On the optimization of heat rectification in graded materials M.G. Naso, E. Vuk, F. Zullo ⇑ DICATAM, Università degli Studi di Brescia, Via Branze 43, Brescia, Italy article info Article history: Received 14 March 2019 Received in revised form 22 July 2019 Accepted 2 August 2019 Available online 12 August 2019 Keywords: Thermal rectification Graded materials Fourier law Porous silicon abstract In this paper we present a method to optimize the performances of a thermal rectifier. The approach, developed from a simple physical observation due to Peyrard (2006), allows to find the particular spatial distribution of the composition along the material giving a reasonably high value of the rectification coef- ficient. The method is applied to porous silicon devices with variable pore sizes. We obtain an improve- ment of the values of the rectification coefficient present in literature. Ó 2019 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded materials are materials designed with a given gradation in the composition in order to achieve specific per- formances or functions. The change in composition along the vol- ume of the material provides non-uniform values of macroscopic properties, such as density, thermal conductivity, specific heat, elasticity, plasticity, and electrical capacitance. Today it is possible to obtain customized structural properties in specific direction with different manufacturing approaches (see e.g. [18] for a recent review on these aspects). An example of graded materials are bin- ary alloys A c B 1c , where A and B are two different atomic species and c 2 0; 1 ð Þ is the content of the specie A, see [4,17]. Another example are porous substances, where the porosity / (i.e. the ratio of the volume of the pores in a region of the device divided by the total volume of the region) changes across the material [2,6]. In this work we are interested in thermal rectification proper- ties of graded materials. Thermal rectifiers are the thermal ana- logues of the electronic diodes acting as rectifier of electric currents: in a thermal rectifier the heat can flow preferably in one direction, meaning that by applying a thermal gradient to the boundaries of the material with opposite directions, a notably difference of the corresponding heat fluxes is obtained. For graded materials the thermal conductivity k is not only a function of the temperature T but also of the variable measuring the gradation in composition. For binary alloys A c B 1c it is a function of T and of the species content c (see e.g. [20]), for porous materials it is a function of T and of the porosity / (see e.g. [2,5]). Further it may depend also on the geometry of the device. For example the ther- mal conductivity of thin wires is influenced by the diameter of the wire (see e.g. [22]). The performance of the thermal rectifier can be evaluated by the rectification coefficient, given by the ratio of the absolute values of the heat fluxes in the two opposite direc- tions (see [4,5,13,19]): R¼ : jq d j jq r j ; ð1Þ where q d is the direct heat flux and q r is the reverse heat flux. To grater values of R there correspond higher performances of the ther- mal rectifier [6,19]. In a certain number of papers different suitable configurations of given materials and the corresponding values of the rectification coefficient (1) have been examined and explored by different points of views: for example in [13] the behavior of the rectification coefficient under transient conditions (not at steady state) for a composite of two different materials has been considered, in [4] the authors give the rectification performances of a graded silicon–germanium alloy Si 1c Ge c for different linear distribution of the composition c and for different geometries of the material, in [5,7,16] the thermal rectification of porous silicon devices with a porosity varying with different power laws and/or a variable radius of the pores have been given. In this work we don’t assume to have, a priori, a given configu- ration of the material but we consider the opposite point of view: given a material whose thermal conductivity k is a known function of the temperature T and of a composition parameter m, we look for a systematic way to choose the spatial distribution of the compo- sition m x ðÞ and the geometry of the device presenting the more https://doi.org/10.1016/j.ijheatmasstransfer.2019.118520 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: mariagrazia.naso@unibs.it (M.G. Naso), elena.vuk@unibs.it (E. Vuk), federico.zullo@unibs.it (F. Zullo). International Journal of Heat and Mass Transfer 143 (2019) 118520 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt