am UTC

I would say that the surface of the ice matters, and stirring, but the actual amount of ice is not that important at all.

If you drop a block of ice in water, than heat in the water goes to melting ice on the outside of the block, but only a small, perhaps even negligible fraction of that heat dissipates to the inside of the cube before the melting itslef has reached the inside. If you put in a larger block with the same surface, the amount of heat going towards ice melting in any fixed time period will stay pretty much the same, so the water cools at the same speed and the amount of ice melted is also nearly the same. The only difference is that the process continues for longer.

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Re: How to make the maximum amount of ice melt?

Postby danpilon54 » Tue Apr 14, 2009 2:46 pm UTC

Mr. Freeman wrote:
Madge wrote:Does this mean that having the drink as cold as possible is the same goal as having as much ice melt as possible?


Not if the ice is below freezing. If the ice is below freezing and you put a sufficient amount in (and let's assume that every point in the ice cube will be at the same temperature), then the ice will heat up, the drink will cool down, and no ice will melt.


In a previous post I mentioned that the ice would have to be at 0 C, but you are correct. You want all of the energy of the drink to go into the phase transition of the ice, so you want it basically at the critical temperature.
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Re: How to make the maximum amount of ice melt?

Postby bigglesworth » Tue Apr 14, 2009 2:50 pm UTC

gmalivuk wrote:(All this is assuming the air isn't having any significant effect, of course. What good's a rough physics estimate if you don't assume everything's taking place in a vacuum?)


Aren't we assuming that the system outside the glass doesn't exist? Because if it were a vaccum, we'd have to account for loss of volume due to liquid evaporating off... I think.
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Re: How to make the maximum amount of ice melt?

Postby Tass » Wed Apr 15, 2009 5:59 am UTC

bigglesworth wrote:
gmalivuk wrote:(All this is assuming the air isn't having any significant effect, of course. What good's a rough physics estimate if you don't assume everything's taking place in a vacuum?)


Aren't we assuming that the system outside the glass doesn't exist? Because if it were a vaccum, we'd have to account for loss of volume due to liquid evaporating off... I think.


More importantly, loss of heat from evaporation. But he was just joking, we asume an isolated system.


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