The extraction of coffee solubles from roasted and ground coffee is a complex operation, the understanding of which is key to the brewing of high quality coffee. This complexity stems from the fact that brewing of coffee is achieved through a wide variety of techniques each of which depends on a large number of process variables. In this paper, we consider a recent, experimentally validated model of coffee extraction, which describes extraction from a coffee bed using a double porosity model. The model incorporates dissolution and transport of coffee in the coffee bed. The model was shown to accurately describe extraction of coffee solubles from grains in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. The full model equations can only be solved numerically. In this work we consider asymptotic solutions, based on the dominant mechanisms, in the case of coffee extraction from a dilute suspension of coffee grains. Extraction in this well mixed system, can be described by a set of ordinary differential equations. This allows analysis of the extraction kinetics from the coffee grains independent of transport processes associated with flow through packed coffee beds. Coffee extraction for an individual grain is controlled by two processes: a rapid dissolution of coffee from the grain surfaces in conjunction with a much slower diffusion of coffee through the tortuous intragranular pore network to the grain surfaces. Utilising a small parameter resulting from the ratio of these two timescales, we construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared with numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters, so the solutions facilitate quick investigation of the influence of various process parameters on the coffee extraction curves.

中文翻译：

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充分混合的系统中的咖啡提取动力学。

从烘焙和研磨咖啡中提取咖啡可溶物是一项复杂的操作，对此的理解是酿造高品质咖啡的关键。这种复杂性源于以下事实：通过多种技术实现咖啡冲泡，每种技术都取决于大量的过程变量。在本文中，我们考虑了最近经过实验验证的咖啡提取模型，该模型描述了使用双孔隙率模型从咖啡床上提取咖啡的过程。该模型结合了咖啡在咖啡床上的溶解和运输。该模型显示出可以准确描述两种情况下从谷物中提取咖啡可溶物的方法：从稀有咖啡粉悬浮液中提取和从包装咖啡床上提取。完整的模型方程只能通过数值求解。在这项工作中，在从稀有咖啡粒悬浮液中提取咖啡的情况下，我们基于主要机理考虑渐近解。可以通过一组常微分方程来描述在这种充分混合的系统中的萃取。这允许分析咖啡粒中的提取动力学，而与与通过包装咖啡床的流动相关的运输过程无关。单个谷物的咖啡提取通过两个过程控制：咖啡从谷物表面快速溶解，以及咖啡通过曲折的颗粒内孔网络到谷物表面的扩散要慢得多。利用由这两个时标之比得出的小参数，我们使用匹配渐近展开的方法构造渐近解。将渐近解与数值解和来自咖啡提取实验的数据进行比较。渐近解取决于少量的无量纲参数，因此该解便于快速研究各种过程参数对咖啡提取曲线的影响。