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Mathematicians Have Devised a Formula That Leads to the Perfect Espresso

Photo credit: Andrew Daniels
Photo credit: Andrew Daniels
  • Research has found that a slightly coarser espresso grind can produce better, faster shots.

  • Commercial burr grinders already make uniform grinds, but too fine and the espresso is slow to make and bitter to drink.

  • The researchers used a complex mathematical model to cover all variables of espresso brewing.


Scientists have identified the golden ratio for ground espresso beans, or more aptly, the Goldilocks ratio: not too small, not too large, but somewhere in the middle.

A team of researchers from around the world studied how espresso behaves at different grind sizes by using a mathematical model.

“We report a novel model, complimented by experiment, that is able to isolate the contributions of several brewing variables, thereby disentangling some of the sources of variation in espresso extraction,” the scientists write. In other words, separating several things about espresso—like water temperature, grind size, and even how hard the ground espresso is “tamped” into the handheld porta-filter.

Co-lead author Jamie Foster told The Guardian that espresso is quixotic: two cups brewed from the same batch of finely ground beans can taste very different, but two cups brewed from the same batch of more coarsely ground beans are more consistent. This goes against conventional coffee wisdom, especially in commercial shops, where a finer grind means more surface area and therefore, we thought, more coffee from fewer beans. But Foster says the tradeoff isn’t worth it if the resulting brewed espresso isn’t predictably good.

Using 24 progressive equations, the research team built an overall model that accounts for all the variables involved in grinding and brewing espresso beans. They named it EY, for espresso yield, “where EY is described as the fraction of solvated mass compared with the total mass of available coffee.” Essentially, it’s calculating how much of the ground espresso is actually being saturated and influencing the brew.

Photo credit: Dr. Jamie Foster et al.
Photo credit: Dr. Jamie Foster et al.

The team’s model predicted that smaller and smaller grind sizes would continue to make higher yields, but in practice, the results were more bell curved, not linear. The smaller grinds ended up falling off in yield. When they looked closely, the team found this was because the particles of espresso were so fine that they smushed together into clogged masses that acted like larger particles, with dry interiors that weren’t available in brewing. The finest grind settings also slowed down the actual time it took to brew an espresso shot.

From the surprising results of their experiments, the team asked three questions. “[F]irst, how can one improve espresso reproducibility given the non-linear dependence of EY on the grind setting? Second, what should one do to reduce shot time or EY variability? And third, can we systematically improve espresso reproducibility while minimizing coffee waste?”

Photo credit: Dr. Jamie Foster et al.
Photo credit: Dr. Jamie Foster et al.

Just Coffee, familiar to any golden age WTF with Marc Maron listeners, recommends a fine, but not too fine, grind. “Too fine a grind will cause a slow, over-extracted shot that can taste bitter and burnt,” the company says on its website. Over-extraction is exactly what the EY research team identified: when a grind is so fine that the tiniest particles are brewed to dissolution while larger clogs aren’t brewed at all. The overall flow is heterogeneous and unpredictable, with too much of the harshest flavor notes from the over-extracted tiny particles.

Foster’s team brainstormed a variety of ways that the economics of espresso could balance with the best outcome, maximizing consistent good results while minimizing the wasted and untouched coffee in a finely ground and “partially clogged” shot. When they introduced a new mix of espresso to a coffee shop in Eugene, Oregon—where one of the researchers is based—they found they saved 25 percent espresso waste and also shortened the time workers spent brewing each shot. Year over year, Foster says, the shop saved thousands of dollars in beans and labor.

Want to nerd out over all the numbers? Check out the full study for the exact mathematical equations the scientists used. Crunch ‘em over a fresh cuppa Joe.

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