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研究生: 吳信賢
Wu, Hsin-Hsien
論文名稱: 反算法於非傅立葉型態熱傳導問題之研究
The Inverse Non-Fourier Type Heat Conduction Problems
指導教授: 黃正弘
Huang, Cheng-Hung
學位類別: 碩士
Master
系所名稱: 工學院 - 系統及船舶機電工程學系
Department of Systems and Naval Mechatronic Engineering
論文出版年: 2006
畢業學年度: 94
語文別: 中文
論文頁數: 97
中文關鍵詞: 脊型及縱向散熱片反算問題共軛梯度法非傅立葉
外文關鍵詞: spine and longitudinal fin, Conjugate Gradient Method, Non-Fourier, Inverse Problem
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  • 摘要
    在實際的工程問題上,經常存在著許多物理量無法由量測或計算之方法獲得其值,因此為了求得這些物理量,往往必須藉由其他可量測之資料反求之,這類問題稱之為逆向或反算問題(Inverse Problem)。
    本論文主要探討所謂的逆向熱傳導問題(Inverse Heat Conduction Problem)。第一章中,吾人以反算法之共軛梯度法(Conjugate Gradient Method)來研究雙曲線熱傳問題表面熱通量之預測,並討論溫度量測誤差對表面熱通量預測值準確度的影響。
    在第二章中,預測非傅立葉散熱片之底部溫度,同樣根據第一章之方法進行預測。
    反算問題亦可稱為最佳化設計問題。在許多複雜的工程問題中,我們可以利用一些已知的設計資料對問題做最佳化的處理,並藉由反算的技巧,吾人可以設計出非傅立葉脊型及縱向散熱片之最佳幾何形狀。在極小時間及小尺度下傳統傅立葉定律將不適用,此時就要從微觀理論來考慮。第三章中主要應用共軛梯度法,針對微觀理論下,求得非傅立葉的脊型及縱向散熱片在極小時間及小尺度下,散熱片之最佳幾何外型之預測及效率分析。

    ABSTRACT
    In practical engineering problems, there exist many physical quantities that are very difficult to measure directly. The techniques for “INVERSE PROBLEM” can be used to solve these kinds of problems. In the present thesis the inverse non-Fourier type heat conduction problems are discussed.
    In chapter one an inverse hyperbolic heat conduction problem is solved by the Conjugate Gradient Method (CGM) in estimating the unknown boundary heat flux based on the boundary temperature measurements. Finally, it is concluded that the drawbacks of the previous study for this similar inverse problem, such as (1) the inverse solution has phase error and (2) the inverse solution is sensitive to measurement error can be avoided in the present algorithm and accurate boundary heat flux can also be estimated.
    In chapter two, the inverse non-Fourier fin problem is examined by CGM in estimating the unknown base temperature of non-Fourier fin based on the boundary temperature measurements. Results show that the drawbacks of previous study for this identical inverse problem, such as (1) the inverse solutions become poor when the frequency of base temperature is increased, (2) the estimations depend strongly on the size of grids, (3) the estimations are sensitive to the measurement errors and (4) the uncertainty of using the concept of future time step, can all be avoided by applying this algorithm.
    In chapter three the inverse design problem is presented to estimate the optimum shapes for the non-Fourier spine and longitudinal fins by using CGM based on the desired fin efficiency and fin volume at the specified time. Results show that CGM can be utilized successfully in determining the optimum shape of the non-Fourier spine and longitudinal fins.

    目錄 摘要 1 ABSTRACT II 誌謝 II 目錄 IV 圖表目錄 VII 符號說明 IX 第一章 反算法於雙曲線熱傳問題表面熱通量之預測 1 1-1研究背景與目的 1 1-2 文獻回顧 2 1-3 直接解問題(Direct Problem) 4 1-4 反算問題(Inverse Problem) 5 1-5 共軛梯度法之極小化過程(Conjugate Gradient Method (CGM) for Minimization) 6 1-6 靈敏性問題與前進步距 (Sensitivity Problem And Search Step Size) 7 1-7 伴隨問題與梯度方程式(Adjoint Problem and Gradient Equation) 8 1-8 收斂條件(Stopping Criterion) 10 1-9 數值計算流程(Computational Procedure) 11 1-10 結果與討論(Results and Discussions) 12 1-11 結論(Conclusions) 18 1-12 參考文獻 29 第二章 反算法於非傅立葉散熱片底部溫度之預測 31 2-1 研究背景與目的 31 2-2 文獻回顧 31 2-3 直接解問題(Direct Problem) 33 2-4 反算問題(Inverse Problem) 35 2-5 共軛梯度法之極小化過程(Conjugate Gradient Method (CGM) for Minimization) 36 2-6 靈敏性問題與前進步距 (Sensitivity Problem And Search Step Size) 37 2-7 伴隨問題與梯度方程式(Adjoint Problem and Gradient Equation) 39 2-8 收斂條件(Stopping Criterion) 41 2-9 數值計算流程(Computational Procedure) 42 2-10 結果與討論(Results and Discussions) 43 2-11 結論(Conclusions) 49 2-12 參考文獻 56 第三章 非傅立葉脊型與縱向散熱片最佳幾何形狀之設計 58 3-1 研究背景與目的 58 3-2 文獻回顧 59 3-3 直接解問題(Direct Problem) 61 3-4 反算設計問題(Inverse Design Problem) 63 3-5 共軛梯度法之極小化過程(Conjugate Gradient Method (CGM) for Minimization) 66 3-6 靈敏性問題與前進步距 (Sensitivity Problem and Search Step Size) 67 3-7 伴隨問題與梯度方程式(Adjoint Problem and Gradient Equation) 69 3-8 數值計算流程(Computational Procedure) 72 3-9 結果與討論(Results and Discussions) 73 3-10 結論(Conclusions) 80 3-11 參考文獻 92 第四章 結語 95

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