After I pour a cup of tea, I can't drink it for 10 minutes. Will it stay warmer if I add the milk at the start, at the end or does it not matter? [duplicate]

Question

So preface, I work in engineering so I'm familiar with the basic principals of most fields in physics, but specialise in Civil Engineering so my knowledge of thermal physics is poor and I haven't studied it since high school.

I have this recurring problem at work where I boil a kettle and pour myself a cup of tea. During the brewing process, I sometimes get interupted by a collegue who wants me to solve a quick problem, taking no more than 10 minutes of my time. I like to drink my tea at my desk while browsing the web so my tea will sit untouched for this time. I take my tea with milk.

So my question:

If I have already brewed my tea and have 10 minutes or so where I know I will not be able to drink it, should I;

1. Add the milk immediately after brewing

2. Add the milk immediately before drinking

To maximise the temperature of the drink at the time of drinking.

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To me this is a closed system and it shouldn't matter what temperature the tea is when the milk is added as the same amount of time will elapse overall and the start values for each liquid is the same.

But also if I add the milk early, it has more time to 'warm up' from fridge temperature.

But if I don't add the milk early, then the tea is at a higher start temperature and intuitively to me will drop in temperature at a faster rate.

The tea can probably be assumed to be at around 95C at the start and the milk 4C from the fridge.

Answer 1

This is a good question and appears in various popular science books, thermodynamics textbooks etc.

I won't give the complete answer because it can easily be looked up or found in the questions posted in the comments. I'll give you a clue to help you solve it yourself.

What is the equation that (at least approximately) describes the rate of cooling? Newton's law of cooling says: $$\frac{dT}{dt} = - \kappa (T - T_0), $$ Where $T_0$ is the temperature of the ambient surroundings (room temperature in this case) and $\kappa$ a constant representing how quickly the system looses heat per unit difference in temperature from $T_0$. I leave it to you to verify the solutions are exponentially decaying towards this final temperature but its not necessary for what follows.

Note that the rate of cooling is greater for greater temperature differences. So to minimise the rate of heat loss you want to reduce the difference. Now compare two situations: leaving a boiling cup of tea for ten minutes to cool, then adding milk or adding the same amount of milk at the same temperature, this lowering the initial temperature, and then allowing to cool for ten minutes. What do you think happens?

Of course you could just do the experiment one day with two cups of tea (equal amount of water and milk at same temperature etc) and a thermometer....