Optimum Experimental Set-Up for Thermal Conductivities Measurement of Composite Materials

The optimum set-up of a two-dimensional (2D) transient heat conduction experiment aimed at estimating simultaneously the out-of- and in-plane thermal conductivities, namely kx and ky, of composite materials is designed. The experimental apparatus here investigated consists of a thin electrical heater put in contact with two larger samples of the same material and thickness, so that the heat diffusion within the specimens is 2D. Moreover, temperature measurements useful for the estimation procedure are obtained only from the unheated surface of the sample which can be subject either to heat losses with the external environment or to heat losses with an insulating material. A D-optimum criterion, known as Δ+ criterion, is applied to design the optimal experiment which is determined not only as the optimum heating and experiment times, but also in terms of the optimum sample aspect ratio and the optimum width of the heated region. One the one hand, this criterion ensures the minimization of the confidence region of the estimated parameters; on the other hand it allows the determinant of the Fisher matrix to be maximized. From a mathematical point of view, the 2D heat conduction problem is modeled through a rectangular plate (i.e., the sample) partially heated at the front boundary through a surface heat flux, whereas the opposite boundary can be subject either to heat transfer by free convection with the surrounding air or to heat losses with an insulating material, through a heat transfer coefficient. All the other boundaries are always kept insulated. Once the orthotropic model is reduced to the common isotropic heat diffusion equation, the exact analytical temperature solution is obtained starting from a generalized solution available in the specialized literature. After that the superposition principle can also be applied to account for a finite heating period. Then, the so-called scaled sensitivity coefficients of temperature with respect to the parameters of interest are computed, and a sensitivity analysis is performed to establish whether correlation between them exists. Finally, the optimal experiment aimed at estimating simultaneously the in- and out-of-plane thermal conductivities as well as the heat transfer coefficient (which represent a disturbance of the thermal model) is designed applying the mentioned criterion.