A Comprehensive Mathematical Model for a Multispecies Flow Through Ground Coffee

Fasano, Antonio; Talamucci, Federico

doi:10.1137/s0036141098336698

Cited by 19 publications

(15 citation statements)

References 4 publications

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“…Actually, with proper source and permeance terms and a certain initial distribution, the groundwater flow naturally satisfies that 0 < s 0 ≤ s(τ ), τ ∈ (0, T 0 ]. Some theoretical results for the solution variables and s(τ ) have been obtained for similar Stefan problems (see, for example, [17][18][19][20][21]). In this article, we will provide condition 0 < s 0 ≤ s(τ ), τ ∈ (0, T 0 ] for a given finite time period T 0 > 0.…”

Section: Remark 22mentioning

confidence: 99%

“…The nonlinear functions f , g, and ω, and their derivatives are required only to be locally uniformly bounded, which allows a wide class of problems to be included in our results. Regarding the existence, uniqueness, and regularity results of the solution, the reader is referred to [17][18][19][20][21], etc.…”

Section: Remark 22mentioning

confidence: 99%

“… 20),. and utilizing Theorem 6.1, one can easily obtain the following error estimates for the fully discrete finite element approximation {H n h , c n h , S n h } of the solution {H , c, s} to the original Problem (P).…”

mentioning

confidence: 99%

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Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Dong

Wang

Sharpley

2006

Numer. Methods Partial Differential Eq.

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In this article, we consider a single-phase coupled nonlinear Stefan problem of the water-head and concentration equations with nonlinear source and permeance terms and a Dirichlet boundary condition depending on the free-boundary function. The problem is very important in subsurface contaminant transport and remediation, seawater intrusion and control, and many other applications. While a Landau type transformation is introduced to immobilize the free boundary, a transformation for the water-head and concentration functions is defined to deal with the nonhomogeneous Dirichlet boundary condition, which depends on the free boundary function. An H 1 -finite element method for the problem is then proposed and analyzed. The existence of the approximation solution is established, and error estimates are obtained for both the semi-discrete schemes and the fully discrete schemes.

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Section: Remark 22mentioning

confidence: 99%

Section: Remark 22mentioning

confidence: 99%

See 1 more Smart Citation

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Dong

Wang

Sharpley

2006

Numer. Methods Partial Differential Eq.

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“…A number of models have been developed which couple flow and extraction in coffee beds. Fasano et al [26–29], developed general multiscale models for the extraction of coffee, primarily focused on the espresso coffee machine. In ref.…”

Section: Introductionmentioning

confidence: 99%

Analysing extraction uniformity from porous coffee beds using mathematical modelling and computational fluid dynamics approaches

et al. 2019

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Achieving a uniform extraction of soluble material from a porous matrix is a generic problem in various separation and filtration operations, with applications in the food processing, chemical and pharmaceutical industries. This paper describes models of fluid flow and transport of soluble material within a packed granular bed in the context of coffee extraction. Coffee extraction is described by diffusion of soluble material from particles of one or more representative sizes into fluid flowing through the packed bed. One-dimensional flow models are compared to computational fluid dynamics (CFD) models. A fine and a coarse coffee grind are considered. Model results are compared to experimental data for a packed cylindrical coffee bed and the influence of a change in geometry to a truncated cone is considered. Non-uniform flow in the truncated cone causes significant variation in the local extraction level. Coffee extraction levels during brewing are analysed using extraction maps and the degree of variation is represented on the industry standard coffee brewing control chart. A high variation in extraction yield can be expected to impart bitter flavours into the brew and thus is an important variable to quantify.

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“…Experiments on the operation and efficiency of the stove-top of Moka pot are detailed in [ 9 , 10 ]. Fasano et al have developed some very general multiscale models of coffee extraction in different situations, with a particular focus on the espresso coffee machine [ 11 – 16 ]. The influence of some brewing parameters on coffee extraction was investigated by Voilley and Simatos [ 17 ].…”

Section: Introductionmentioning

confidence: 99%

Coffee extraction kinetics in a well mixed system

et al. 2016

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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.

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A Comprehensive Mathematical Model for a Multispecies Flow Through Ground Coffee

Cited by 19 publications

References 4 publications

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Analysing extraction uniformity from porous coffee beds using mathematical modelling and computational fluid dynamics approaches

Coffee extraction kinetics in a well mixed system

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