Linear, trigonometric and hyperbolic profiles of thermal effusivity in the Liouville space and related quadrupoles: Simple analytical tools for modeling graded layers and multilayers

Abstract

This paper presents a series of new high-level analytical tools for the modeling of one-dimensional, transient or periodic, heat transfer in media presenting graded thermal properties (conductivity and/or specific heat), possibly in a layered configuration. They capitalize on a recent work on sequences of analytically solvable profiles and the related exact temperature solutions. These profiles describe the square root of thermal effusivity or its inverse; the independent variable is the square root of the integrated diffusion time along the considered path, as obtained after a Liouville transformation. The profiles addressed here are linear, hyperbolic or trigonometric functions of this variable. A systematic presentation is given on how to build these profiles in both partner forms, on the so-called Liouville inverse transformation to step back into the physical-depth space, and on the three quadrupole formulations. As compared to other graded profiles from the literature, the three transfer matrices are very easy to compute (only elementary functions are involved). The quadrupole approach is particularly suitable for modeling multilayers. We apply it to calculate the thermal response of a two-layer system with a graded coating of one or other of the three classes. The modulated and the transient regime have been considered. The ease of obtaining these results indicates that, upon proper arrangement, these three classes of solvable profiles may be used to compute the thermal response of continuously (or piece-wise continuously) heterogeneous media of arbitrary complexity (e.g., functionally graded materials). This also paves the way to new methods for photothermal inversion. Other research fields could benefit from these tools insofar as the evolution equation governing the observed phenomena involves variable coefficients (e.g., advection-diffusion equation, wave equation, etc.).