Modelling roasting coffee beans using mathematics: now full-bodied and robust

Coffee is one of the most traded commodities in the world, valued at more than $100 billion annually. Even if you’re not an entrepreneur looking for the next big coffee venture, you’ll probably still care about how to make the 2.25 billion cups of coffee globally consumed every day as delicious as possible. Fortunately, scientists and researchers have partnered with large coffee companies in the hopes of understanding some of the complexities behind making a good cup of coffee: specifically, how coffee beans are roasted.

During the roasting process, partially dried coffee beans turn from green to yellow to various shades of brown, depending on the length of the roast. Once the residual moisture content within the bean dries up in the yellowing phase, crucial aromas and flavours are developed. However, the associated chemical reactions that produce these desirable coffee traits are highly complex and not well understood. This is partially due to the fact that the browning reactions linked to aroma and flavour development, called the Maillard reactions, is comprised of a large network of individual chemical reactions, where only preliminary understanding of the network’s construction exists.

To tackle the challenges involved with creating mathematical models for the Maillard reactions, along with other chemical reaction groups in a roasting coffee bean, we use the concept of a Distributed Activation Energy Model (DAEM), originally developed to describe the pyrolysis of coal. Not dissimilar to the Maillard reactions, the pyrolysis of coal involves large numbers of parallel chemical reactions and, using the DAEM, can be simplified to a single global reaction rate that describes the overall process. Crucially, however, the DAEM relies on knowing the distribution of individual chemical reactions beforehand. While the overall distributions associated with the Maillard chemical reactions remain unknown, we can reasonably approximate the reaction kinetics of the majority of the Maillard chemical reaction group.

However, the DAEM approach to chemical reaction groups only works when each of the reactions is happening parallel of one another. Because of this, we examine a simplified pathway of reactions involving sugars (which are linked to the formation of Maillard products) and separate groups of reactions to follow a progression of reactants to products. Specifically, we examine how sucrose first hydrolyses into reducing sugars, which in turn become either Maillard products or products of caramelisation. This division of this sugar pathway network allows us not only to fit each reaction subgroup with different parameter values, but also to determine that the hydrolysis of sucrose creates a “bottleneck” in the sugar pathway and prevents Maillard products from forming too early in the roast.

To model the local moisture content and temperature of the bean, two variables that crucially change which chemical reactions can occur during the roast, we use multiphase physics to describe how the solid, liquid, and gas components within the coffee bean evolve. This is a crucial difference to what has previously been done to model coffee roasting, as existing models often treat the coffee bean as a single “bulk” material. Additionally, unlike in previous multiphase models for roasting coffee beans, we allow the porosity of the bean to change according to the consumption of products in the sugar pathway chemical reaction groups. We also incorporate a sorption isotherm, an equilibrium vapour pressure specific to the evaporation mechanisms present in coffee bean roasting, in our model. Finally, to reduce the system variables to functions of a single spatial variable and time, we model a whole coffee bean as a spherical “shell”, while modelling a chunk of a coffee bean as a solid sphere.  This is another improvement to previous multiphase models, which disagreed with recent experimental data describing the moisture content in both roasting coffee chunks and roasting whole beans.

Numerical simulations of this improved multiphase model (referred to as the Sugar Pathway Model) provide several key conclusions. Firstly, the use of spherical shells and solid spheres to describe whole and broken coffee beans, respectively, allows for good agreement with experimental data while simplifying the mathematical model’s structure. Secondly, due to the large number unknowns in the model, the Sugar Pathway Model can be fit to experimental data using a variety of parameter values. While this could be viewed as a drawback to the Sugar Pathway Model, we also show that small changes in parameter values do not drastically change the model’s predictions. Hence, the Sugar Pathway Model provides a reasonable qualitative understanding of how to model key chemical reactions that occur in the coffee bean, as well as how to model coffee bean chunks differently to whole coffee beans.

While largely theoretical, the Sugar Pathway Model provides a balance between the immensely complicated underlying physical processes occurring in a real-life coffee bean roast and its dominant qualitative features predicted by multiphase mathematical models. Additionally, industrial researchers can cheaply and efficiently use these multiphase mathematical models to determine the important features at play within a coffee bean under a variety of roasting configurations. While a basic framework for the roasting of a coffee bean is presented here, understanding the qualitative features of key chemical reaction groups allows us to get one step closer to that perfect cup of coffee.

Featured image credit: Coffee by fxxu. CC0 via Pixabay.