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研究生： 蔡佳晟
Tsai, Chia-Cheng
論文名稱： 四端點為時變性之平板的熱傳分析
Nonhomogeneous Time-dependent Boundary Condition
指導教授： 李森墉
Lee, Sen-Yung
學位類別： 碩士
Master
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 121
中文關鍵詞： 熱傳導 、二維平板 、第一類邊界條件 、時變性 、溫度 、解析解
外文關鍵詞： heat conduction, rectangular plate, Dirichlet boundary condition, nonhomegeneous time-dependent, analytical solution
相關次數： 點閱：107  下載：3 
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    • 本文探討二維平板四端點為時變性溫度之熱傳問題。首先使用轉換函數將原系統之時變性溫度端點轉換為零，使用疊加原理與廣義傅立葉係數，將二維問題化簡為兩組一維問題，再使用移位函數法，將一維問題之非齊次邊界轉換成齊次邊界，最後利用特徵函數展開法求解。針對具時變性邊界的第一類型邊界條件且四端點亦皆為時變性之問題發展出一套簡單求解的流程，過程中無須積分變換，即可在不同案例中求出解析解或近似解。文中舉相關案例與現有文獻比較，說明此方法的正確性，並探討此方法在級數解流程中的收斂速度。除了第一類邊界條件外，本文亦將此套求解流程延伸至第三類邊界條件。

    This thesis discusses the solution method for heat conduction problems of two dimensional plate with nonhomogeneous time-dependent boundary conditions. Firstly, using transformation to let time-dependent endpoints be zero and then reducing the two-dimensional problem into two one-dimensional subsystems by means of the principle of superposition and generalized Fourier coefficient. With help of shifting function method, the non-homogeneous boundary conditions problem can be converted into the transformed function associated with homogeneous boundary conditions. Eventually, the transformed function can be determined by the method of eigenfunction expansion. For first kind boundary conditions and with time-dependent endpoints, this thesis has developed the simple solution method that doesn’t require any integral transformation. Finally, some heat conduction problems are studied by the proposed method.

    摘要..........I ABSTRACT..........II 誌謝..........III CONTENTS..........IV LIST OF TABLES..........VIII LIST OF FIGURES..........IX NOMENCLATURE..........XII CHAPTER 1 INTRODUCTION..........1 1.1 BACKGROUND..........1 1.1.1 Boundary condition of the first kind..........1 1.1.2 Boundary condition of the second kind..........2 1.1.3 Boundary condition of the third kind..........3 1.2 LITERATURE REVIEW..........6 1.3 RESEARCH MOTIVATION AND METHOD..........7 CHAPTER 2 ANALYSIS OF HEAT CONDUCTION OF PLATES WITH TIME-DEPENDENT DIRICHLET BOUNDARY CONDITION..........9 2.1 MATHEMATICAL MODELING..........9 2.1.1 The matching condition of the plate..........10 2.2 SOLUTION METHOD..........10 2.2.1 Transformation..........10 2.2.2 Principle of superposition..........12 2.2.3 Reduce to one-dimensional problem..........15 2.2.4 Change of variable..........17 2.2.5 Shifting function method..........17 2.2.6 Orthogonality condition..........18 2.2.7 Eigenfunction expansion..........19 2.3 ANALYTICAL SOLUTION FORM..........21 CHAPTER 3 VERIFICATION AND EXAMPLES..........26 3.1 EXAMPLE Ⅰ..........26 3.2 EXAMPLE Ⅱ..........33 CHAPTER 4 NUMERICAL RESULTS AND DISCUSSION..........46 4.1 EXAMPLE Ⅲ WITHOUT SOURCE TERM..........46 4.2 EXAMPLE Ⅳ WITH SOURCE TERM..........63 CHAPTER 5 ANALYSIS OF HEAT CONDUCTION OF PLATES WITH TIME-DEPENDENT BOUNDARY CONDITIONS OF THE THIRD KIND..........80 5.1 MATHEMATICAL MODELING..........80 5.2 SOLUTION METHOD..........81 5.2.1 Principle of superposition..........81 5.2.2 Reduce to one-dimensional problem..........85 5.2.3 Change of variable..........87 5.2.4 Shifting function method..........88 5.2.5 Eigenfunction expansion..........89 5.3 ANALYTICAL SOLUTION FORM..........91 CHAPTER 6 VERIFICATION AND EXAMPLE..........95 6.1 EXMAPLE Ⅴ..........95 CHAPTER 7 CONCLUSIONS..........116 7.1 SUMMARY..........116 7.2 FUTURE PROSPECTS..........117 REFERENCE..........118 VITA..........121

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