二維平板熱物理量之預測
施明陽 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
本文應用混合拉氏轉換法（Laplace transform technique）和有限差分法（Finite difference method）的數值方法，並配合最小平方法（Least squares scheme）來預測二維平板之熱物理量。而平板之熱物理量吾人假設其為常數形式與溫度函數形式兩種。首先利用拉氏轉換法處理系統之時間域，而後再以有限差分法處理系統轉換後之空間域，最後再以數值逆拉氏轉換法求取二維平板之溫度場分佈。拉氏轉換法的優點是可以求得在某一特定時間的溫度值，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地收斂。本文將探討熱電偶（Thermocouple）的安置位置及其數量對預測結果的影響。本文之預測結果顯示本文之數值方法能夠精確地、有效地預測出未知材料的熱物理量。本文也將加以探討量測誤差對預測值的影響。預測結果也顯示量測誤差對其影響並不是很敏感。因此本文之混合數值方法可成功地被應用來解析二維之逆向熱傳導問題。

The present study introduces a hybrid numerical method to analyze the inverse heat conduction problem concerning the prediction of the thermal properties of a two-dimensional plate. The thermal properties are assumed as two types. One type is constant, and the other is a function of temperature. This algorithm combines the Laplace transform technique and the finite-difference method in conjunction with the least-squares scheme. Time-dependent terms in the governing equation are removed by using the Laplace transform technique, and then the resulting differential equation is solved by using the finite-difference method. Temperature distributions in the domain are obtained by using the numerical inversion of Laplace transform. Due to the application of the Laplace transform technique, the temperature can be calculated at a 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 this thesis, various examples are illustrated to show the applicability and efficiency of the present numerical method. The effect of time-step, measurmant error and thermocouple location is investigated. It can be seen from various illustrated examples that the present numerical method can accurately and efficiently estimate the unknown boundary temperature even though the thermocouple is located far from the estimated surface. Results also show that the estimations are not very sensitive to the measurement error. Thus, it can be concluded that the present numerical method can successfully be applied to analyze the inverse heat conduction problem .

目 錄
摘要…………………………………………………………Ⅰ
英文摘要……………………………………………………Ⅱ
誌謝…………………………………………………………Ⅲ
目錄…………………………………………………………Ⅳ
表目錄………………………………………………………Ⅵ
圖目錄………………………………………………………VIII
符號說明……………………………………………………XI
第一章 緒論…………………………………………………1
1-1 研究背景與目的……………………………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 二維均質材料之常數型熱傳導係數與單位
體積熱容預測………………………………………6
2-1 簡介…………………………………………6
2-2 理論分析……………………………………7
2-3 直接熱傳導問題……………………………9
2-3-1 實例說明………………………………12
2-4 逆向熱傳導問題……………………………14
2-5 結果與討論…………………………………18
第三章 二維均質材料之非常數型熱傳導係數與
單位體積熱容量預測………………………………36
3-1 簡介…………………………………………36
3-2 理論分析……………………………………37
3-3 直接熱傳導問題……………………………39
3-3-1 實例說明………………………………42
3-4 逆向熱傳導問題……………………………44
3-5 結果與討論…………………………………47
第四章 綜合結論與建議……………………………………73
4-1 綜合結論……………………………………73
4-2 未來發展方向………………………………74
參考文獻……………………………………………………75
自述…………………………………………………………79

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