A class of mathematical models are proposed for modelling and numerical simulation of coupled radiative and conductive heat transfer in non-grey absorbing and emitting media under phase change. Progress in this area of mathematical modelling would contribute to a sustainable future manufacturing involving high temperature and phase change. Accurately predicting phase-change interface is the crucial step for these applications in non-grey semi-transparent media. In the present study, the conduction and radiation effects are analyzed by a set of nonlinear partial differential equations and a linear integral equation, respectively. The proposed model forms a system of nonlinear integro-differential equations and it accounts for both thermal radiation and phase change in the design. For non-grey media, the spectrum is divided into a sequence of finite intervals of frequency bands with averaged absorption coefficients resulting in coupled systems to be solved for each frequency band. The coupled equations are approximated using a second-order method in both time and space. Using discrete ordinates for the angular discretization of the integral equation for the radiation effects, a Newton-type algorithm is used to deal with the nonlinear systems. Numerical results are presented for several test problems in both grey and non-grey media, and comparisons to simulations without radiation are also shown in this study. The findings here could be used to understand effects of thermal radiation in non-grey absorbing and emitting media under phase change.

中文翻译：

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相变下非灰色吸收和发射介质中辐射传热的建模和模拟


提出了一类数学模型，用于对相变下非灰色吸收和发射介质中耦合的辐射和传导传热进行建模和数值模拟。数学建模这一领域的进步将有助于涉及高温和相变的可持续未来制造。准确预测相变界面是非灰色半透明介质中这些应用的关键步骤。在本研究中，分别通过一组非线性偏微分方程和线性积分方程分析了传导和辐射效应。所提出的模型形成了一个非线性积分微分方程组，它在设计中同时考虑了热辐射和相变。对于非灰色介质，频谱被划分为具有平均吸收系数的有限频带间隔序列，从而为每个频带求解耦合系统。耦合方程在时间和空间上都使用二阶方法进行近似。使用离散坐标对辐射效应的积分方程进行角度离散化，使用牛顿型算法来处理非线性系统。本文介绍了灰色和非灰色介质中几个测试问题的数值结果，并且还显示了与无辐射模拟的比较。这里的发现可用于了解相变下非灰色吸收和发射介质中的热辐射的影响。