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A Minimal Thermo-Fluid Model for Pressure-Driven Extraction in a Moka Pot

The moka pot provides a familiar example of a thermally driven flow system in which heating, vapor pressure generation, and fluid extraction are strongly coupled. We present a minimal, dimensionless dynamical model describing the evolution of temperature, pressure, and extracted volume during moka pot brewing. The model consists of a small set of coupled ordinary differential equations incorporating constant heating, heat loss, vapor pressure buildup, and pressure-driven flow through the coffee bed. The heating stage of the model is quantitatively compared with published experimental temperature time data, allowing the characteristic thermal timescale to be fixed independently. Using the experimentally constrained temperature evolution as input, the model predicts the pressure rise and identifies the onset of extraction without additional fitting parameters. Despite its simplicity, the model exhibits several qualitatively distinct extraction regimes, including delayed onset of flow, smooth extraction, and rapid extraction driven by nonlinear feedback between temperature and pressure. These regimes are governed by a small number of dimensionless parameters with clear physical interpretation. Rather than providing detailed quantitative predictions for specific devices, the model is intended as a transparent pedagogical framework for illustrating how physicists construct, simplify, and test coupled thermo-fluid models using experimentally accessible data in an everyday physical system in an everyday physical context.