The Mathematics of a Perfect Cup of Coffee

Black coffee in a white porcelain cup sitting on a saucer with roasted coffee beans scattered around.

Brewers have developed a wide range of methods to make the perfect cup of coffee, but few have ever posed the question mathematically. (Image: Valentyn75 via Dreamstime)

Researchers are a step closer to understanding how to prepare a perfect cup of coffee with the help of mathematics. Coffee is one of the most widely consumed drinks in the world, with estimates of more than a couple of billion cups consumed each day worldwide. Brewers have developed many methods to make the perfect cup of coffee, but few have ever posed the question mathematically.

To better understand the advanced mathematics involved, a group of researchers, led by Dr. Kevin Moroney at the University of Limerick in Ireland and Dr. William Lee at the University of Portsmouth in the UK, put forward a brand new model of coffee extraction from drip filter machines.

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Ground coffee ready to go in the espresso maker.
The researchers looked at the effect of grain size on the way that coffee comes out of a drip filter machine. (Image: Nicholas Lundgaard via Flickr)

Drip filter coffee is made by pouring hot water over a bed of ground coffee beans, otherwise known as grains, housed in a filter. Gravity pulls water through the filter, dissolving the coffee into the liquid on its way through.

Grain size does matter

The researchers looked at the effect of grain size on the way that coffee comes out of a drip filter machine. They found that coffee is extracted from the grains in a two-step process.

The first step involves the rapid extraction of coffee from the surface of the grain. The second step is slower and consists of extracting coffee from the center of the grain.

It had previously been known that grinding coffee beans too finely could result in a cup of coffee that is strong and bitter. But, on the other hand, not grinding them enough can make the result watery and bland. The research took that observation and made it quantitative by explaining why grain size plays such an essential role in determining the taste of coffee. It also quantifies that relationship through formulas.

Grinding coffee beans too finely could result in a cup of coffee that is strong and bitter, while not grinding them enough can make the end result watery and bland.
Grinding coffee beans too finely could result in a cup of coffee that is strong and bitter, while not grinding enough can make the result watery and bland. (Image: Nailia Schwarz via Dreamstime)

Their research noted a trade-off between water flowing over the surface and through the interior of the grains. When the grains are large, the overall surface area is less. However, the water also flows more quickly through large grains, and because the water spends less time in contact with the coffee, the amount of extraction is less.

On the other hand, refined grains increase the amount of surface area. Also, when the grains are small, the water flow is slowed, increasing the amount of coffee extraction.

Shape and flow also contribute to taste

The research discovered that other factors also come into play when making a great cup of coffee, including the shape of the coffee bed in the filter and the flow pattern of the water.

The shape of the coffee bed in the filter is deformed as the coffee is brewed. When first starting to brew, the coffee grains are sitting flat at the bottom of the filter, but at the end of the brewing cycle, they are coating the walls of the filter, which can affect the taste of the coffee.

The flow pattern of water into the filter also can affect the taste. A single jet of water down the center, like water coming out of a tap or a broad pattern like a showerhead, has different effects on the shape of the coffee bed in the filter and, thus, the cup of coffee’s taste.

The research is of great interest to manufacturers of drip filter coffee machines, as it could allow users to fine-tune a machine for a particular grain size or flow pattern of water, resulting in a perfect cup of coffee every time!

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