thermodynamics - Would a Cup of Tea Be Hotter If you Add the Milk Before or After Boiling Water?

This is a bit of dispute between work colleagues. An answer would be greatly appreciated.

My argument is as follows:

If you add X amount of milk at a temperature of M to a mug at room temperature R before adding X amount of water at temp W, the result would be a cooler cup of tea than if you'd added the hot water first. This would be due to the milk reducing the overall temperature of the mug in the time that has been in it resulting in the addition of boiling water having less of an effect and resulting in a slightly cooler cup of tea/coffee.

Conversely, if you add the hot water first, due to the mug being heated, the effect of adding the cool milk will be less and therefore the end result will be a hotter cup of tea.

My Colleagues arguement is simply that it would not make a difference but without any form of justification.

If anyone can give real scientific light to this issue it would be greatly appreciated :)

Answer

We can make the following simplifying assumptions:

1)The the milk/water cool according to Newton's law of cooling/Fourier's Law(http://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling).

2)The effect of adding milk is an instantaneous drop in temperature of solution by a fixed amount $\Delta T$.

Let the initial temperature of the water be $T_0$ and the temperature of the solution(water or water+milk) as function of time be T.

Case 1:Milk is added at the end of the experiment

$\frac{dT}{dt}=k(T_{env} - T)$

where $T_{env}$ is the temperature of the environment.

The solution for $T$ is $T(t) = T_{env} + (T_0 - T_{env}) e^{-k t}$

After adding milk at time $\tau$ according to our assumption 2,we get the temperature of the solution as:

$T_1(\tau) = T_{env} + (T_0 - T_{env}) e^{-k t} - \Delta T$

Case 2:Milk is added at the beginning of the experiment

The only change from case 1 would be that the initial temperature of the solution would now be $T_0 - \Delta T$ instead of $T_0$.

So the solution fot $T$ will be :$T(t) = T_{env} + (T_0 -\Delta T- T_{env}) e^{-k t}$

Therefore after time $\tau$ the temperature of the solution will be:

$T_2(\tau) = T_{env} + (T_0 -\Delta T- T_{env}) e^{-k t}$

So finally we have,$T_2(\tau)-T_1(\tau)=\Delta T (1-e^{-k t})$

Now for all $t>0$,$(1-e^{-k t})$ is always positive.

So $T_2(\tau)>T_1(\tau)$ always.

Moral of the story:"If you want hot tea,add milk in the beginning!"

Note:Here we assumed Newton's Law of Cooling was valid which was somewhat a simplistic assumption which may not be true in the real world.