Fractional models for measuring sap velocities in trees

The compensation heat-pulse (CHP) method is founded on the measurement of the temperature difference between sensors located above and below a heater inserted in tree trunks. This technique assumes that the temperature in each probe follows an exponential function and, as a result, an expression to calculate the heat-pulse velocity (v) is deduced. The current work investigates the application of recent theory of fractional calculus (FC) to model such thermal processes in trees. In particular, a generalisation of the mentioned exponential function, given by the two-parameter Mittag-Leffler function, is considered to model temperature dynamics. Several temperature curves, corresponding to zero and nonzero values of v, taken from real trees of different species are given to show that the proposed fractional order models are able to better describe such thermal flows, especially when velocities are different from zero.