Heat Transfer Process with Solid-solid Interface: Analytical and Numerical Solutions

Diana Rubio,Guillermo F Umbricht,Domingo A Tarzia

doi:10.37394/23206.2021.20.42

Abstract

This work is aimed at the study and analysis of the heat transport on a metal bar of length L with a solid-solid interface. The process is assumed to be developed along one direction, across two homogeneous and isotropic materials. Analytical and numerical solutions are obtained under continuity conditions at the interface, that is a perfect assembly. The lateral side is assumed to be isolated and a constant thermal source is located at the left-boundary while the right-end stays free allowing the heat to transfer to the surrounding fluid by a convective process. The differences between the analytic solution and temperature measurements at any point on the right would indicate the presence of discontinuities. The greater these differences, the greater the discontinuity in the interface due to thermal resistances, providing a measure of its propagation from the interface and they could be modeled as temperature perturbations. The problem of interest may be described by a parabolic equation with initial, interface and boundary conditions, where the thermal properties, the conductivity and diffusivity coefficients, are piecewise constant functions. The analytic solution is derived by using Fourier methods. Special attention is given to the Sturm-Liouville problem that arises when deriving the solution, since a complicated eigenvalue equation must to be solved. Numerical simulations are conducted by using finite difference schemes where its convergence and stability properties are discussed along with physical interpretations of the results.

Highlights

Heat transfer problems in multilayer or solid-solid interface materials have been arisen in a several applications in science and engineering [3]
This work focus on the analytical solution to a heat transfer problem that it is assumed to occur along a bar composed by two different materials with continuity conditions at the solid-solid interface
A perfect assembly between the two parts are considered, so that differences between the analytical solution and experimental measurements will provide an amount of thermal dissipation between the two materials, that would be useful to model tension and roughness at the interface as well as solid-solid thermal resistance

Summary

Introduction

Heat transfer problems in multilayer or solid-solid interface materials have been arisen in a several applications in science and engineering [3]. A large number of articles are devote to the study of thermal, electromagnetic and/or optical properties of composed materials, among them [2]-[3], [5], [9]-[13], [17]-[23] These types of problems are generally approached experimentally or through numerical simulations. This work focus on the analytical solution to a heat transfer problem that it is assumed to occur along a bar composed by two different materials with continuity conditions at the solid-solid interface. Analytical and numerical solutions to this problem are useful to predict temperatures profiles under different situations assuming perfect assembly between materials and to detect discontinuities at the interface.

Mathematical Framework

The steady-state problem

The transient problem

Numerical simulations

Conclusion

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