The RBF Collocation Method to Design a Digital Twin for Coffee Percolation

TL;DR

Problem addressed: modeling espresso coffee percolation as a coupled PDE system of Darcy flow, heat transfer, and multi-species transport. Main approach: a meshless Radial Basis Function collocation method (Kansa's approach) is used to discretize the PDEs on a cylindrical coffee pod domain, yielding a system of DAEs $A\dot{\bm u}(t)=\bm b(t,\bm u)$ solved in MATLAB and benchmarked against FEFLOW. Findings: hydraulic head, solid caffeine, and temperature are captured with high fidelity (solid caffeine error ≈ 1.4%), while liquid caffeine is less accurate and requires refinement; overall, the method shows robustness and good agreement with the reference. Significance: provides a flexible, mesh-free framework for simulating complex poro-thermal-chemical processes in espresso extraction and related porous-media problems, enabling rapid prototyping and design of digital twins in industrial coffee applications.

Abstract

Espresso coffee extraction is a complex physico-chemical process and can be modeled through a system of coupled partial differential equations. We present a numerical solution based on a meshless Collocation Method using Radial Basis Functions and Kansa's approach, which reveals to be accurate and robust in comparison to a reference numerical solution provided by a well-known simulation software.

Figures (8)

• Figure 1: Domain $\mathcal{D}= \mathcal{C} \cup\Gamma_1 \cup\Gamma_2 \cup\Gamma_3$ of the Percolation Model, i.e., schematization of the coffee pod.
• Figure 2: Upper face discretization (a). Cross section view and curves where the horizontal discretization is repeated (b).
• Figure 3: Collocation nodes of the Percolation Model when $\mathcal{D}$ is a cylinder with height $L=1.388$ and $R=3$.
• Figure 4: Trend of the approximate Hydraulic Head over time.
• Figure 5: Trend of the approximate Solid Caffeine over time.