This paper presents a theoretical analysis of a one-dimensional multilayer heat transfer problem with diffusion, advection and linear temperature-dependent heat generation occurring in each layer. A general solution of the problem is derived. Orthogonality of eigenfunctions is proved, and an explicit expression for the eigenequation is derived. The special case of a two-layer body is discussed. It is shown that, under specific conditions, this problem admits two types of imaginary eigenvalues, one of which is related to divergence of temperature at large times, corresponding to the thermal runaway phenomenon in batteries. The impact of various problem parameters related to diffusion, advection and heat generation on the appearance of imaginary eigenvalues is discussed. Specifically, due to the directional nature of fluid flow, advection in each layer of a two-layer body has opposing impact on the occurrence of imaginary eigenvalues. It is also shown that a balance between heat generation, diffusion and advection determines whether an imaginary eigenvalue is encountered, and consequently, whether thermal runaway occurs. Results presented here expand the theoretical understanding of multilayer heat transfer, and may also contribute towards improved thermal design of multilayer engineering systems such as flow batteries.

本文对一维多层传热问题进行了理论分析，该问题具有在每一层中发生的扩散、平流和线性温度相关的发热。推导出问题的一般解决方案。证明了本征函数的正交性，并推导出本征方程的显式表达式。讨论了两层体的特殊情况。结果表明，在特定条件下，该问题存在两种虚特征值，其中一种与大时间温度发散有关，对应于电池中的热失控现象。讨论了与扩散、平流和发热相关的各种问题参数对虚特征值出现的影响。具体来说，由于流体流动的方向性，两层体的每一层中的平流对虚特征值的出现具有相反的影响。还表明，热量产生、扩散和平流之间的平衡决定了是否遇到虚特征值，从而确定是否发生热失控。这里介绍的结果扩展了对多层传热的理论理解，也可能有助于改进多层工程系统（如液流电池）的热设计。