基于非傅里叶瞬态热阻抗分析一维导热中的热弛豫效应

Analysis of thermal relaxation effect in one-dimensional heat conduction based on non-Fourier transient thermal impedance

中文关键词: 热弛豫  非傅里叶导热  CV模型  瞬态热阻抗

英文关键词: Thermal relaxation  Non-Fourier thermal conductivity  CV model  Transient thermal impedance

中文摘要:

      温度场的重建滞后于热扰动发生的时间被称为热传导的弛豫效应。现有基于傅里叶定律得出的瞬态热阻抗中不含反映热弛豫效应的时变项，因此无法用于描述当热量以有限速度传播时的一维纳尺度导热问题。本文基于显含热弛豫时间项的Cattaneo-Vernotte模型（CV模型），针对一维有限长平板在第一类边界条件下的双曲型导热问题，首先通过变量代换及分离变量法获得了温度时空分布的解析解，在解的构成中区分了热扩散项与考虑有限热传播速度的波动项的贡献；其次，给出了考虑热弛豫效应的瞬态热阻抗修正表达式，并阐明了对经典瞬态热阻抗定义进行理论扩展的合理性；最后，以典型的一维硅纳米输入功率器件导热问题为对象，分别将基于傅里叶瞬态热阻抗与非傅里叶瞬态热阻抗求解得出的最高温度与CV模型有限元求解进行对比，证明了非傅里叶瞬态热阻抗在揭示热弛豫影响中的可行性。这一方法的提出为纳米尺度下一维化复杂热网络的高效计算提供了可能。

英文摘要:

      The delay in the reconstruction of the temperature field compared to the time of thermal disturbance is called the relaxation effect of thermal conduction. The existing transient thermal impedance based on Fourier's law does not contain a time-varying term reflecting thermal relaxation effects, so it cannot be used to describe one-dimensional nanoscale thermal conductivity problems when heat propagates at a finite speed. This article is based on the Cattaneo-Vernotte model (CV model) with an explicit thermal relaxation time term. For the hyperbolic heat conduction problem of a one-dimensional finite length flat plate under the first type of boundary conditions, an analytical solution of the temperature spatiotemporal distribution is first obtained through variable substitution and separation of variables. In the composition of the solution, the contribution of the thermal diffusion term and the wave term considering finite heat propagation velocity are distinguished; Secondly, a transient thermal impedance correction expression considering thermal relaxation effect was provided, and the rationality of theoretical extension of the classical definition of transient thermal impedance was elucidated; Finally, taking the typical thermal conductivity problem of one-dimensional silicon nano-input power devices as the object, the highest temperature obtained based on Fourier transient thermal impedance and non-Fourier transient thermal impedance solutions were compared with the finite element solution of the CV model, demonstrating the feasibility of non-Fourier transient thermal impedance in revealing the effects of thermal relaxation. The proposal of this method provides the possibility for efficient computation of one-dimensional complex thermal networks at the nanoscale.