- 期刊
凝固問題之混合拉式轉換法分析
Analysis of Hybrid Laplace Transform Method on Solidification Problems
摘要
於鑄造過程中，凝固變化是相當重要的物理現象，處理凝固問題的數值模式不少，一般是直接將能量方程式以有限差分離散或有限元素積分來求解。本文提出一個不一樣的求解方法，也就是應用混合拉氏轉換法，並搭配數種潛熱效應的處理方法，來求解非線性凝固相變化問題。文中將介紹混合拉氏轉換法的原理，以及處理潛熱效應的數種方法，包含割線法、比熱法、熱焓法、等效比熱－熱焓法。本文以所提出的方法求解有正解的凝固問題，一維史蒂芬問題、紐曼問題和二維Rathjen問題；並將數值解與正解進行準確性和誤差比較。由分析的結果發現，在求解史蒂芬問題與紐曼問題時，混合拉氏轉換法搭配熱焓法有最好的準確性。比起有限差分法，混合拉氏轉換法搭配比熱法在相同的時間步伐與空間間隔下，明顯地有較佳的準確性，因此本文的方法可以有效地簡化數值求解的過程。最後，本文將所提出的方法運用於一實際鑄造過程的凝固問題。
並列摘要
Solidification is an important physical phenomenon in the casting process. Generally, finite difference and finite element methods are frequently used to solve the governing equation of solidification problems. The present study employs a method involving the combined use of the hybrid Laplace transform and various ways to deal with the effect of latent heat to investigate nonlinear phase-change solidification problems. The theorems of hybrid Laplace transform method are presented. Secant method, effective specific heat method, enthalpy method and effective specific heat-enthalpy method are used to estimate the effect of latent heat in phase-change problems. The present numerical methods are used to solve one-dimensional Stefan and Neumann phase-change problems and two-dimensional Rathjen phase-change problem. Comparisons of the present numerical and analytical solutions are performed. Results show that the present numerical results agree well with the analytical ones. The numerical scheme combining the hybrid Laplace transform and enthalpy method has the most accuracy when dealing with the Stefan and Neumann phase-change problems. In the same time step and spaced at intervals, the numerical scheme combining the hybrid Laplace transform and effective specific heat method has the more accuracy than finite difference method. The present methods are valid for simplifying the numerical solving process of phase-change problem. Eventually, the present methods are applied to a practical solidification problem.