Abstract The present paper addresses uni-directional, unsteady, heat conduction in a plane wall with uniform initial temperature and equal temperature prescribed at the surfaces. The thermal diffusivity of the solid is assumed to be nearly invariant with temperature. A hybrid analytical /numerical procedure named the Transversal Method Of Lines (TMOL) transforms the one-dimensional, unsteady heat conduction equation along with the initial temperature in rectangular coordinates into an equivalent one-dimensional, “quasi-steady” heat conduction equation applied at a pre set time. In other words, a partial differential equation of parabolic type is converted into an ordinary differential equation of second order with the space coordinate as the independent variable and the time absorbed as an embedded parameter. In this work, the analytical/numerical mid-plane and mean temperature profiles, as well as the analytical-numerical total heat transfer are produced by various time levels within TMOL framework, namely 1−TMOL, 2−TMOL and 3−TMOL. Overall, from a qualitative standpoint, the analytical/numerical mid-plane and mean temperature profiles and total heat transfer with the the 3−TMOL approach exhibit excellent quality over the crucial “small time” sub-domain evidencing relative errors that stay within a 5% margin.

中文翻译：

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具有相同表面温度的平面壁中非定常热传导的半解析解：线的横向方法 (TMOL) 划定到“小时间”子域

摘要 本文讨论了具有均匀初始温度和表面规定温度相等的平面壁中的单向、非定常热传导。假定固体的热扩散率几乎不随温度变化。称为横向线法 (TMOL) 的混合分析/数值程序将一维非定常热传导方程与直角坐标中的初始温度一起转换为等效的一维“准定常”热传导方程。预先设定的时间。换言之，将抛物型偏微分方程转化为以空间坐标为自变量、吸收时间为嵌入参数的二阶常微分方程。在这项工作中，分析/数值中平面和平均温度分布，以及分析-数值总传热是由 TMOL 框架内的不同时间水平产生的，即 1-TMOL、2-TMOL 和 3-TMOL。总体而言，从定性的角度来看，使用 3-TMOL 方法的分析/数值中平面和平均温度分布以及总热传递在关键的“小时间”子域中表现出出色的质量，证明相对误差保持在 5 ％ 利润。