Determining the most efficient way to cool down a drink that is too hot.
THE MOST COMMON EVERYDAY EXPERIENCES give rise to practical questions. Is it true that running, rather than walking, under the rain will keep you drier? Does it really feel cooler to wear white clothes on a sunny day? Does water always drain out from a bathtub in a clockwise direction in the Northern Hemisphere? Testing such questions experimentally is a simple matter, but we are rarely willing to go to the trouble.
Small mysteries of this sort abound in cooking. Take the coffee we drink every morning. It is always boiling hot, and we are never sure how to cool it down. Some great minds have taken an interest in the question, including the British physicist Stephen Hawking. According to Hawking, if you want to avoid burning your mouth you should wait a few moments before adding sugar, assuming you sweeten your coffee, because it will cool down more rapidly than if you add the sugar right away.
What is the theoretical basis for this prediction? Stefan’s law, which stipulates that the heat radiated by a body per unit of time is proportional to the fourth power of its absolute temperature, tells us that hot bodies radiate more energy than cold bodies. In allowing the coffee to cool down by itself one profits from its higher temperature, and therefore from its more intense radiation; dissolving the sugar afterward completes the process. Putting the sugar in first, on the other hand, has the effect of immediately cooling the coffee, so less advantage is derived from its radiation. This is plausible enough, but is it true?
Theory to the Test
Experiments (all the ones mentioned here were done under controlled conditions that very closely reproduce actual experience) show that the effect predicted by Hawking is too weak to be observed. Moreover, they reveal that the cup plays an important role: As cooling begins, the cup is reheated, but because it gives off heat only by conduction its temperature falls less rapidly than that of the liquid, whose cooling it subsequently retards.
What if one were to add cold milk instead of sugar? Once again the experiment could hardly be simpler: Make a very hot cup of coffee, then add milk at room temperature and measure the rate of cooling. Next, compare this result with what happens when a very hot cup of coffee is allowed to cool beforehand and room temperature milk added at a later stage.
This time the theoretical prediction is correct. For 20 centiliters (6.75 oz.) of boiling coffee that is cooled initially with 7.5 centiliters (2.5 oz.) of room temperature milk, a comfortable drinking temperature of 55°C (131°F) is obtained after about 10 minutes, but if one waits for the coffee to reach 75°C (167°F) before adding the milk, one obtains the same result after only four minutes. Knowing an elementary law of physics reduces the wait by more than half.
Teaspoon and Radiator
Where does this leave those who take neither sugar nor milk? Should they put a teaspoon in their coffee, reasoning that the heat of the coffee will travel through the metal of the spoon and thence radiate into the atmosphere?
This time let’s skip theoretical calculations, however interesting they may be (for the first time in my life I was able to use Clairaut’s equation, which as a student I had worked so hard to learn) and use a thermocouple to measure the actual effect instead. We will see that the effect of the teaspoon is virtually nil: When one takes two cups containing coffee at an initial temperature of 100°C (212°F) and puts a teaspoon in one of them, the difference in temperature after ten minutes is less than 1°C (or 1.8°F). A teaspoon is not an efficient radiator, even when it is made of silver.
Coffee cools slowly, but what about tea? The difference in color between coffee and tea affects the degree of radiation in principle, but in practice it does not affect the rate of cooling because the effect is too small to be experimentally observed: Assuming identical cups and identical quantities of liquid, the cooling curves coincide exactly. On the other hand, a large bowl of tea may cool more slowly than a small cup of tea, but that is another story.
Let’s conclude by examining the intuitive behavior of drinkers of hot beverages everywhere. Is blowing on coffee an efficient way of cooling it? What about stirring it with a spoon? Experiment shows that a boiling liquid that cools spontaneously by 6°C (11°F) per minute cools by 11°C (20°F) per minute when one stirs and one blows on it at the same time. Which is more effective: stirring or blowing?
Stirring has two effects: It makes the temperature of the liquid throughout the cup uniform and, by bringing the hottest part up to the surface, accelerates cooling. Stirring therefore increases the area over which energy is exchanged between the hot liquid and the cold air. A further consequence of this ventilation is that it expels from the liquid the fastest-moving molecules, which have now entered into the steam phase. This means that they are not recycled through the rest of the liquid, with the result that its average molecular kinetic energy is reduced. Because temperature measures exactly this, the average velocity of molecular agitation, a lower velocity means that the liquid has cooled. Let’s blow vigorously on the coffee in one cup while vigorously stirring the contents of another cup, trying in the latter case to maximize the surface area of contact between air and liquid, as the theoretical description of the problem requires.
Question: What will we observe? Answer: Blowing is much more efficient than stirring. The same coffee that loses 6°C (11°F) per minute when one blows on it cools down only by 3.5°C (6°F) when it is stirred.