Espresso as a 3-D surface

The setup

For a given bean at a given roast, the barista's fundamental choices are grind size, ratio, and shot time. Temperature and the pressure profile are manipulable too on some machines. I wanted to plot the ‘shape' of espresso variables and where on that shape good espresso sits.

The variables involved in espresso are not independent. When one of the fundamental variables is adjusted it constrains the other two. This is the reason that there is a geometric plane of possible espressos.

The trilemma

These variables create a trilemma. That is: the barista can only ever choose two of them ahead of time.

This is because of how grind size constrains shot time and ratio, so they are not three independent choices. The mechanisms behind grind size are what is called puck resistance, and surface area. The lower the grind size, the smaller the spaces are for water to travel through, causing greater resistance for finer ground coffee than coarser ground. This leads to increased pressure, and longer shot times to achieve a given yield. Additionally finer grinds create more surface area of coffee exposed to water, which leads to greater extraction.

That leaves three variables and one constraint: two degrees of freedom. Choose any two and the third is determined. The standard espresso settings make this explicit. Fixing the ratio at 2:1 and the time at around 30 seconds, then experiment with the grind setting that satisfies both at once. If you used pre-ground coffee, conversely, the grind is fixed for you, so you control either time or ratio, never both.

The relation has been scientifically quantified. Darcy's law for flow through a porous bed, combined with the Kozeny–Carman estimate of permeability, makes flow rate scale with the square of particle diameter. Integrate over the shot and you get, roughly, ratio ≈ C · d² · t. Take logarithms and the feasible set is a plane in (log d, log t, log ratio) space. Everything off that plane is impossible to produce. A coarse grind pulled to 2:1 in fifty seconds does not exist; the water is long gone through the puck before the clock gets there.

Which of these variables you actually control is a property of the machine, not of espresso. A pump machine at a fixed over-pressure valve pins the pressure created by water forced out of the machine meeting the resistance of the puck. A lever machine, or a machine with pressure profiles, lets you set pressure directly, but flow is still an emergent property. A flow-control paddle machine changes that: you set flow, and pressure is the emergent property instead.

Secondary variables

Extraction, intensity, and the pressure at the puck are all secondary variables, caused by the combination of the choices above. Extraction yield is the fraction of the ground coffee that dissolved. Intensity, measured by total dissolved solids (TDS), is just extraction divided by ratio, which is the same dissolved coffee spread across more or less water. So once you know extraction and ratio, you can derive intensity.

Extraction and intensity are the axes of the standard brewing control chart, and from those two numbers you can get a very complete picture of the cup.

Holding the machine and any other exogenous variables fixed, the whole space flattens onto one plane, with grind size against shot time. Ratio, extraction, intensity, all become a field over that plane. Here is that field, with pressure still adjustable.

Drop the pressure to 6 bar and watch the channelling risk at 300 µm nearly vanish, but the surface sinks with it. At 6 bar, 300 µm now needs about 45 seconds to reach 2:1, and much of the plane falls out of the good band.

Traditional versus turbo

Traditional and turbo shots sit at roughly the same extraction, at approximately 21%. The difference is intensity. Traditional runs near 2:1, so TDS lands around 21/2 ≈ 10.5%. Turbo runs longer and more dilute, near 3:1, so TDS is around 21/3 ≈ 7%. The same overall mass of coffee is dissolved, but by a greater amount of water.

Important to remember is these are just two points on one continuous band of ~21% extraction shots, and the labels are conventions about where on the band the best coffee clusters. These are just two conventions for two different styles of espresso. Traditional espresso maximizes concentration and accepts more channelling risk at the fine grind it needs, while turbo sacrifices concentration for consistency at a coarser grind.

A nice product of the model is that the constant C is calibrated on a single traditional shot — 300 µm, 30 s, 2:1. With that constant fixed, the same relation predicts turbo's 20 seconds at 3:1 for a 450 µm grind, without the model being calibrated with any turbo shots. The geometry carries over.

The last piece is that channelling highlights an increase in shot variance. Shots in the channelling range aren't reliably mediocre, rather sometimes are fine and sometimes are poor, depending on where the water broke through. So the fine-grind end of the band carries both the higher peak and the fatter tail. The expected-score surface in the plot is just the clean surface, pulled down unevenly by that risk. However, for realized enjoyment most espresso drinkers place a value in consistency, so staying out of this range is a better idea than the score may suggest.

The load-bearing assumption in all of this is the d² exponent, that permeability goes with the square of particle size. It's the standard Kozeny–Carman result, but a grinder's real particle distribution could put the true exponent anywhere from about 1.7 to 2.3, and that would bend the plane. Whether there is also a genuine quality dip somewhere between the shot styles is an empirical question the model can't settle. But I would need a refractometer reading to confirm anything.