An insight into Newton's cooling law using fractional calculus

Abstract

For small temperature differences between a heated body and its environment, Newton's law of cooling predicts that the instantaneous rate of change of temperature of any heated body with respect to time is proportional to the difference in temperature of the body with the ambient, time being measured in integer units. Our experiments on the cooling of different liquids (water, mustard oil, and mercury) did not fit the theoretical predictions of Newton's law of cooling in this form. The solution was done using both Caputo and Riemann-Liouville type fractional derivatives to check if natural phenomena showed any preference in mathematics. In both cases, we find that cooling of liquids has an identical value of the fractional derivative of time that increases with the viscosity of the liquid. On the other hand, the cooling studies on metal alloys could be fitted exactly by integer order time derivative equations. The proportionality constant between heat flux and temperature difference was examined with respect to variations in the depth of liquid and exposed surface area. A critical combination of these two parameters signals a change in the mode of heat transfer within liquids. The equivalence between the proportionality constants for the Caputo and Riemann-Liouville type derivatives is established.

A Fractional Derivative Modeling of Heating and Cooling of LED Luminaires

“…Recently, it has been argued that a simple generalization of Newton's law of cooling, in which the conventional time derivative is replaced by a fractional derivative, properly reproduces the results of measurements in certain liquids [19]. Motivated by this finding, we apply this fractional model to the present case with rather successful results, as we show below.…”

“…The value of α should be related to the strength of the interplay between the different modes, approaching 1 when one of them is largely dominant. In fact, α = 1 was found to be fine for metal alloy slabs in [19]. In our case, the "fractionalization" does not come from the material itself but from the nontrivial geometry of the system under study [28].…”

“…Checking the validity of these qualitative statements would be an interesting problem for the future. The possibility of fractional calculus playing a role in the thermal evolution of nontrivial systems is a compelling result, see also the aforementioned reference [19]. This can be specially important for procedures with industrial relevance.…”

“…A relevant question arises: why should the fractional derivative lead to better results than the conventional derivative in this setup? Following [19], our conjecture is that this happens because there are several relaxation rates at play. The conventional derivative is local in time but the fractional derivatives integrate over a time interval, opening the possibility of taking into account a more complex type of temporal dynamics.…”

“…This expression depends on a real parameter α ∈ (0, 1) such that the conventional derivative is recovered in the α → 1 limit. We postulate the following form of fractional Newton's law of cooling (heating) [19] for the temporal evolution of the temperature of the LED chip:…”