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研究生： 張宇良
Chang, Yu-Liang
論文名稱： 利用有限元素法求解相變化熱傳問題
Phase-Change Heat Transfer Analysis Using Finite Element Method
指導教授： 趙隆山
Chao, Long-Sun
學位類別： 碩士
Master
系所名稱： 工學院 - 工程科學系
Department of Engineering Science
論文出版年： 2012
畢業學年度： 100
語文別： 中文
論文頁數： 107
中文關鍵詞： 相變化 、潛熱 、熱傳
外文關鍵詞： Phase change, latent heat, Heat Transfer
相關次數： 點閱：98  下載：0 
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    • 凝固在材料製程中扮演著很重要的角色，例如金屬材料之鑄造、熱電材料製程、半導體材料長晶等。其熱傳行為探討是應該被重視的。本文希望利用數值方法分析凝固過程中之溫度場變化情形。

    本文研究對象是材料在凝固過程中潛熱的影響。數值方法採用有限元素法，建立自我滿足的網格形狀，並使用帶狀存取模式，直接利用高斯消去法求解溫度場。
    在計算過程中，數學模型利用有限元素法做數值運算。使用單區法之等效比熱法來處理潛熱的效應，以一維史蒂芬問題(Stefan problem)和二維Rathjen相變化問題來作測試與分析。在有限元素法中，通常都是先將邊界帶入統御方程式再一同推導出欲求解之方程式。而在非線性的相變化熱傳問題中，分割成矩形元素，可利用高斯積分法離散處理潛熱問題，但三角形元素之閉合式積分公式不利於潛熱釋放之計算。

    本文提出了利用數值積分中的積分點來解決三角形元素之問題。並針對不同積分點Case來比較總誤差(total error)，而對稱的數值積分點有較高的準確性。並把總誤差繪製成機率密度曲線，有效的比較各種不同Case之積分點處理之準確性。

    Solidification plays a very important role in material processes, for example, the casting process of a metal material, the fabrication process of thermoelectric materials, and the crystal growth of semiconductor materials. The behavior of the heat transfer in solidification processes should be taken seriously for the studies. In this thesis, the temperature fields of solidification processes are analyzed by using the numerical method.

    The study is to handle the latent-heat influence in solidification processes of materials. The numerical scheme is the finite element method. The banded storage mode is used to store the coefficient matrix and the Gaussian elimination method is utilized to solve the temperature fields.

    In the thesis, the effective specific heat method of the single-domain schemes is employed to deal with the latent-heat effects. The one-dimensional Stefan problem and the two-dimensional Rathjen one are applied to test and analyze. In the finite element method, the finite-element equations are derived from the combination of the governing equation and the boundary conditions generally. For the nonlinear phase-change problems, the Gauss integration method can be applied to calculate the latent heat releases of rectangle elements. However, the closed-form integration formula is not good for triangle elements to compute the latent heat effects.

    In the research, the integration methods of integration points are used to deal with the problem of triangle elements mentioned above. The total errors are employed to compare and analyze the numerical results for different integration methods with different number of integration points. The symmetrical integrations points have the better temperature solutions. The computing results are also investigated by using the PDF (Probability Density Function), whose analysis is more effective than the total-error one.

    摘要.......................................I ABSTRACT..................................II 致謝.......................................III 目錄.......................................IV 表目錄......................................VII 圖目錄......................................VIII 符號說明....................................XII 第一章 緒論..................................1 1-1 前言....................................1 1-2文獻回顧..................................2 1-3 研究方法與目的............................4 第二章 熱傳問題之數學模式.......................6 2-1 暫態線性熱傳問題...........................6 2-2史蒂芬問題(STEFAN PROBLEM).................7 2-2-1等效比熱法(EFFECTIVE SPECIFIC HEAT METHOD).....8 2-3RATHJEN問題..............................10 第三章 有限元素法之理論分析.....................15 3-1 基本元素形狀.............................15 3-2加勒金法(GALERKIN’S METHOD)..............16 3-3內插函數(INTERPOLATION FUNCTION).........17 3-4數值積分處理.............................18 第四章 數值分析..............................23 4-1四邊形元素...............................23 4-1-1四邊形元素之內插函數.....................25 4-1-2矩形元素的計算..........................26 4-2三角形元素...............................28 4-3溫度場之元素方程式........................30 4-4三角形高斯積分法..........................33 4-4-1 三角形積分點積分.......................36 4-4-2 內插函數與座標轉換關係..................37 4-4-3 比熱項之計算模型.......................39 4-5求解流程與收斂條件.........................40 4-6解聯立方程式..............................41 第五章 結果與討論.............................55 5-1 一維暫態熱傳問題..........................56 5-1-1 矩形元素數值積分........................56 5-1-2 三角形元素數值積分.......................56 5-2 史蒂芬問題................................58 5-2-1 矩形元素數值積分.........................58 5-2-2 三角形元素數值積分........................58 5-3 史蒂芬問題討論..............................60 5-4 RATHJEN問題及二維鑄模之凝固問題...............61 第六章 結論.....................................90 參考文獻........................................91 附錄 程式範例....................................95

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