A mathematical coffee filter

Coffee drinkers have a myriad of methods for getting their fix, espresso machines, drip filters, percolators, the French press. And, as true connoisseurs know there is an art to the roasting, grinding and brewing of this complex bio-matrix into the perfect hot beverage. As with many areas of materials science, albeit extraction of natural products from said bio-matrix, also known as the coffee bean, mathematics can assist in optimizing thre process.

There are almost 2000 different chemicals present in the average coffee bean, some of them are critical to aroma and flavor as well as the allegedly stimulating effects of a cup of the "average Joe". Many of the components of the bean are soluble in water, but many are not and most approaches to brewing have a straightforward filtration approach to keeping the particulates out of the brew. For example, espresso coffee is made by blasting pressurized hot water through a compacted bed of finely ground coffee held in a metal filter cup. By contrast, drip filter brewing, as the name suggests, involves pouring hot water over a loose bed of more coarsely ground coffee through a paper or metal gauze filter. In both methods, it is the flow of water that leaches the soluble coffee components from the grains leaving behind used grounds in the filter.

Such solid-liquid extraction lends itself to a mathematical analysis given that there are so many chemical components in a coffee and so many variables to consider in the brewing process, not least water temperature, flow rate, and filtration technology. Writing in the SIAM Journal on Applied Mathematics, researchers at the University of Limerick, Ireland, describe a new multiscale model of coffee extraction from a coffee bed [Kevin M. Moroney et al, SIAM J Appl Math (2016) 76(6), 2196-2217; DOI: 10.1137/15M1036658].

Moroney explains that most of the mathematical models of coffee extraction in the scientific literature focus on batch extraction and consider a well-mixed system, either that or they derive a general transport equation without experimental validation. "Our model describes flow and extraction in a coffee bed, specifies extraction mechanisms in terms of the coffee grain properties, and compares the model's performance with experiment," Moroney explains. "Our initial focus on the flow-through cylindrical brewing chamber [in drip filter coffee] allowed us to consider the model in one spatial coordinate and ensure that the model assumption of a static bed was valid."

Drip filter coffee machines account for some 10 million of the more than 18 million coffee machines sold each across Europe, so understanding their pros and cons mathematically is important to manufacturers hoping to impress coffee drinkers.

David Bradley blogs at Sciencebase Science Blog and tweets @sciencebase, he is author of the popular science book "Deceived Wisdom".