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標題: 以逆算法預測平板之時變熱傳導
Prediction of transient thermal conductivity in a slab using the inverse method
作者: 劉宗彥
Liou, Zong-Yan
關鍵字: Keywords;關鍵字;heat transfer;inverse problem;thermal conductivity;digital filter;polynomial regression;熱傳;逆問題;熱傳導係數;數位濾波器;多項式迴歸
出版社: 應用數學系所
引用: 中文參考文獻 [ 1] 戴伯臣,應用 Borukhov-Vabishchevich 法預測未知熱源,國立中興大學應用數學系碩士論文,2005。 [ 2] 張浮明,應用微分值積法(DQM)探討逆向熱傳導問題,國立中興大學應用數學系博士論文,2008。 [ 3] 王俊雄,以逆算法預測細長圓柱之時變熱源,國立中興大學應用數學系中等教師在職進修數學教學碩士班碩士學位論文,2011。 英文參考文獻 [ 4] G. Tr. Stolz, Numerical Solution to an Inverse Problem of Heat Condition for Simple Shapes, ASME J. Heat Transfer, 82, pp. 20-26, 1960. [ 5] E. M. Sparrow, A. Haji-Sheikh and T. S Lundgren, The Inverse Problem in Transient Heat Conduction, J. Appl. Mech. 86e, pp. 369-375, 1964. [ 6] N. V. Shumakov, A Method for the Experimental Study of the Process of Heating a Solid body, Sov. Phys. Tech. Phys. (Translated by American Institute of Physiscs) 2, pp. 771-781, 1957. [ 7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, Landon, 2nd , 1959. [ 8] O. M. Alifanov, Solution of an Inverse Problem of Heat Conduction by Iteration Methods, J. Eng. Phys. 26, (4), pp. 471-476, 1974. [ 9] O. M. Alifanov and V. V. Mikhailov, Solution of the Non-linear Inverse Thermal Conductivity Problem by the Iteration Method, J. Engineering Physics, 35, pp. 1501-1506, 1978. [10] R. G. Arledge and A. Haji-Sheikh, An Iterative Approach to the Solution of Inverse Heat Conduction Problem, Numerical Heat Transfer, 1, pp. 365-376, 1978. [11] J. V. Beck, B. Litkouhi and C. R. St. Clair, Efficient Sequential Solution of Nonlinear Inverse Heat Conduction Problem, Numerical Heat Transfer, 5, pp. 275-286, 1982. [12] J. V. Beck and H. Wolf, The Non-linear Inverse Heat Conduction Problem, ASME J. Heat Transfer, No. 65-HT-40, 1965. [13] T. T. Lam and W. K. Yeung, Inverse Determination of Thermal Conductivity for One-dimensional Problems, J. Thermophys. Heat Transfer, 9, (2), pp. 335-344, 1995. [14] C. L. Chang and M. Chang, Non-iteration Estimation of Thermal Conductivity Using Finite Volume Method, Int. Commun. Heat Mass Transfer, 33, pp. 1013-1020, 2006. [15] R. Bellman, J. Casti, Differential Quadrature and Long Term Integration. J. Math. Anal. Appl. 34, pp. 235-238, 1971. [16] R. Bellman, B. G. Kashef and J. Casti, Differential Quadrature : A Technique for the Rapid Solution of Nonlinear Partial Differential Equations, J. Comput. Phys. 10, pp. 40-52, 1972. [17] M. I. Char, F. P. Chang and B. C. Tai, Inverse Determination of Thermal Conductivity by Differential Quadrature Method, Int. Commun. Heat Mass Transfer 35, pp. 113-119, 2008. [18] V. T. Borukhov and P. N. Vabishchevich, Numerical Solution of the Inverse Problem of Reconstructing a Distributed Right-Hand Side of a Parabolic Equation, Computer Physics Communications, 126, pp. 32-36, 2000. [19] J. V. Beck, Calculation of Surface Heat Flux from an Integral Temperature History, ASME J. Heat Transfer, 62-HT-46, 1962. [20] J. V. Beck, Surface Heat Flux Determination Using an Integral Method, Nucl. Eng. Des. 7, pp. 170-178, 1968. [21] I. Frank, An Application of Least Squares Method to the Solution of the Inverse Problem of Heat Conduction, ASME J. Heat Transfer, 85C, pp. 378-379, 1963. [22] N. Al-Khalidy, On the Solution of Praabolic and Hyperbolic Inverse Heat Conduction Problems, Int. J. Mass Trasfer, 41, pp. 3731-3740, 1998. [23] H. T. Chen and S. M. Chang, Application of the Hybrid Method to Inverse Heat Conduction Problems, Int. J. Heat Mass Transfer, 33, (4), pp. 621-628, 1990. [24] H. T. Chen and J. Y. Lin, Hybrid Laplace Transfer Technique for Non-linear Transient Thermal Problems, Int. J. Heat Mass Transfer, 34(4/5), pp. 1301-1308, 1991. [25] H. T. Chen, J. Y. Lin, H. R. Wang and L. C. Fang, Estimation of Two- Sided Boundary Conditions for Two-Dimensional Inverse Heat Conduction Problems, Int. J. Heat Mass Transfer, 45, pp.15 -23, 2002. [26] A. Savitzky, and M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedure, Anal. Chem. 36, pp. 1627-1638, 1964.
本論文係以數值逆算法預測平板之時變熱傳導係數k(thermal conductivity)。文中使用二階有限差分法(second-order finite difference method),結合Savitzky-Golay digital filter數位濾波器,將含測量誤差之溫度平滑化,然後以逆運算法求其熱傳導係數,最後再以多項式迴歸得到熱傳導係數之經驗公式。文中探討了量測誤差大小ω(1%,3%與5%)、量測時間(t=0.3、0.5及1秒)、平滑視窗寬度N等因素對預測熱傳導係數的影響。

結果顯示:(1)量測時間越短,逆運算法求得熱傳導係數之誤差會較小;(2)當量測誤差ω較小時,經由平滑後所逆算的預測k值誤差會較小;(3)平滑視窗寬度越寬,得到的逆運算熱傳導係數越逼近真實值,因此利用Savitzky-Golay digital filter數位濾波器搭配有限差分法逆算熱傳導係數,再加上多項式迴歸,是一個預測熱傳導係數簡單而有效的方法。

A hybrid numerical method is employed to predict the unknown transient thermal conductivity k in a slab. The Savitzky-Golay digital filter is used to smoothen the error of the measured temperature, and the second-order finite difference method is to calcualte inversely the value of k. Finally, using the polynomial regression gets the empirical formula for the value of k. This article discusses the influences of the memasurement errors ω (1%, 3% and 5%), the measurement time ( t=0.3, 0.5 and 1 second), and the breadth of the smooth window N(7, 11 and 15)on predicting the value of k.

The results show that the shorter the measurement time, the lower is the error of estimations. The accuracy of the calculated thermal conductivity increases with decrease in the measurement errors or increase in the breadth of the smooth window. So, using the hybrid method including the Savitzky-Golay digital filter, the finite difference method, and polynomial regression to predict the value of k is a simple and effective way.
其他識別: U0005-0207201210315200
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