Trivial Question - rate of heat loss from hot drink

[Francescos123]

Hello people of PhysicsForum

Apologies in advance if my prefix selection is inaccurate and for my absolute physics noobness. I know no physics jargon so I'm sure my question will be phrased in the same way a 10 year old might.

So here is my silly, trivial question:

First of all, remove all variables (apart from the ones stated). Same time to pour and same materials etc.

You have 2 cups of hot water (making cups of tea is what brought this thought to me)

Cup 1 & 2 - Pour hot water in (same time etc. (no variables))

Cup 1 - Pour in precise amount of milk (or other cold substance straight out of fridge) let's just say 30 ml. I'm not sure if precise temperature of milk matters, but let's say 5 celsius.

Cup 2 - Leave, do nothing.

* After 5 minutes *

Cup 2: Pour in milk, same as poured into cup 1

At this point, would there be any difference in temperature between the liquid in the cups? I think Cup 2 would end up cooler.

I've asked a few people and am getting mixed views, majority say no difference. Something along the lines of you can't create energy you can only distribute it or something.. Again, as I'm sure you can tell, I am fizix noob.

My logic in thinking that the liquid in the cups would have a different temperature is that the rate of cooling is determined by the difference in the room temperature and the temperature of the liquid, so Cup 2 would be cooler in the end because it would have cooled down faster as there was no cold liquid to drop the temperature, until it had cooled for 5 mins. Then with the milk added it would drop below the temperature of Cup 1. But maybe the milk would affect the temperature less as after 5 minutes the temperature difference between the milk and the liquid would be lower so it wouldn't drop the temperature much?

separately:

Surely if i had 2 cups with water at temp. 70C in each and placed one in a room at 45C and the other in a room at 0C the cup in the hotter room would cool down slower...

Thanks for reading, don't laugh at me!

Any responses would be much appreciated

 

[Bystander]

You have 2 cups of hot water (making cups of tea is what brought this thought to me)
Cup 1 & 2 - Pour hot water in (same time etc. (no variables))
Cup 1 - Pour in precise amount of milk (or other cold substance straight out of fridge) let's just say 30 ml. I'm not sure if precise temperature of milk matters, but let's say 5 celsius.
Cup 2 - Leave, do nothing.
* After 5 minutes *
Cup 2: Pour in milk, same as poured into cup 1
At this point, would there be any difference in temperature between the liquid in the cups? I think Cup 2 would end up[insert]be[/insert] cooler[insert]at the five minutes[/insert].

 

[PeroK]

Hello people of PhysicsForum

Apologies in advance if my prefix selection is inaccurate and for my absolute physics noobness. I know no physics jargon so I'm sure my question will be phrased in the same way a 10 year old might.

So here is my silly, trivial question:

First of all, remove all variables (apart from the ones stated). Same time to pour and same materials etc.

You have 2 cups of hot water (making cups of tea is what brought this thought to me)

Cup 1 & 2 - Pour hot water in (same time etc. (no variables))

Cup 1 - Pour in precise amount of milk (or other cold substance straight out of fridge) let's just say 30 ml. I'm not sure if precise temperature of milk matters, but let's say 5 celsius.

Cup 2 - Leave, do nothing.

* After 5 minutes *

Cup 2: Pour in milk, same as poured into cup 1

At this point, would there be any difference in temperature between the liquid in the cups? I think Cup 2 would end up cooler.

I've asked a few people and am getting mixed views, majority say no difference. Something along the lines of you can't create energy you can only distribute it or something.. Again, as I'm sure you can tell, I am fizix noob.

My logic in thinking that the liquid in the cups would have a different temperature is that the rate of cooling is determined by the difference in the room temperature and the temperature of the liquid, so Cup 2 would be cooler in the end because it would have cooled down faster as there was no cold liquid to drop the temperature, until it had cooled for 5 mins. Then with the milk added it would drop below the temperature of Cup 1. But maybe the milk would affect the temperature less as after 5 minutes the temperature difference between the milk and the liquid would be lower so it wouldn't drop the temperature much?

separately:

Surely if i had 2 cups with water at temp. 70C in each and placed one in a room at 45C and the other in a room at 0C the cup in the hotter room would cool down slower...

Thanks for reading, don't laugh at me!

Any responses would be much appreciated

I think there was actually one couple who got divorced because they couldn't agree on this question!

Does the milk stay in the fridge?

 

[russ_watters]

Thanks for reading, don't laugh at me!

Looks to me like everything you said is correct, so no laughter needed!

 

[Francescos123]

Looks to me like everything you said is correct, so no laughter needed!

Thanks for the response, nice to hear!

So.. You're saying cup 2 would be cooler?

 

[Francescos123]

I think there was actually one couple who got divorced because they couldn't agree on this question!

Does the milk stay in the fridge?

Yeah, milk stays in the fridge, exactly same temperature when poured into each.

 

[PeroK]

Yeah, milk stays in the fridge, exactly same temperature when poured into each.

In that case, cup 2 would be cooler, because the milk is kept cool in the fridge. Consider, instead, these scenarios:

Cup of hot water with small portion of cold milk added, left for 5 mins.

Cup of hot water and small portion of cold milk, left for 5 mins (outside the fridge), then added.

Which mixture would be cooler, if any?

 

[Francescos123]

In that case, cup 2 would be cooler, because the milk is kept cool in the fridge. Consider, instead, these scenarios:

Cup of hot water with small portion of cold milk added, left for 5 mins.

Cup of hot water and small portion of cold milk, left for 5 mins (outside the fridge), then added.

Which mixture would be cooler, if any?

They would be the same temperature, as the rise in milk temperature from being left outside of the fridge would negate the faster fall of water with no milk added ?

 

[PeroK]

They would be the same temperature, as the rise in milk temperature from being left outside of the fridge would negate the faster fall of water with no milk added ?

Yes, exactly.

You do have to make an assumption that the mixture cools at the weighted average of the two ingredients - which is a fairly natural assumption. It's possible, of course, that a mixture of water and milk might be better or worse at holding onto its heat than the two ingredients separately. In which case, it would obviously make a difference.

 

[Cutter Ketch]

The second cup will definitely be colder. Consider an extreme case: 2 thimbles worth of hot tea and two gallons worth of milk. The tea is irrelevant, and the milk that sat out on the counter for 5 minutes is clearly warmer than the milk that sat in the refrigerator for that time. Now take the opposite extreme case with almost no milk. The milk is irrelevant and both teas are at the same temperature. At all other ratios the temperature difference varies linearly between these two extreme cases. The second cup is always colder and you only approach equal as the amount of milk goes to zero.

This can be proven mathematically, but not when you are typing on an iPhone!

 

[jbriggs444]

Yes, exactly.

You do have to make an assumption that the mixture cools at the weighted average of the two ingredients - which is a fairly natural assumption. It's possible, of course, that a mixture of water and milk might be better or worse at holding onto its heat than the two ingredients separately. In which case, it would obviously make a difference.

[This relates to the scenario where one compares the final temperature for one cup of mixture sitting on the counter against that of one cup of mixture from two cups sitting separately on the counter. It does not relate to the scenario where the milk stays in the fridge]

The surface area exposed to heating or cooling is smaller for the single cup of mixture than for the aggregate for the two cups separately. That factor tends to increase the final temperature of the single cup. It equilibriates with room temperature more slowly since its surface to volume ratio is smaller.

The cooling rate resulting from evaporation is non-linear. It is more than proportional to temperature. Accordingly, the separate cup of hot water will lose disproportionately more heat. This factor tends to decrease the average final temperature of the separate cups.

Both factors favor the premix. Accordingly, I would predict that in this scenario, the pre-mixed beverage stays hotter longer.

 

[PeroK]

The cooling rate resulting from evaporation is non-linear. It is more than proportional to temperature. Accordingly, the separate cup of hot water will lose disproportionately more heat. This factor tends to decrease the average final temperature of the separate cups.

Even so, if all other factors are equal, if you do the maths for the exponential rate of cooling, the time of mixing does not affect the final temperature.

 

[jbriggs444]

Even so, if all other factors are equal, if you do the maths for the exponential rate of cooling, the time of mixing does not affect the final temperature.

The exponential rate of cooling assumes that the incremental heat emission rate is a linear function of temperature difference. I am telling you that the actual heat emission rate in this case is not a linear function of temperature difference.

 

[PeroK]

The exponential rate of cooling assumes that the incremental heat emission rate is a linear function of temperature difference. I am telling you that the actual heat emission rate in this case is not a linear function of temperature difference.

Sorry, I misunderstood.

It's easy to see how this issue of when to add milk to your spouse's tea or coffee can end in divorce!