# Computer Percolation Models for Espresso Coffee: State of the Art, Results and Future Perspectives
## 2 Materials and Methods
### The Percolation Model  
The EC extraction consists of the following physico-chemical process: hot water with prescribed pressure pours into the basket and hits the tamped coffee powder within the basket, it flows into the powder passing by the void spaces in the coffee grains and dissolves several chemical substances from the wet grains of the coffee pod. The water also removes a certain amount of fine particles from the ground coffee and transports it downward. This extraction process can be described as a fluid-dynamics process that is the well-known percolation process. From the fluid-dynamics point of view, these components are involved: the dynamics of the fluid, the fine particles worn away by the flow from the porous matrix and the chemical species dissolved in the fluid, as well as the transfer of heat between the solid medium and the fluid. The books [22; 23] provide a general formalisation of percolation processes. A preliminary study of the percolation model used in this work can be found in [17]. This model is based on some key hypotheses:  
1. The porous medium is isotropic and homogeneous. 2. A chemical species is called liquid if it undergoes the transport process; in contrast, a chemical species is called solid if it is not involved in the transport process so it is bound to the porous medium. Thus, being liquid or solid is independent of the actual physical phase. 3. The first 5 s of the real extraction process are called imbibition, when the porous medium gets wet. After imbibition, there is no gaseous phase in the porous medium, the porous medium is saturated and a local thermal balance occurs between coffee powder and water. The percolation process is investigated after imbibition. The percolation problem has the circular cylinder \(\mathcal{C}\) in Figure 1 as spatial domain. We fix a 3D Cartesian coordinate system and we suppose that the z-axis of such system and the symmetry axis of \(\mathcal{C}\) coincide, \(\Gamma_{3}\) with radius \(R\) lays on the plane \(z=-H\) while \(\Gamma_{1}\) lays on the plane \(z=0\), thus the height of \(\mathcal{C}\) is \(H>0\). The following model describes the physico-chemical process of EC percolation, under the previous assumptions:  
\[ \begin{cases} S_{0}\dfrac{\partial h}{\partial t}+\
abla\cdot \mathbf{q}=Q\\ \mathbf{q}=-\mathbf{K}f_{h}\cdot(\
abla h+\chi\mathbf{e})\\ \varepsilon\dfrac{\partial C_{k}}{\partial t}+\mathbf{q}\cdot \
abla C_{k}+\
abla\cdot\mathbf{j}_{k}=R_{k}-C_{k}Q,&k=1,\ldots,N_{l\text{-}s },\\ \varepsilon_{s}\dfrac{\partial C_{m}^{s}}{\partial t}=R_{m}^{s},&m=1,\ldots,N_ {s},\\ (\varepsilon pc+\varepsilon_{s}\rho^{s}c^{s})\dfrac{\partial T}{\partial t}+ \rho c\mathbf{q}\cdot\
abla T-\
abla\cdot(\mathbf{\Lambda}\cdot\
abla T)=H_{ e}-\rho c(T-T_{0})Q,\end{cases}  \tag{1}\]  
prescribing all the equations in the spatial domain \(\mathcal{C}\) and for \(t\in(0,\tau)\), with \(\tau>0\) the percolation time. We note that the fluid flow in porous media that are saturated is described in (1) by the first equation, which combined with the second one, is known as Richards Equation [24], formulated for the unknown hydraulic head \(h\). Here, \(S_{0}\) is the specific storage coefficient and the term \(Q\) accounts for sources and/or sinks of liquid mass. \(h\) depends on the pressure head \(\psi\) and the elevation \(z\), more precisely, \(h=\psi+z\), where \(\psi=p/\rho_{0}g\), with \(g\) the gravity acceleration, \(\rho_{0}\) a fluid mass density of reference and \(p\) the pressure. The second equation is the Darcy law for the unknown Darcy flux \(\mathbf{q}\). Here, \(\mathbf{e}\) is the coordinate unit vector \((0,0,1)^{T}\), \(\mathbf{K}\) is the hydraulic conductivity tensor and \(\chi_{\ell}\),\(f_{h}\) the buoyancy coefficient and the viscosity relation function, respectively, which account for the effects of temperature and pressure on density and viscosity. The third equation is the advective-diffusive-reactive equation written for the chemical species \(k\) with unknown mass concentration \(C_{k}\).