Cooling Hot Coffee Using Nonlinear Equation Numerical Methods
Keywords:
temperature, time, numerical methodAbstract
Freshly brewed hot coffee requires time to reach the ideal temperature for comfortable consumption, typically around 58–66 °C. Consumers generally rely on estimation or experience when waiting, without a precise scientific approach. This study examines the cooling process of coffee as a heat transfer phenomenon that can be modeled using Newton’s Law of Cooling, in which the rate of temperature decrease is proportional to the temperature difference between the object and its environment. Since the governing equation is nonlinear, solving it requires a numerical method. The Newton-Raphson method was selected for its efficiency in solving single-variable nonlinear equations and its fast convergence.
The simulation was conducted using Python, with the following parameters: an initial temperature of 80.57 °C, ambient temperature of 27 °C, target consumption temperature of 62.99 °C, and a cooling constant of 22.24×10⁻³ s⁻¹ based on previous experimental data. The results showed that the ideal consumption temperature is reached in approximately 17.88 minutes. The iteration graph demonstrated a rapid decrease in function values, requiring only four iterations to converge. While the simulation showed high accuracy during the initial cooling phase, minor deviations occurred as the temperature approached ambient levels. This discrepancy is likely due to the assumption of a constant cooling coefficient, whereas in reality it may vary depending on the solution temperature and air convection conditions.
This model can be used to objectively predict the optimal time to enjoy coffee based on scientific calculations. In addition to benefiting home consumers seeking the best coffee-drinking experience, the findings of this research can be applied in the culinary and hospitality industries to consistently and efficiently serve hot beverages at the right temperature. By providing predicted consumption times, the risk of burns from excessively hot drinks can be minimized, thereby enhancing customer satisfaction. This approach can also be adapted for other hot beverages requiring temperature control, offering broader applicability in the fields of food engineering and thermal design.
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