Numerical Computation and Analysis of the Numerical Scheme Order of the Two-Dimensional Temperature Field of Thermoelectric Coolers Cold Substrate

Abstract

This paper presents a numerical computation and analysis of the numerical scheme order of the two-dimensional temperature distribution in cold substrate of thermoelectric coolers with the presence of a discrete heat source. The finite difference method is applied to obtain the linear system of equations and the temperature at each node of the domain. The numerical code is implemented in the free software package GNU Octave. In the first part, a mesh independency study is carried on in order to establish a relation between accuracy and computational cost, comparing the simulated results with results from an approximate analytical solution available in the literature. It is observed excellent agreement between numerical and the analytical results. In the second part, an analysis of the overall order of the numerical scheme is carried on by considering numerical solutions for three different meshes and different formulations of the boundary conditions. Results show that the overall order of the numerical scheme is first-order even if full second-order discretization scheme is applied for all grid points, due to the presence of the discrete heat source. Those results are observed for the considered discretization schemes and are confirmed after using a continuous formulation for the heat source term.