The physics of a stove-top espresso machine

The operation of a common type of stove-top espresso machine is analyzed to determine the initial fill conditions required to have the coffee extracted in the optimum temperature range of $90°C–95°C$⁠. By using the gas laws and some benchtop experiments, it is shown that the pressure in the vessel increases much more slowly than does the saturated vapor pressure of water at the temperature of the vessel. For any given final temperature, the volume of coffee that can be extracted is linearly proportional to the initial volume of air in the pressure vessel; that is, the higher the fill level, the smaller the volume of coffee that can be extracted. It is also shown that for typical operating conditions for which the water is initially at room temperature, half of the coffee is extracted when the water temperature is below $70°C$⁠, which is much less than the desirable temperature, and that hotter coffee extraction temperatures will result if the water is preheated to about $70°C$ before the pressure vessel is sealed and at least $100ml$ of air space is left in the vessel. Experiments confirming the analysis use easily obtained equipment and are appropriate for undergraduate laboratory work, with the added attraction that students can enjoy consuming the results of the experiments.