Mathematical Problems in Engineering

Volume 2015, Article ID 697639, 10 pages

http://dx.doi.org/10.1155/2015/697639

## An Analytical Development of the Hyperbolic Behaviour of Micro Thermoelectric Coolers

^{1}Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio, Località Coppito, 67100 L’Aquila, Italy^{2}Department of Industrial and Information Engineering and Economics, University of L’Aquila, Via Giovanni Gronchi 18, 67100 L’Aquila, Italy

Received 9 March 2015; Revised 7 October 2015; Accepted 8 October 2015

Academic Editor: John D. Clayton

Copyright © 2015 Giulia De Aloysio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The transient behaviour of a micro thermoelectric cooler is described in the present paper through the theory of the thermal wave, involving the relaxation time. The internal heat generation due to the sudden application of the electric current is taken into account by means of the Heaviside function. The governing equations of both the semiconductors are solved by a modified Separation of Variables Method that allows us to have a better description of the device which can be obtained at early times. As regards the performances, the cooling load shows discontinuities due to the contributions of the travelling wave fronts of both the semiconductors. The results show that the coefficient of performance (COP) of the device obtained with the hyperbolic model is lower than that provided by the parabolic model at early times.

#### 1. Introduction

In the literature many models related to the transient behaviour of thermoelectric devices have been presented [1–9]. These systems have large application as coolers [9–12], thanks to the possibility of maintaining the junction temperature as low as required. Most of the models make use of the parabolic thermal diffusion equation: three-dimensional numerical coupled models have been proposed recently, in which the coupling of the thermal and the electric field is considered, [4, 10]. In particular, the coupling of both the fields is analyzed under unsteady and steady states for coolers in [4, 10].

However, the parabolic modelling is inadequate for devices with small dimensions, in particular in the analysis of their transient behaviour, during the start-up phase and the shut-down phase and when changing the operational parameters. In all of these cases it is more adequate to use the hyperbolic heat conduction model, which considers the relaxation time of the material [6–8]. This is a crucial parameter in the heat conduction theory because it takes into account the finite speed of the heat propagation. The relaxation time is absent in the classical theory of heat diffusion based on the Fourier law, whose intrinsic underlying assumption is that heat propagates in the material with an infinite speed.

Unlike the previously proposed model, in the present paper we describe the sudden application of the electric current by means of the Heaviside function. Following Haji-Sheikh and Beck’s approach [13], the thermal field in both the semiconductors is obtained analytically by means of a modified Separation of Variables (SoV, for short) Method for time-independent boundary conditions of the first kind. According to this method, the solution of the hyperbolic problem is obtained by modifying the solution of the corresponding parabolic problem through an appropriate time-dependent function. This function is due to the presence of the relaxation time and may be derived by imposing the modified solution which will be the solution of the hyperbolic problem. Then MATLAB ambient is used to implement the solution. Once the temperature distributions are known, the performances of the device, cooling load, and COP are evaluated. The results of the analysis show that the COP obtained by means of the hyperbolic model is lower than that provided by the parabolic model at early times.

#### 2. Analytical Formulation

The elementary unit of a thermoelectric cooler (TEC) module consists of two semiconductors, P and N, connected electrically in series, as shown in Figure 1. Because of the Peltier effect, by applying an electric current, heat is absorbed at the cold junction, whose temperature is denoted by , and heat is released at the hot junction, whose temperature is . The heat absorption at the cold junction is counteracted by Joule and Fourier’s effects. It is worth noting that the Thomson effect is neglected in this analysis. The following assumptions are made to perform the analytical transient modelling of the device:(a)The thermal conductivity and the electric resistivity are considered temperature-independent and computed at an average temperature , by using (23) (see ahead). This assumption is valid for low electric applied currents, as shown in [10].(b)Adiabatic side surfaces of the TEC are taken into account.(c)Semiconductors are employed in the analysis, which have the same geometrical characteristics, the cross-sectional areas and , and the lengths of the thermoelements, and . We assume that .(d)At the instant both the semiconductors have an ambient uniform temperature and a sudden constant uniform current is applied to the system.