Mathematical model for coffee extraction based on the volume averaging theory

• Published in: Journal of food engineering 2019-12, Vol.263, p.1-12
• Main Authors: Sano, Yoshihiko, Kubota, Shun, Kawarazaki, Akito, Kawamura, Kazuhiko, Kashiwai, Hajime, Kuwahara, Fujio
• Format: Article
• Language: English
• Subjects: Coffee extraction; Drip-brewed coffee; Espresso coffee; Mathematical model; Stirring extraction; Volume averaging theory
• ISSN: 0260-8774; 1873-5770
• DOI: 10.1016/j.jfoodeng.2019.05.025

Coffee extraction is a complex mass transfer process that takes place between hot water and ground coffee beans when the water passes through a bed of coffee grounds. In this study, a general set of macroscopic governing equations for coffee extraction was derived using the volume averaging theory. Moreover, lumped parameter analytical solutions for the extraction of drip coffee, espresso coffee, and immersion-brewed coffee (e.g., siphon coffee) were obtained by integrating the macroscopic governing equations. A dimensionless number associated with controllable parameters in the coffee brewing techniques, namely the flow rate of hot water, the amount of coffee grounds, and the size of the ground coffee particles, is proposed based on the appropriate normalization of the derived equations. It was found that when brewing an espresso coffee, this dimensionless number is sufficiently high so that the extraction depends on the flow rate of hot water and the amount of beans. In contrast, drip coffee extraction can be controlled by this dimensionless number. A set of exhaustive experiments brewing each considered type of coffee was conducted to examine the validity of the proposed models. It was found that the present analytical solutions agree well with experimental data, indicating that the proposed models can be used to accurately predict coffee extraction. •The 3D coffee extraction model was derived based on volume averaging theory.•Analytical solutions were proposed for drip, espresso and immersed coffee.•Experiments were conducted to examine validity of the proposed models.•It was found that the presented models can predict each coffee extraction.