Darcy's equation

Pure Percolation

Pure over boxed
The Pure Over in its box. The glass base is designed with an inbuilt filter, avoiding the need for disposable paper filters but making the physics of percolation unavoidable.

It was entirely appropriate that the first coffee I tried in the Pure Over coffee brewer was the directly traded La Lomita Colombian from Ricardo Canal via Amoret Coffee. Ricardo was a special guest at one of the Coffee and Science evenings we held at Amoret Coffee in Notting Hill (pre-pandemic) where, among other things, he spoke about how he is using Biochar on his coffee farm. Biochar is a porous, charcoal based material that can help to provide the coffee plants with nutrients as well as water, thereby reducing the amount of fertiliser the plants need. To understand how it works, we need to understand a bit about percolation, which of course we also need to understand in order to brew better coffee in the Pure Over. Indeed, there are enough similarities, and an extension to a quirk of how espressos are brewed, that it is worth spending a little more time thinking about this process and the connections revealed as we brew our coffee.

Percolation recurs in many of the brew methods we use for making coffee. The V60, Chemex, Kalita wave, percolators and the espresso itself, all rely at some point on water flowing through a bed of ground coffee. The flavour of the resultant cup is dependent on the amount of coffee surface that the flowing water is exposed to together with the time that it is in contact with the coffee. What this means is that grain size, or the degree to which you grind your coffee, is critical.

Playing with brewing coffee, we know some things by experience. Firstly, frequently, the flow through a coarse grind of coffee will be quite fast (probably too fast to make a good cup). Secondly, we know that for any particular brew method, the more water we pour into the brewer, the faster the water initially comes through. We also know that we can affect the flow rate of water through the coffee if we increase the area of the coffee bed, or decrease its thickness. These observations were quantified into an equation by Henri Darcy in 1856. Darcy’s work had been as an engineer, designing and building public works such as the aqueduct that brought drinking water into the city of Dijon in the 1840s. Darcy received significant recognition at the time for his work including the L├ęgion d’honneur, but it is more for a later set of experiments and particularly for his equation that we remember him today. In the 1850s Darcy was working on the problem of water purification. Passing water through a bed of sand is still used as a method of purifying the water today. Darcy used a series of cylinders filled with sand to investigate how quickly water trickled through the sand bed in order to come up with a proper quantification of those things that we too know by experience with our coffee filters. You can read about the mathematics of Darcy’s equation here.

espresso puck
An espresso puck. The compact structure nonetheless allows water to percolate through it at high pressure.

Darcy found that the flow rate of water through the sand bed increased when the porosity of the bed was higher (fine, dense sand would delay the flow of the water more than coarse, loosely packed sand). If there was a greater pressure on the water at the top of the bed (ie. more water is on top of the sand), the flow rate through the bed would increase too. Conversely the flow would get slower as the water was made more viscous. This is something we too know from experience: try to pour honey through the coffee grounds and it just won’t work.

For us to apply Darcy’s insights into making better coffee, it means that we need to think about the grind size. Too coarse and there will be lots of empty space through the bed of grounds: the porosity is high, and the water will flow straight through. Too fine and the flow rate will decrease so much that rather than just the sweet and slightly acidic solubles that first come out of any coffee extraction*, there could be too much of the bitter organic compounds that come out later, changing the character of the cup. With coffee we have an additional concern. Unlike sand, coffee grinds will swell, and splinter, as water is added to them, closing up any narrower paths and lengthening the brew time. This also means that, unless we properly wet the grounds prior to filtering our coffee, the extraction will be non-uniform and not reproducible. Another reason to bloom coffee thoroughly before brewing.

There is one more factor in brewing our coffee however that Darcy’s equation, which is valid for more stable systems, overlooks. Darcy assumed a constant flow rate of water through the sand bed, but coffee is different. In his book about espresso*, Illy showed that the flow of the water through an espresso puck was not constant over time. Something really interesting was happening when you looked carefully at an espresso puck. Ground coffee can come in a large distribution of sizes. In addition to the grind that we are aiming for, we also get a whole load of smaller particles called ‘fines’. Sometimes this is desirable, but with espresso, and by extension with our filter coffees, these fines add a twist to the physics of the percolation. As the espresso water is pushed through the coffee puck, the fines get pushed down through the puck between the ‘grains’ of the coffee grinds. This reduces the flow rate of the water until the point at which they get stuck. This will have the effect of increasing the contact time between the coffee and the water and so allowing more flavour solubles to be extracted. But crucially, these fines remain somewhat mobile. If you were to turn the whole espresso puck upside down (and Illy had a machine that allowed him to do this in-situ), the fines would again go on the move. Migrating from the new top of the puck to the new bottom. Filling the voids between the slightly too coarse grains. Complicating the simplifications in Darcy’s equation, but adding flavour to our brew.

Watch House coffee Bermondsey
There is a fountain on the wall (right hand side) of the Watch House cafe in Bermondsey. Many public fountains in London date from the 1850s emphasising just what a problem access to drinkable water once was.

Which leaves the connection between the farming method and the coffee. Biochar is formed by burning carbon containing waste (such as plant matter) in a low oxygen environment. Burying the resultant charcoal is therefore a way of storing carbon, and preventing its release into the atmosphere, for many years. But it is not just good for carbon storage. The buried charcoal is highly porous and traps nutrients within its structure so that the plants growing near it can be fertilised more efficiently. Moreover, the fact that it is porous, just like the coffee or sand beds, means that it traps water for a long time. Consider how long it takes a used filter full of coffee grounds to completely dry out! The water gets trapped within the porous structure and does not evaporate easily. This aspect of the biochar means that, as well as nutrients, the plants that grow nearby get a good source of reliable water. The ancient civilisations of the Amazon region used something similar to biochar in their farming techniques resulting in soil now known as “Terra Preta”, an extremely rich form of soil that improves plant growth. On his farm, Ricardo is going fully circular and making his biochar out of old coffee trees. The old trees thereby giving new opportunities to the fresh growth. It is a carbon capture scheme that reduces the need for fertilisers and that relies on percolation physics to work to best effect for the plants.

It seemed a moment of perfect coffee-physics poetry to use coffee grown on a farm using these techniques while initially experimenting with my own, percolation sensitive, Pure Over brewer. Percolation physics and interconnectedness all in one cup.

*Illy and Viani (Eds), “Espresso Coffee”, 2nd Edition, (2005)