## Tea with (Almond) Milk

Making a cup of tea in a hurry is a challenge. I want the tea to be as drinkable (cold) as possible after a short amount of time. Say, 5 minutes. What should I do: should I add milk to the tea at the beginning of the 5 minutes or at the end?

The rule we will use to work this out is Newton’s Law of Cooling. It says “the rate of heat loss of the tea is proportional to the difference in temperature between the tea and its surroundings”.

This means the temperature of the tea follows the differential equation $T' = -k (T - T_s)$, where the constant $k$ is a positive constant of proportionality. The minus sign is there because the tea is warmer than the room – so it is losing heat. Solving this differential equation, we get $T = T_s + (A - T_s) e^{-kt}$, where $A$ is the initial temperature of the tea.

We’ll start by defining some variables, to set the question up mathematically. Most of them we won’t end up needing. Let’s say the tea, straight from the kettle, has temperature $T_0$. The cold milk has temperature $m$. We want to mix tea and milk in the ratio $L:l$. The temperature of the surrounding room is $T_s$.

Option 1: Add the milk at the start

We begin by immediately mixing the tea with the milk. This leaves us with a mixture whose temperature is $\frac{T_0 L + m l }{L + l}$. Now we leave the tea to cool. Its cooling follows the equation $T = T_s +\left( \frac{T_0 L + m l }{L + l} - T_s \right) e^{-kt}$. After five minutes, the temperature is

Option 1 $= T_s +\left( \frac{T_0 L + m l }{L + l}- T_s \right) e^{-5k} .$

Option 2: Add the milk at the end

For this option, we first leave the tea to cool. Its cooling follows the equation $T = T_s + (T_0 - T_s) e^{-kt}$. After five minutes, it has temperature $T = T_s + (T_0 - T_s) e^{-5k}$. Then, we add the milk in the specified ratio. The final concoction has temperature

Option 2 $= \frac{(T_s + (T_0 - T_s) e^{-5k}) L + m l }{L + l}.$

So which temperature is lower: the “Option 1” temperature or the “Option 2” temperature?

It turns out that most of the terms in the two expressions cancel out, and the inequality boils down to a comparison of $e^{-5k} (T_s L - ml)$ (from Option 2) with $(T_s L - ml)$ (from Option 1). The answer depends on whether $T_s L - ml > 0$. For our cup of tea, it will be: there’s more tea than milk ($L > l$) and the milk is colder than the surroundings ($m < T_s$). [What does this quantity represent?] Hence, since $k$ is positive, we have $e^{-5k} < 1$, and option 2 wins: add the milk at the end.

But, does it really make a difference? (What’s the point of calculus?)

Well, we could plug in reasonable values for all the letters ($T_0 = 95^o C$, etc.) and see how different the two expressions are.

So, why tea with Almond milk?

My co-blogger Rachael is vegan. She inspires me to make my tea each morning with Almond milk.

Finally, here’s a picture of an empirical experiment from other people (thenakedscientists) tackling this important question: