Analytical and Numerical Study of a Convection-Diffusion-Reaction-Source Problem in Multilayered Materials

TL;DR

The paper develops a 1D transient heat-transfer model for multilayer media governed by a Convection-Diffusion-Reaction-Source (CDRS) equation, incorporating diffusion, advection, temperature-proportional generation, external sources, and interfacial thermal contact resistance. An explicit analytical solution is obtained via Fourier methods after nondimensionalization and advection removal, producing an infinite real eigen-spectrum from a transcendental eigenvalue condition and an orthogonality relation for the eigenfunctions. The authors extend prior two-layer results to $m$ layers and introduce a convergent explicit finite-difference scheme for numerical simulation, demonstrated on a four-layer Ni-Al-Cu-Ag example that shows interface temperature jumps and layer ordering consistent with material properties. This work enhances theoretical understanding and provides a practical tool for the thermal design of multilayer engineering systems, enabling accurate temperature profiles under realistic features such as external sources and interfacial resistance.

Abstract

In this work, a thermal energy transfer problem in a one-dimensional multilayer body is theoretically analyzed, considering diffusion, advection, internal heat generation or loss linearly dependent on temperature in each layer, as well as heat generation due to external sources. Additionally, the thermal contact resistance at the interfaces between each pair of materials is taken into account. The problem is mathematically modeled, and explicit analytical solutions are derived using Fourier techniques. A convergent finite difference scheme is also formulated to simulate specific cases. The solution is consistent with previous results. A numerical example is provided, demonstrating the coherence between the obtained results and the physical behavior of the problem. This work was recently published for a two-layer body; the generalization to m-layer bodies allows for conclusions that enhance the theoretical understanding of heat transfer in multilayer materials and may contribute to improving the thermal design of multilayer engineering systems.

Figures (4)

• Figure 1: General scheme of the problem of interest.
• Figure 2: Heat source.
• Figure 3: Temperature distribution for Ni-Al-Cu-Ag
• Figure 4: Scheme of intersection of the functions $q(x)$ and $r(x)$. To the left for case \ref{['case1']} and to the right for case \ref{['case2']}.

Theorems & Definitions (1)

• Example 1