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系統識別號 U0026-1808201609222500
論文名稱(中文) 利用有限元素法及可變時間步伐探討相變化之熱傳問題
論文名稱(英文) Analysis of Phase Change Problems by Finite Element Method and Adaptive Time Step Scheme
校院名稱 成功大學
系所名稱(中) 工程科學系
系所名稱(英) Department of Engineering Science
學年度 104
學期 2
出版年 105
研究生(中文) 王姿婷
研究生(英文) Tzu-Ting Wang
學號 N96031287
學位類別 碩士
語文別 中文
論文頁數 111頁
口試委員 指導教授-趙隆山
口試委員-張建宏
口試委員-彭勳章
中文關鍵字 有限元素法  可變時間步伐  相變化  等效比熱法  等效比熱/熱焓法 
英文關鍵字 finite element method  adaptive time step  phase change  effective specific heat method  enthalpy/specific heat method 
學科別分類
中文摘要 潛熱釋放在凝固過程中的相變化是極為重要的物理現象,本文以有限元素法搭配等效比熱法、等效比熱/熱焓法、可變時間步伐之等效比熱法及可變時間步伐之等效比熱/熱焓法來處理潛熱效應,比較上述四種方法之所需計算時間、潛熱釋放多寡及溫度分佈準確度,並探討一維史帝芬問題、一維紐曼問題及二維瑞特延問題。其中利用溫度分佈之數值解與正解的總誤差(total-error)來比較各數值方法及節點的準確度。為使溫度分佈更精準,本研究藉由變換所採用之數值方法及調整參數來達到此目的,例如:縮小時間步伐、針對單位元素增加節點數、以不同形狀之元素組成求解區域及增加求解區域之節點數等等。
研究結果發現史帝芬問題加上可變時間步伐的等效比熱法溫度分佈及潛熱釋放的計算皆較準確,且電腦運算時間皆有減少,而等效比熱/熱焓法本身就很精準,因此加上可變時間步伐無法再提升溫度計算準確度,但可有效地減少運算時間;紐曼問題加上可變時間步伐亦不能增加準確度,但可提升計算效率;瑞特延問題使用三種元素節點所得液固界面皆很接近數值解,其中以四邊形元素平均誤差較小。
英文摘要 The phase change involving latent heat effect in a solidification process is an important physical phenomenon because it would affect the accuracy of the temperature distribution. In the thesis, FORTRAN programs of finite element method and adaptive time stepping scheme are written to simulate the heat transfer problems with phase change including one-dimensional Stefan and Neumann problems and two-dimensional Rathjen problem. The effective specific heat and the enthalpy/specific heat methods are applied to the calculation of the latent heat released during the phase change, which is related to the adaptive or uniform time step scheme and the node number and the geometry of the element, such as four-node quadrilateral and nine-node quadrilateral elements and three-node triangular element. Gaussian method is employed to solve the integral in the element equations. To compare the numerical methods, the accuracy of temperature, the release of latent heat, and the CPU time are utilized as the comparison base. From the analysis results, the quadrilateral element has the better accuracy of temperature than the triangular one. The adaptive time step of effect specific heat method could obtain greater accuracy of temperature and calculation efficiency than the uniform time step scheme. However, since the high accuracy is inherent in enthalpy/specific heat method, the adaptive time step is not more accurate than the uniform one. The calculated results of Neumann problem is similar to those of Stefan problem with the effective specific heat method. Whereas, the adaptive time step effective specific heat method could only reduce the calculation time but not improve the accuracy of temperature. For Rathjen problem, using triangular element brings out higher average error than applying quadrilateral element.
論文目次 目錄
摘要 I
Abstract II
誌謝 VII
表目錄 XII
圖目錄 XIII
符號說明 XXI
第一章 緒論 1
1-1前言 1
1-2文獻回顧 3
1-3研究方法與目的 6
第二章 相變化熱傳問題之數學模式 7
2-1暫態線性熱傳問題 7
2-2史帝芬問題(Stefan Problem) 8
2-3紐曼問題(Neumann problem) 10
2-4等效比熱法(Effective specific heat method) 13
2-5等效比熱/熱焓法 14
2-6瑞特延問題(Rathjen problem) 15
2-7可變時間步伐(Adaptive time step method) 17
第三章 有限元素法理論分析 22
3-1有限元素法基本概念 22
3-2加權殘值法(Weighted residuals approach) 24
3-3加勒金法(Galerkin’s method) 25
3-4內插函數 26
3-5元素形狀 26
第四章 有限元素計算及數值分析 32
4-1元素方程式 32
4-2四邊形元素 34
4-2-1四邊形元素之內插函數 36
4-2-2四邊形元素計算 39
4-3三角形元素 42
4-3-1三角形座標轉換[31]及內插函數推導 44
4-3-2三角形高斯積分法 46
4-4比熱項之元素矩陣 47
4-5潛熱釋放計算 49
4-6求解流程 51
第五章 結果與討論 57
5-1一維暫態熱傳問題 57
5-1-1四邊形元素求解 57
5-1-2三角形元素求解 58
5-2史帝芬問題(Stefan Problem) 58
5-2-1等效比熱法 59
5-2-2可變時間步伐之等效比熱法 60
5-2-3等效比熱法vs可變時間步伐之等效比熱法 60
5-2-4等效比熱/熱焓法 61
5-2-5可變時間步伐之等效比熱/熱焓法 61
5-2-6等效比熱/熱焓法vs可變時間步伐之等效比熱/熱焓法 62
5-2-7等效比熱法vs等效比熱/熱焓法 62
5-3紐曼問題(Neumann Problem) 63
5-3-1等效比熱法 63
5-3-2可變時間步伐之等效比熱法 63
5-4潛熱釋放分析 64
5-4-1史帝芬問題 65
5-5運算時間測試 66
5-6瑞特延問題 67
第六章 結論 107
6-1史帝芬問題 107
6-2紐曼問題 108
6-3瑞特延問題 108
參考文獻 109
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