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# Coffee Roasting, Mathematical Modelling, and Asymptotic Analysis

If you’re sipping away at one of the 2.25 billion cups of coffee that are consumed daily throughout the world, chances are you’ll want your cup of joy to be as delicious as possible. However, achieving that perfect cup of coffee depends on a myriad of variables: the origin of the coffee beans, how the beans are roasted, and even the hardness of the water used for brewing. This challenge appears rather daunting at first glance, so how do scientists and researchers determine what makes a good cup of coffee?

One key factor in the processing of coffee beans is how they are roasted. During this phase, partially dried coffee beans turn from green to yellow to various shades of brown, residual moisture content within the bean dries up, and crucial aromas and flavours develop. The associated chemical reactions that produce these desirable coffee traits are often linked to the presence (or absence) of water. Therefore, understanding the moisture content in a coffee bean at a specific time during the roasting process can also provide insight into that flavours and aromas that will be generated during the roast.

Our mathematical model uses multiphase physics to describe the evolution of the solid, liquid, and gas components within the coffee bean. This is a crucial difference from previous attempts to model coffee roasting, as existing models treated the coffee bean as a single “bulk” material. While such an approach yields great mathematical simplifications to the model, bulk models lack the ability to capture local dynamics between the different phases. As such, if we are trying to gain further insight into how a coffee bean loses water during a roast, then we must employ a model that takes into account the interactions that occur between the gas and liquid phases.

Typical parameter values indicate that, due to the coffee bean’s rigid cellulose structure, the evaporation timescale of the multiphase model is extremely small when compared to the corresponding vapour diffusive timescale. In consequence, we anticipate observing an internal buildup of vapour pressure until the local water content has completely evaporated. Additionally, the thermal diffusive timescale of the multiphase model is significantly smaller than the corresponding vapour diffusive timescale. This implies that the coffee bean will heat to its roasting temperature significantly faster than the vapour can diffuse out of the bean. While a full temperature-dependent model has been considered and analyzed, the salient features of drying are retained when considering an isothermal simplification of the multiphase model.

Solution of the moisture content (left: % water/% voids) and vapour pressure (right: atm) using the isothermal multiphase model and a spherically symmetric coffee bean of 4mm radius. The “drying front,” which propagates from the surface of the bean towards the centre, divides the coffee bean into an interior region with high vapour pressure and a dry outer region with negligible water content.

Numerical solutions of the isothermal multiphase model confirm that two large-scale regions exist where the evaporation rate is turned off, along with the predicted buildup of vapour pressure inside the bean. Curiously however, we also observe a thin “drying front” connecting these two regions, whose position moves in time and where the bulk of evaporation occurs. While an explicit solution of the multiphase model remains too difficult to obtain, we can use asymptotic analysis to determine approximate solutions in each of these three regions. The first region admits a constant vapour pressure (due to constant temperature) to turn off evaporation, thus yielding a constant moisture profile. However, this forces evaporation to be triggered in a thin transition region due to the much lower vapour pressure in the roasting environment. In this thin region, centred about the drying front, a small perturbation in vapour pressure drives the moisture to negligible quantities. While the drying front’s explicit form remains undetermined at this step, the transition layer admits a Stefan-like condition, which relates the flux of vapour pressure to the speed of the drying front. Finally, in the dry region (where negligible water content forces the evaporation to stop), we are able to simultaneously solve the pressure profile and drying front’s shape.

Comparison of the drying front determined via asymptotic analysis and the numerical solution of the isothermal multiphase model.
Our asymptotic solution for the drying front agrees well with the drying front predicted by numerics. This implies that solving the leading-order equations determined via asymptotic analysis, which are significantly easier to solve than the full multiphase model, captures all the salient features of the observed drying phenomena. Additionally, the computation time of these approximations takes a fraction of the time to solve relative to the full numerical solution; this is mainly due to stiff time solvers having to resolve the sharp drying front dynamics—seen in the full multiphase model—at high precision. Given these advantages, industrial researchers can cheaply use these asymptotic approximations to determine the important features of a coffee bean under a variety of roasting settings. While we only provide a basic framework to roasting models in this study, understanding qualitative features of these key processes will allow us to get one step closer to that perfect cup of coffee.