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When to Add Cream & Sugar - Coffee Thermodynamics

December 2001

In these winter months people start drinking more hot beverages, and I get more questions about the best ways to keep them hot. One such question is: when you will be drinking your coffee some time after pouring it (e.g. buy it in the cafeteria, then take it to your office to drink), should you add your cream and sugar immediately, or just before drinking it?

While I must admire the spirit of those who have thought this out and consequently take the extra effort to carry sugar and cream packages off with them to add later... I must say that every argument I've been able to apply says they shouldn't bother. The basic math says you find the temperature immediately after the mixing by T=(mcoffeeTcoffee+mcreamTcream)/(mcoffee+mcream) where the m & T refer to mass and temperature of each (for simplicity, this is considering the thermal properties of the cream & coffee to be identical).

For example, consider the coffee at 80oC and the cream at 20oC, and add 1/2 oz cream to 7.5 oz coffee:
T = (7.5*80 + 0.5*20) / (7.5 + 0.5) = (600 + 10) / (8) = 610/8 = 76.25oC

Consider the same problem with the coffee starting a bit colder - 70oC:
T = (7.5*70 + 0.5*20) / (7.5 + 0.5) = (525 + 10) / (8) = 535/8 = 66.875oC

In the first instance the cream dropped the temperature by 3.75 degrees, in the second instance by only 3.125 degrees. This should seem sensible - if the temperatures are closer, the cream will have less effect on the coffee's temperature. At first this seems an argument for adding the cream later, it has less effect since the coffee has already cooled somewhat. However, this argument fails to treat the cream and coffee as a single system properly.

The cream and coffee have a certain total mass which is losing energy to the surrounds, thus lowering the temperature. The temperature of the mixture will be given by the total energy remaining. So, the critical factor is not when they are mixed but how fast energy is being lost to heating the environment in each case. Examine the various contributions (evaporation, black body radiation, conduction, convection)... in each case energy is transfered faster from a hotter object than from a colder object. So, by adding cream/sugar at the beginning the mixture achieves a lower temperature immediately, which means a slower rate of transfering energy to the surrounding environment. Losing energy slower means the average temperature of the combined coffee/cream/sugar will be higher.

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