So I’m making tea for my lunch, and the thought occurs to me: Should I pour in the milk right away, or should I let it cool first? The idea being to reduce the time until the tea is at a drinkable temperature. The heat transfer to the surrounding atmosphere depends on the difference between the tea and ambient temperatures, so hot tea cools faster in terms of joules transferred per second. On the other hand, milk-cooled tea starts closer to the target temperature.
We’ll suppose I have about 9 units of tea to one of milk, that the tea starts at 100 degrees, and that I want to bring it down to say 50 – hotter than blood but not uncomfortable to the touch. Milk begins at something like 5 degrees. So, if I mix in the milk right away, I’m starting at 90.5 degrees. Temperature as a function of time is
T(t) = T_0 * exp(-kt)
where k is an unknown constant which depends on the geometry of my cup. Taking T_0 as 90.5 degrees and solving for T(t)=50, we find t=0.5933k.
Now, if I let the tea cool first and then pour the milk, the target temperature from cooling is no longer 50 degrees, but rather that temperature which will average to 50 when a unit of milk at 5 degrees is added to it, which comes to 55 degrees. So we’re solving the same equation with T_0 of 100 and T(t) at 55, giving t=0.5978k.
Well then! Pour the milk in first, it’ll save a good three seconds off the cooling time!
But wait, I assumed a constant k, and that may not be true. When I add milk, the shape of the heat-transferring volume changes a bit, and who knows what that’ll do to k? Not me; this is the sort of problem one studies with very finely-grained simulations. To first order, however, my cup is a cylinder. When I add to the level of liquid in it, the surface-to-colume ratio decreases. (For sufficiently long cylinders, one can ignore the endcaps and the S/V ratio depends only on the radius, but my cup is quite squat so this approximation does not work.) So adding milk probably increases k, and makes pouring the milk in later the superior option after all.
Now then, lunch. And my tea is exactly at the right temperature.