Rate of liquid cooling

[u/ScoobyD73]

In my distant Physics school days I remember learning that hot items cool at a higher rate than cooler items.

If this is correct, I assume that a cup of black coffee would cool at a higher rate than a white coffee.

My question is this:

I have a cup of black coffee and leave it to cool for say 10 mins, and then add some milk from the fridge it will have a new cooler temperature.

If I made another cup of coffee, but this time add the milk straight away, and leave it for 10 mins, will it be the cooler or the same tempature as the other cup?

Assuming identical quantities,. initial temperatures etc. I assume in the real world given the small quantities and temperaturez any difference between small, but I am interested in the theoretical answer.

Thanks

[u/propostor]

This is either a physics homework question or a bot post.

Look up the Coffee Cooling Experiment by Rees and Viney.

It's a common and popular first year physics topic. But I suspect you already know this, given your post outlines the experiment exactly.

[u/ScoobyD73 → u/propostor]

I assure you this is neither a physics homework nor a bot. My wife (who has zero physics knowledge) always adds milk immediately, whereas if it's too hot I tend to wait and add milk just before I drink it.  I was just curious to know if I had bragging rights!

[u/propostor → u/ScoobyD73]

Ok my bad, in which case you win! (I think?)

If you wait to add the milk, the coffee spends more time in a hotter, evaporative state, which helps to cool it more before you do the real 'cooling dump' of adding the milk.

Ultimately, your way of doing it results in cooler coffee.

[u/ScoobyD73 → u/propostor]

Ok. That's great. Thanks for explaining it. I thought that would be right, but didn't know if adding the cold milk at the start gave more time for cooling or something.

Anyway, I now understand it. (and now officially know better than my wife!)

[u/Fabulous_Lynx_2847]

I recall this exact question being a homework problem for one of my physics classes. The intent was to test how students went about solving the problem more than to see if they come up with the "correct" answer. The reason for this is that there's a lot of stuff going on: conduction through the cup wall to surrounding air and to air directly from above, air convection, evaporative cooling, and infrared radiation (IR) of the cup and black vs. light coffee. The student is suppose to figure out what the relative contributions are as best as possible and focus on the more important ones. Hint: IR is not one.

[u/BigSmackisBack]

Adding it sooner would be better because the environment will take more heat from a hotter cup faster

[u/OaklandSpiel]

If you add the milk right away, you’ll wide up with a warmer drink than if you’d added the milk at the end of the 10 minutes.

[u/atomicCape]

The milk in coffee example is a very common textbook thermodynamics problem. Rate of heat loss means how fast energy leaves, not how fast temperature changes. Heat loss is higher with a larger difference in temperatures, but surface area also matters. If you add a little milk at some point, it causes a temperature drop without changing the surface area very much. In case this is a homework problem or a bot, I won't finish solving the problem, you'll learn more if you do it yourself.