Job interview question: 2 glasses of coffee (Heat Transfer)

[berko1]

I tried searching for this but i couldn't find an answer so i hope this is hasn't been discussed and if so - sorry- and id appreciate a link..

Here we go:

Lets say we have to glasses, A and B. In glass A we pour boiling water and add milk straight afterwards. Now for glass B we pour the boiling water then wait 3 min and then add the milk. Now we take temps of both.
Which of the two mixtures will be hotter after the three min? Explain.

My solution:
We all know that the heat transfer is dependent on the temp diff - therefor the glass without the milk (B) losses more heat to its surroundings during the 3 min.
But- now that we add the milk to glass B, the temperature difference is lower and the milk "cools" the water less then it cooled the hot water in A (At the beginning the temp diff was bigger).
- At this point i just picked one and didn't really back up my choice, i said that it depends which is has a more crucial effect..

Question:
Which has more of a effect? the higher temp between the glass and surroundings or the milk cooling that relies on the temp diff?

**The milk is stirred, the same amounts and relations for A and B, the milk stays in the same temp (refrigerated).

 

[verty]

This is a good question. But think about what effect the milk has. It just increases the volume and adds a set amount of heat energy. It does this whether added at the start or the end. Why should the milk have greater cooling effect?

 

[berko1]

The milk heat transfer is also dependent on the temp diff.
Q=mCp(Twater-Tmilk)
If we add the milk at the start: Twater=100 degrees c and Tmilk =15 degrees c. The diff is 85.
If we wait three min the Twater will be less, let's say 90 degrees while the milk will still be 15 degrees, and so the difference between Twater and Tmilk will be smaller thus having less heat transfer resulting in less "cooling".

Right?

 

[Delphi51]

I had a little trouble with this, too. Let me try to give verty's thought another way.
There is a certain amount of heat energy in the original water + milk. The final temperature will be determined by how much heat is lost to the outside world. As you said, in A more heat is lost due to the higher delta T. It doesn't matter how much heat is transferred from water to milk in either case because the two are mixed together - only their total heat energy divided by the total mass matters.

 

[BruceW]

Delphi's got the right idea. It doesn't matter how much heat is transferred from the water to the milk, because both end up as the coffee mixture.

 

[berko1]

Got it. Nice.

Thanks!

 

[verty]

How I think of it is, adding the same milk should have the same change. We should not think that the same milk could effect different change.

So something changes by the exact same amount in both cases, what could it be? The total heat energy of the mixture, of course.

So let A be the energy of the water, B the energy of the milk. In the first case, we have (A+B) - losses. In the second case (A - losses) + B. This looks like it should be equal, but the losses are not the same.