Fractional-Order Thermal Energy Transport for Small-Scale Engineering Devices

Fractional-order thermodynamics has proved to be an efficient tool to describe several small-scale and/or high-frequency thermodynamic processes, as shown in many engineering and physics applications. The main idea beyond fractional-order physics and engineering relies on replacing the integer-order operators of classical differential calculus with their real-order counterparts. In this study, the authors aim to extend a recently proposed physical picture of fractional-order thermodynamics to a generic 3D rigid heat conductor where the thermal energy transfer is due to two phenomena: a short-range heat flux ruled by stationary and nonstationary transport equations, and a long-range thermal energy transport representing a ballistic effects among thermal energy propagators. Thermodynamic consistency of the model is investigated introducing the state function of the temperature field, namely the entropy, and obtaining the thermodynamic restrictions on the signs of the coefficients involved in the proposed model of fractional-order thermodynamics. Finally, numerical applications are presented for both 1D and 2D rigid bodies.