Construction Of Empirical Models Via A Stepwise Fitting Of A Fractional Newtonian Cooling Law

The cooling curves of some natural and artificial systems show a series of anomalies (such as sudden drops in temperature or non-homogeneous behavior) that can hardly be modeled with simple laws covering their entire duration. Via three algorithms, we developed a procedure which consists of a stepwise fitting of a fractional Newtonian law of cooling model to empirical curves that show such anomalies. We used the Caputo fractional derivative to construct models for phenomena which show such anomalies — Mpemba effect and the cooling performance of household utensils — which we present in detail. In this manner, we were able to construct continuous functions that represent quite well the experimental data, as opposed to the functions commonly obtained by an integer-order model of Newton’s cooling law. Although these piecewise models are purely empirical descriptions, they can be predictive tools of great interest to product designers in the food industry and might provide elements to gain a deeper understanding of an unintelligible phenomenon such as the Mpemba effect.