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首页> 外文期刊>International Journal of Heat and Mass Transfer >Heat conduction in a semi-infinite medium with a spherical inhomogeneity and time-periodic boundary temperature
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Heat conduction in a semi-infinite medium with a spherical inhomogeneity and time-periodic boundary temperature

机译：具有球形非均匀性和时间周期边界温度的半无限介质中的热传导

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摘要

We solve the problem of heat conduction in a homogeneous media below a planar boundary subjected to time-periodic temperature (of frequency ω), in the presence of a spherical inhomogeneity (of radius R), whose center is at distance d > R from the boundary. In the absence of the sphere, the well known one dimensional solution can be regarded as an oscillating thermal boundary layer of displacement thickness δ= (2α/ω)~(1/2), where α is the heat diffusivity. The general solution depends on four dimensionless parameters: d/R, δ/R, the heat conductivity ratio k and the heat capacity ratio C. An analytical solution is derived as an infinite series of Bessel functions, which converges quickly. The results are illustrated and analyzed for a given accuracy and for a few values of the governing parameters. The general solution can be simplified considerably for asymptotic values of the parameters. A first approximation, obtained for R/d «1, pertains to an unbounded domain. A further approximate solution, for R/δ «1, while k and C are fixed, can be regarded as pertaining to a quasi-steady regime, and is similar in structure to Maxwell's solution for steady state. However, its accuracy deteriorates for k «1, and a solution, coined as the insulated sphere approximation, is derived for this case. Comparison with the exact solution shows that these approximations are accurate for a wide range of parameter values. Besides providing insight, they can be employed for solving in a simple manner more complex problems, e.g. effective properties of a heterogeneous medium made of an ensemble of spherical inclusions.
机译：我们解决了在球形边界不均匀（半径R），中心距d> R的球形不均匀性的情况下，在平面边界下方受时间周期温度（频率ω）影响的均匀介质中的导热问题。边界。在没有球体的情况下，可以将众所周知的一维解视为位移厚度为δ=（2α/ω）〜（1/2）的振荡热边界层，其中α为热扩散率。通用解取决于四个无量纲参数：d / R，δ/ R，热导率k和热容比C。解析解是由无穷贝塞尔函数级数导出的，它们迅速收敛。对于给定的精度和控制参数的一些值，将对结果进行说明和分析。对于参数的渐近值，可以大大简化一般解决方案。对于R / d«1获得的第一近似值与无界域有关。当k和C固定时，对于R /δ«1的另一种近似解可以视为属于准稳态方案，其结构与稳态的Maxwell解相似。但是，其精度在k«1时变差，在这种情况下，得出了一个称为绝缘球近似的解决方案。与精确解决方案的比较表明，这些近似值对于各种参数值都是准确的。除了提供洞察力，它们还可以用于以简单的方式解决更复杂的问题，例如球形包裹体组成的异质介质的有效特性。

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