Newton’S Law Of Cooling With Caputo Derivative: Consistent Dimensionality To Compare With Experiments

The dimensional homogeneity of different non-integer order versions of Newton’s law of cooling was studied. The Caputo derivative was considered for the analysis, as well as an auxiliary parameter σ with a time dimension. Usually, in the open literature this auxiliary parameter is arbitrarily defined by restricting its value or the range of values that it can take. In this investigation, many experimental data from different sources were collected and the most probable values of σ parameter and the parameter representing the non-integer order of derivatives were determined using the standard least-squares fitting technique. The results for the typical system in which an amount of hot water in a container is cooled by natural convection show that the order of the derivative takes values smaller but close to one, implying that the fractional behavior of the cooling curves deviates slightly from that perceived in the integer order solutions. It was also confirmed that the auxiliary parameter σ can be represented as the product between a real number n and the inverse of the cooling constant k of the classic model. Thus, the parameter σ can be interpreted as a kind of time constant, which can be considered reduced if n < 1 and enlarged if n > 1.