The calming humming of the machine, maybe some clanking of metal and ceramic. A small cup filled with a dark liquid that is covered by the light brown crema. The acoustical and visual impressions that come with a cup of espresso are described rather easily. Its smell and especially its taste however, are multifaceted and vary for example from earthy over chocolatey or spicy to fruity and floral. This is due to a broad range of options in the preparation and the hundreds of different chemicals that can be extracted from the coffee grounds. Brewing espressos (or, espressi, for the Italophile) with consistent flavour is still a challenge even for the experienced barista and thus, it is hardly surprising that it is a vivid field of research.

A quantity that is not too complex to model, but still very useful to control the flavour, is the concentration of coffee solubles: During the brewing process of an espresso, hot water flows through a cylindrical filter that is densely packed with coffee grounds. Along the way, chemicals are extracted such that the liquid leaving the filter is not pure water anymore, but espresso. The concentration of dissolved chemicals in the liquid that comes out of the machine is the quantity of interest in a wide range of mathematical models for espresso brewing.

In their recent paper “A multiscale model for espresso brewing: Asymptotic analysis and numerical simulation”, the authors Yoana Grudeva, Kevin M. Moroney and Jamie M. Foster argue that a major part of the extraction is happening in the five to ten seconds from when the water enters the coffee bed until the first drop exits, a time span that has often been ignored in previous works. Performing an asymptotic analysis with respect to the speed of diffusion, they identify four time-dependent regions in the coffee bed that can be distinguished by the rate of concentration increase.

The proposed model consists of a system of partial differential equations that describe the flow of the incoming water through the filter, the transport of solubles within the coffee grains, the transfer from the grain surface to the liquid outside of the grains, and the transport of solubles through the liquid between the grains. It further incorporates the common distinction of coffee grains into boulders, which are large and porous, and fines, which are small and non-porous. The asymptotic analysis is then driven by the following assumptions on the different diffusivities: In fines, the diffusivity is very large, such that the transport within fines is very fast. The diffusivity in boulders is smaller and the transport within them is on a similar timescale to that of the infiltration. Between the grains, diffusion is very slow and dominated by the convection caused by the liquid flow. Thus, in the asymptotic limit solubles within fines are instantly available at the grain surface and transport of solubles within the liquid between the grains is determined entirely by convection.

Considering a reduced model based on the described asymptotic limit, the authors can distinguish four different regions inside of the espresso filter. They are vertically stacked on top of each other and the lowest region is particulary easy to describe: Since the incoming water flows from top to bottom, there is a dry region between the vertically moving wetting front and the bottom of the filter.

To divide the already wet part into subregions, the authors first pose the following question: Does saturation occur? In other words, this asks if the first drops of liquid leaving the filter are saturated, meaning that they cannot absorb any more solubles. From a hands-on brewing experiment, the authors conclude that yes is a reasonable answer to this question (however, the implication of answering the question with no are discussed in the supplementary material). Incorporating saturation into the model, one obtains a characterization of the layer directly above the dry region: A saturated layer, in which the concentration of solubles is almost constant.

Now, there are only two regions left to describe. For these, the different diffusivities in fines and boulders finally come into play: Since extraction from the fines is so fast and solubles are limited, there are almost no solubles in fines left to extract in a region adjacent to the top of the filter. Here, a slow extraction from boulders is taking place, causing only a flat increase of concentration from top to bottom. But of course, extraction from fines also has to happen somewhere: In a small strip between the region with slow extraction from boulders and the saturated region, solubles are extracted rapidly from the fines. Consequently, in this last region, the concentration changes drastically.

The authors complete their work with numerical experiments that indicate that their reduced model captures the main characteristics of the detailed model, while being significantly faster to evaluate. They point out that this could be especially interesting in the context of optimisation studies which aim to improve certain parameters like the brew strength. Further announced perspectives we can already look forward to are the comparison of model solutions to experimental data and advancements towards taste and aroma prediction.

The paper ‘A multiscale model for espresso brewing: Asymptotic analysis and numerical simulation‘ by Yoana Grudeva, Kevin M. Moroney and Jamie M. Foster appears in European Journal of Applied Mathematics and is available open access.

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