Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world. The seeds (coffee beans) from the plant of the same name are roasted and ground, allowing a flow of hot water to extract their soluble content. Undissolved solids are filtered from the dissolved particles, and the resulting liquid becomes the concoction that much of the population drinks every day.
While past studies have investigated the mathematics of coffee extraction, researchers have previously paid little attention to the drip filter brewing system. Drip filter machines make up about 10 million of the 18+ million coffee machines sold yearly in Europe, and involve pouring hot water over a bed of coffee grounds housed in a filter. Gravity pulls the water through the filter, extracting coffee solubles from the grains during the flow.
The authors' earlier paper presents the derivation of this general model, which considers bed dimensions, flow rates, grind size distribution, and pressure drop. They assume isothermal conditions (constant temperature), because optimal brewing circumstances require a narrow temperature range of 91-94 degrees Celsius. They also assume that coffee bed properties remain homogeneous in any cross section and that water saturates all pores in the coffee bed, eliminating the need to model unsaturated flow. A set of conservation equations on the bed scale monitor the transport of coffee and liquid throughout the coffee bed.
Because coffee brewing involves so many components, simplifying the model becomes necessary. "In modelling a complicated physical process such as coffee brewing, one attempts to write down a system of equations which captures the essence of the process," O'Brien said. "In doing so, we initially make some simplifications, which neglect some aspects of the real problem. For example, real coffee contains a large number of dissolved substances; we simplify our model by considering the case of a single such substance. The mathematical model then comprises conservation laws (mass momentum), which in their complete form cannot be solved exactly."
Recognizing these approximate solutions helps the authors easily identify typical trends. "Approximate solutions are formed based on the dominant processes in the coffee bed during different stages of the extraction process," Moroney said. "Initially, the concentration of coffee in the bed is determined by the balance between a rapid extraction from the surfaces of coffee grains and the rate at which coffee is removed from the coffee bed by the extracting water. Later in the process, the extraction is dominated by slow diffusion of coffee from the kernels of larger grains, which was initially negligible." Although the timescales of the aforementioned extraction methods are much shorter in fine coffee grinds rather than coarse grinds, the authors can still construct approximate solutions because of the timescale ratios' small size. "The value of the solutions lies in the ability to explicitly relate the performance of a brewing system with the properties of the coffee, water and equipment used," Moroney said. These solutions help predict the coffee quality for specific brewing configurations.
A possible next step involves incorporating the changing coffee bed shape that occurs while water flows through the conical filter holder of a drip filter machine. "This causes both the extraction and the flow rate through the coffee bed to become a function of position," Marra said. The authors' research also has the potential to inspire further models on different extraction processes, including unsaturated flow and the trapping of air pockets in a coffee bed, in the never-ending quest for a perfect cup of coffee.
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About the authors: Kevin M. Moroney is a Ph.D. researcher with the Mathematics Applications Consortium for Science and Industry (MACSI) in the Department of Mathematics and Statistics at the University of Limerick. William T. Lee is a lecturer in the Department of Mathematics and Statistics at the University of Limerick, and is a part of MACSI. Stephen B.G. O'Brien is director of MACSI and a professor of applied mathematics at the University of Limerick. Freek Suijver is a program manager and senior director of the Program Management Team at Philips Research Laboratories. Johan Marra is a principal scientist and chemical engineer at Philips Research Laboratories.
SIAM Journal on Applied Mathematics