用糊状层法验证了求解Stefan问题的数学模型

R.S. Yurkov, L. Knysh
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引用次数: 1

摘要

由于太阳能的周期性可用性,它的使用有局限性:太阳能发电厂不能在夜间运行,在沉闷的天气里也不起作用。解决这一问题需要在转换回路中引入能量存储和复制系统。在储能系统中,固液相变模块具有显著的能量、生态和成本优势。这种类型的模块中的物理过程由一组具有相界面特定边界条件的非平稳非线性偏微分方程来描述。本文给出了一种求解储热材料斯特芬问题的方法的验证。糊状层方法的使用使得将Stefan问题的经典数学模型简化为具有考虑相变潜热的隐式热源的非平稳热传导问题成为可能。相变被认为发生在由固相温度和液相温度决定的中间区域,而不是发生在无限区域内。为了开发Python代码,使用了一种隐式计算方案,其中固体和液体温度保持恒定,并在数值实验过程中确定。计算机模拟和算法验证所选择的物理模型是恒定环境温度下水面上冰层形成的过程。数值计算结果可以确定固、液相温度场和相界面位置,并计算其推进速度。通过分析一维情况下等相界面推进速度下Stefan问题的经典解析解,验证了所提出的算法。验证系数的值是使用特殊的内置Python函数从非线性方程的数值解确定的。将所考虑的物理模型的数据代入解析解,并与采用糊状层法得到的数值模拟数据进行比较,结果非常吻合,从而证明了所开发的计算机算法的正确性。这些研究将允许人们根据糊状层方法开发的Python代码来计算具有固-液相变的储热系统,并考虑其几何特征、温度水平和实际边界条件。
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Verification of a mathematical model for the solution of the Stefan problem using the mushy layer method
The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.
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