It is not known how much espresso the mathematician William Lee drinks a day. But the brew obviously appealed to the University of Huddersfield researcher. With the statistician Ann Smith, the scientist has repeatedly analyzed how coffee can be brewed in the best possible way. And now the two are targeting what appears to be a decisive factor: the so-called extraction, i.e. the extraction of the soluble ingredients from the ground coffee beans. It takes place when pressurized water flows through the bed of coffee powder. As Lee and Smith now, together with young researcher Arsalaan Arshad im journal physics of fluids report, this extraction is often uneven. And so – probably a nightmare for mathematicians – the result becomes difficult to calculate.
Mathematically modeling espresso brewing is not trivial, on the contrary: With more than 2000 ingredients and aromas, coffee is one of the most complex beverages there is. It is almost as difficult to tease out the best flavors from the beans with a portafilter machine. A total of 30 percent of coffee is water-soluble, mainly caffeine and essential oils. With an espresso, however, only 18 to 22 percent should be extracted. According to Lee and Smith, the espresso tastes either bland or bitter depending on whether the brewing water has flowed through too slowly or too quickly.
The coffee beans must not be ground too finely
In order to prevent this, there is a common recipe in practice: An espresso shot is made from roasted beans of the Robusta or Arabica coffee plant by running water at a temperature of 95 degrees Celsius at around ten times the atmospheric pressure through a bed of fine – but not too fine! – presses ground coffee powder. Only about 25 milliliters of water go into a simple espresso, Baristi recommend eight to nine grams of powder. “The optimal ratio of espresso powder and water is 1 to 2.5,” explains Felix Killmayer, head of the Dallmayr Academy in Munich, where barista courses are offered. “The flow time should be 20 to 30 seconds.”
In order to decode the physics behind it, the two mathematicians chose a simplified model: It is based on only two possible paths that the water can take when flowing through. They assigned different porosity factors to the paths, which means that the coffee powder is more densely packed in one of the two path regions than in the other. The result: The extraction yield depends on the grain size. Accordingly, it is not conducive to enjoying the espresso if the coffee beans are ground too finely. In addition, the two scientists came across a kind of tipping point in the coffee. If the water flows quickly through the powder, it draws out the flavorings faster, which in turn reduces the resistance to flow: a feedback loop.
Another thing their results show is that even if the espresso tastes good, i.e. if the overall yield of the extraction seems to be on the right level, it does not necessarily mean that the brewing process was optimal. It could also be an accidental product of underdeveloped and bitter coffee, Lee says. The scientists therefore want to further improve their model in order to be able to prevent uneven flow. You might have to pay attention to a balanced aspect ratio or change the dimensions of the sieve in order to make even better coffee in the future – and save coffee powder at the same time.