臺灣博碩士論文加值系統

二維非傅立葉熱傳問題之研究
盧基業* 陳寒濤**
國立成功大學機械工程研究所
中華民國台灣省台南市
中文摘要
本文應用混合拉氏轉換法(Laplace transform technique)與控制體積法(Control volume method)並配合最小平方法(Least-squares scheme)來求解二維非傅立葉(non-Fourier)熱傳導問題。本混合數值法首先利用拉氏轉換法來處理系統之時間域，而後再以控制體積法來處理系統轉換後之空間域，最後再以高斯消去法和數值逆拉氏轉換法來求解系統溫度值。拉氏轉換法的優點是可以求得在任意特定時間的溫度值，而不需要由初始時間慢慢地求解。最小平方法的應用在於使數值結果能較快收斂。
於分析非傅立葉熱傳導問題時，由於熱波前端附近的溫度會發生急遽的變化，因而很容易引起數值振盪的現象，為了獲得穩定的數值解，故以控制體積法來描述系統的空間域。由研究結果得知，本數值方法解析非傅立葉熱傳導問題，所需之格點數目不多，且又能求得精確的數值結果。另外，本文將在第二章建立新數值方法的精確性，再由此方法推得第三章的圓柱非傅立葉熱傳導問題。
至今已有相當多的專家學者以各種不同的方法求解二維逆向熱傳導問題，但皆僅局限於傅立葉的熱傳導問題上，故本文根據所量取的溫度值來預估二維非傅立葉熱傳導問題的熱源，在反算的過程中，熱源和時間的函數型態事先為未知。因此，本文將整個時間域分割成幾個時間區間，而每個時間區間皆以多項式來近似。結果顯示，本文的反算法精確性很高。
*：研究生
**：指導教授

Study of Two-Dimension Non-Fourier Heat Conduction Problems
Ji-Ye Lu* Han-Taw Chen**
Department of Mechanical Engineering
National Cheng Kung University
Tainan, Taiwan, R.O.C
ABSTRACT
The hybrid application of the Laplace transform technique and control volume method in conjunction with the least-squares scheme is applied to analyze two-dimension non-Fourier heat conduction problems. In this hybrid numerical scheme, Time-dependent terms in the governing equations are removed by using the Laplace transform technique and then the resulting differential equations are solved by using the control volume method. Temperature distributions in the physical domain are obtained by using numerical inversion of the Laplace transform and the Gaussian elimination method. Due to the application of the Laplace transform technique, the temperature can calculated at any specific time without step-by-step computation in the time domain. By the least-squares scheme, the convergence of iteration can become fast and stable.
In the analysis of non-Fourier heat conduction problems, since the temperature distributions exhibit the phenomena of steep jumps at thermal wave fronts. The present study employs hyperbolic shape functions in the control volume formulation to suppress numerical oscillations in the vicinities of thermal wave fronts. According to the studying results, solving the non-Fourier heat conduction problems with the present numerical method not only needs fewer node-number but also gets the accurate solution. In addition, this thesis proves the accuracy of this new numerical method in chapter two and than applied it to the cylinder non-Fourier heat conduction problems in chapter three.
Up to now, many scholars have studied the two-dimension inverse heat conduction problems, limited with the Fourier heat conduction problems. This thesis predict the heat source of two-dimension non-Fourier heat conduction problems with the measured temperatures. The function model between the heat source and time is unknown in the inverse process. As a result, the whole time domain will been divided several time intervals and they will been approximated by polynomials. In result , the accuracy of this inverse method is well.
*：Author
**：Advisor

摘要 ……………………………………………………….I
英文摘要 ………………………………………………….Ⅱ
誌謝 ……………………………………………………….Ⅳ
目錄 ……………………………………………………….Ⅴ
表目錄 …………………………………………………….Ⅶ
圖目錄 …………………………………………………….Ⅷ
符號說明 ………………………………………………….Ⅹ
第一章 緒論 ………………………………………….1
1-1 研究背景 ………………………………….1
1-2 文獻回顧 ………………………………….2
1-3 研究目的…………………………………3
1-4 研究重點與架構 ………………………….4
第二章 於直角座標系統下之二維非傅立葉熱傳導問題
………………………………………………….5
2-1 簡介 ……………………………………….5
2-2 理論分析 ………………………………….6
2-2-1數學模式建立 …………………………….6
2-2-2熱源點之數學模式 ……………………….12
2-3 結果與討論 ……………………………….15
2-3-1驗講本文方法之準確性 ………………….15
2-3-2具有熱源點作用之熱傳問題 …………….16
2-4 結論 ……………………………………….18
第三章 於圓柱座標系統下之二維非傅立葉熱傳導問題
………………………………………………….44
3-1 簡介 ……………………………………….44
3-2 數學模式建立 …………………………….44
3-3 結果與討論 ……………………………….50
3-4結論 ……………………………………….51
第四章 二維非傅立葉熱傳導之逆向問題 …………..60
4-1 簡介 ……………………………………….60
4-2 反算法的運算過程 ……………………….60
4-3 結果與討論 ……………………………….63
4-4 結論 ……………………………………….64
第五章 綜合結論與未來發展方向 …………………..68
5-1 綜合結論 …………………………………68
5-2 未來發展方向 ……………………………69
參考文獻 ………………………………………………….71
自述 ……………………………………………………….75

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