Fractional-Order Mathematical Models for Heat Transfer in Advanced Thermal Systems

Abstract

Proper modeling of the heat transfer in sophisticated thermal systems is of the essence to improve the performance, assurance of reliability, and effective design in a wide variety of engineering applications including microelectronics, aerospace, renewable energy storage, and biomedical. Standard integer-order models of heat conduction, mainly employing the Fourier law, give reasonable predictions at macroscopic steady-state regimes but tend to fail at the complex systems with memory effects, non-local thermal transport, and anomalous diffusion. Such restrictions are particularly intense in micro/nanoscale devices, porous materials, phase change systems and high temperature reactors where heat propagation is subdiffusive or superdiffusive. In a bid to overcome these difficulties, this paper constructs and studies fractional-order mathematical models of heat transfer using the concept of fractional calculus as a generalization of classical Fourier and non-Fourier models. The proposed framework incorporates long-range temporal correlations, fractal spatial dynamics and intrinsic non-locality in thermal processes by including derivatives of non-integer order in the time and space domains, which provide extra degrees of freedom that better match simulations to experimental data. The derivation of numerical formulations is performed based on Caputo operator, RiemannLiouville operator and integration schemes are approximated based on the GrunaldLetnikov approximation. Comparative studies on the traditional Fourier and Cattaneo-Vernotte-type non-Fourier models have shown that, fractional-order models are much more accurate, especially predicting transient thermal responses, energy storage hysteresis and non-Gaussian diffusion in porous media. The solidity and stability of the fractional framework is validated by simulation studies where the fractional parameters serve as a tunable index in order to recapitulate material-specific memory and diffusion properties. These results have demonstrated that fractional-order models provide better flexibility, stability and physical interpretability and as such can be an effective tool in developing next generation thermal systems design. The suggested framework also provides a route to incorporate data-based approaches to identify the parameters and combine multi-physics coupling in hybrid ways and make fractional modeling a perspective to be developed in the future in terms of energy-efficient and adaptive thermal managements systems.