臺灣博碩士論文加值系統

於於鑄造過程中，相變化或凝固是相當重要的物理現象，處理相變化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，來求解凝固過程之溫度場分佈。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。

Phase change or 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 phase change 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 problems. In the paper, 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 utilized to solve the one-dimensional Stefan and Neumann phase-change problems and the two-dimensional Rathjen phase-change problem. The comparison results show that the numerical solutions agree well with the analytical ones. The numerical scheme incorporating the hybrid Laplace transform and enthalpy methods is the most accurate when dealing with the Stefan and Neumann problems. Combined with the effective specific heat method, the hybrid Laplace transform scheme is more accurate than finite difference method. As a result, the proposed approach applying the hybrid Laplace transform method could provide an effective way in solving phase change problems.

摘要 I
致謝 III
目錄 IV
圖目錄 VI
符號說明 VII
第一章 緒論 1
1.1 文獻回顧 2
1.2 研究方法與目的 4
第二章 逆拉氏轉換之理論分析 5
2.1 傅氏積分轉換在逆拉氏轉換之應用 5
2.2 Dubner與Abate之逆拉氏轉換法 6
2.3 Durbin之逆拉氏轉換法 8
2.4 截尾誤差之分析 9
2.5 數值運算 11
2.6 求取最佳v值 12
第三章 相變化熱傳問題數值方法與分析 17
3.1 一維高低溫熱傳問題 18
3.2 史蒂芬問題 (Stefan Problem) 18
3.3 求解史帝芬問題之數值方法 20
3.3.1 割線法(Secant method) 20
3.3.2 等效比熱法 (Effective specific heat method) 23
3.3.3 熱焓法(Eethalpy method) 24
3.3.4 等效比熱-熱焓法 (Effective specific heat - Eethalpy method) 27
3.4 紐曼問題 (Neumann Problem) 28
3.5 求解紐曼問題之數值方法 29
3.5.1 等效比熱法 (Effective specific heat method) 29
3.5.2 熱焓法(Eethalpy method) 30
3.5.3 等效比熱-熱焓法 (Effective specific heat - Eethalpy method) 31
3.6 Rathjen問題 32
第四章 結果與討論 39
4.1 史蒂芬問題(Stefan Problem) 39
4.2 紐曼問題(Neumann Problem) 40
4.3 Rathjen問題 41
4.4 有限差分法(Finite Difference Method)與混合拉氏轉換法(Hybrid laplace transform method)比較 41
第五章 結論 60
參考文獻 62

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