Optimize Apple Juice Coldness w/ Minimal Added Water

[GladScientist]

Hi PF. I'm not asking this for academic or professional reasons, but merely out of curiosity.

If I add ONE ice cube to a large glass of warm apple juice, two things will happen
-the cube will melt, fairly quickly, and add its water content to the juice
-the apple juice will be somewhat cooler than before

If I instead add EIGHT ice cubes to the same glass of apple juice, two things will happen
-they will begin to melt, but the melting will be slowed and nearly stopped as the juice cools
-the juice will cool until it is nearly as cold as the ice

Suppose I want my apple juice as cold as possible, but with as little added water as possible. If I add 8 ice cubes instead of 1 ice cube, will I have
-a higher ratio of coldness to melted ice water,
-the same ratio of coldness to melted ice water,
-a lower ratio of coldness to melted ice water

My intuition says that adding more ice cubes will make the juice cool faster, and the ice melt slower, and will thus yield the best result. But I worry that I may be missing variables here.

 

[BiGyElLoWhAt]

I think it would be pretty close to the same. Yea the ice cubes will melt slower, but you have 8 ice cubes melting at the slower rate instead of one at the faster. In order for the juice to give off 30 degrees worth of thermal energy, the ice as a whole (8 cubes or 1 cube or 23 cubes) must take that 30 degrees worth of thermal energy. I would think that this will result in approximately the same amount of melted ice.
However, there is the latent heat of melting to consider. If all the cubes are melting at the same rate, uniformly, you should have more heat going into adding heat at the melting point in order for it to start to melt.

So in summation, probably pretty close to the same, but maybe a little better (for more ice).

 

[Erik S]

Just a quick look at the problem and I would expect it to be the same result at any give cooled temp.

1 cube will melt fast yes, 8 cubes will melt less each, but 8 cubes gives 48 faces that are exposed to the juice, so a little melt off each will quickly Equal the same mass of the one melted cube.

I'm pretty sure the heat transfer balances out, giving the same result. (Neglecting the idea that you take a looong time to drink the juice so that you ahve a lot of heat transfer through the glass walls)

 

[Chestermiller]

Are you asking what happens if I replace (a) a single ice cube of mass m with n ice cubes, each of mass m/n
or
(b) a single ice cube of mass m with n ice cubes each of mass m?

Chet

 

[Khashishi]

Depends how cold the ice cubes and juice were to begin with. Very cold ice cubes will cool down the juice quite a bit even before they reach melting point.

 

[GladScientist]

Are you asking what happens if I replace (a) a single ice cube of mass m with n ice cubes, each of mass m/n
or
(b) a single ice cube of mass m with n ice cubes each of mass m?

Chet

I'm asking the latter of your two questions. Assuming that all "ice cubes" in this question are of equal mass and other properties.

 

[BiGyElLoWhAt]

Drawing from kashashi's point, you would be better off with more ice cubes if the ice cubes are below melting or 0 thermal energy has been expended as latent heat in the melting process, otherwise the scenarios won't differ.

 

[Chestermiller]

Based on my professional experience with heat transfer, the more ice cubes you have, the faster the cooling will take place, primarily because of the increased surface area. The greater the ratio of the ice cube surface area to the volume of juice, the higher the rate of heat transfer, assuming constant heat transfer coefficient between the cubes and the juice. Basically, the amount of heat transferred to lower the temperature of the juice to a certain value will be the same for a single ice cube as for multiple ice cubes, but the amount of time it takes to bring about the transfer of this amount of heat will vary inversely with the amount heat transfer area. This is definitely the first order picture. If you would like quantitative details, I can provide the heat transfer equations and the solution to these equations.

Chet

 

[sophiecentaur]

When trying to produce the coldest cocktails, it is usual to crush the ice (big surface area for heat transfer), shake the mixture and then strain the remaining ice cubes away. Ideally (for the least dilution by water), you would use a lot of ice at as low a temperature as possible and you would make sure that the ice is just at 0C or below. You can buy fancy cooling blocks that are not made of ice and they do not melt.