以修正GLS法與有限差分法分析凝固熱傳問題

Numerical Analysis of Solidification Heat Transfer Problems By Using Modified GLS Scheme and Finite Difference Methods

劉宇岍(Yu-Chien Liu) ； 趙隆山(Long-Shan Chao) ； 武易璋(I-Chang Wu)

《鑄造工程學刊》 46卷3期 (2020/09) Pp. 23-32

摘要

在凝固熱傳問題中，相變化的過程會有潛熱釋放的效應，而潛熱釋放會影響溫度場分布導致材料的微結構改變，所以潛熱釋放的影響對溫度場的分布就顯得相當重要。本文是以有限差分法和不同的數值方法與時間步伐方法模擬凝固熱傳問題。其中主要探討的有一維暫態性熱傳、一維史蒂芬問題和一維紐曼問題，處理潛熱釋放的數值方法有等效比熱法和等效比熱／熱焓法，時間步伐方法有固定時間步伐、GLS法（ε=1e-3）、修正GLS法（ε＜1e-3）和修正截尾誤差（修正LTE）。透過不同的數值方法與時間步伐方法來比較溫度場分布的準確性、潛熱釋放量和CPU運算時間，其中溫度場的準確性以總誤差（Total error）當作比較依據。從分析的結果發現，在求解一維暫態線性熱傳問題時，GLS法可以有效的節省CPU運算時間和維持一定的精準度；在求解一維史蒂芬問題時，等效比熱法搭配修正GLS法和修正截尾誤差確實能比GLS法求得更精準的數值解，而搭配等效比熱／熱焓法時，一階的Euler法卻比GLS法來的精準；在求解一維紐曼問題時，等效比熱法和等效比熱／熱焓法搭配修正GLS法和修正截尾誤差比GLS法精準。

關鍵字

有限差分法 ； 可變時間步伐 ； 修正GLS ； 等效比熱法 ； 史蒂芬問題 ； 紐曼問題

並列摘要

In the solidification heat transfer problems, the latent heat is released during the phase change process and affects the distribution of temperature field. In this study, the solidification heat transfer problems are analyzed by using finite difference methods, different numerical methods and time discretization techniques. Furthermore, the one-dimensional transient heat transfer problem, Stefan problem and Neumann problem will be discussed. The numerical methods for calculating latent heat release are effective specific heat method and effective specific heat/enthalpy method. As for the time discretization techniques, the uniform time step, GLS methods(ε=1e-3), modified GLS methods(ε＜1e-3) and modified local time truncation error scheme are utilized. In order to compare the accuracy of various numerical methods, the temperature field distribution, latent heat release and CPU computation time are important basis. From the analysis results, it is found that the GLS method can effectively save the CPU computation time and maintain precision when solving the one-dimensional transient heat transfer problem. In solving the Stefan problem, the effective specific heat method combined with the modified GLS method and modified local time truncation error scheme can be more accurate than GLS methods. For the effective specific heat/enthalpy method, the first-order Euler methods is more precise than the GLS methods. However, when solving the Neumann problem, the effective specific heat method and the effective specific heat/enthalpy method combined with the modified GLS method and the modified local time truncation error scheme are more accurate than the GLS methods.

並列關鍵字

GLS method ； effective specific heat method ； effective specific heat/enthalpy method ； adaptive time step