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Prediction of transient thermal boundary condition of a plate using the inverse method
|關鍵字:||heat transfer problem;熱傳問題;inverse problem;boundary condition;digital filter;逆問題;邊界條件;數位濾波法||出版社:||應用數學系所||引用:|| 戴伯臣，應用 Borukhov-Vabishchevich 法預測未知熱源，中興大學應用數學系碩士論文，2005  張浮明，應用微分值積法(DQM)探討逆向熱傳導問題，中興大學應用數學系博士論文，2008 O. M. Alifanov, E.A. Artyukhin and S.V. Rumyantsev, Extreme Method for Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell house, Inc NY ,1995.  M. N. Özisik, Boundary Value Problem of Heat Conduction, Dover Publication, INC, NY, 1989.  M. N. Özisik, Heat Conduction , 2 nd , John Wiley and Sons, NY,1993.  D. N. Ghosh Roy, Methods of Inverse Problems in Physics, CRC press, 1991.  C. R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. O. M. Alifanov, Inverse Heat Transfer Problems, Springer – Verlag, NY, 1994.  A. N. Tikhnov and V. Y. Arsenin, Solution of Ill-posed Problems, Winston and Washington DC, 1977. J. V. Beck, B Blackwell and C. R. St. Clair, Inverse Heat Conduction ill-posed Problems, Wiley, NY, 1985. 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 (1957) 771-781.  G. Tr. Stolz, Numerical Solution to An Inverse Problem of Heat Condition for Simple Shapes, ASME Journal of Heat Transfer, 82, pp. 20-26, 1960. J. V. Beck, Surface Heat Flux Determination Using an Integral Method, Nucl. Eng. Des. 7, pp170-178, 1968.  J. V. Beck, Calculation of Surface Heat Flux from an Integral Temperature History, ASME J. Heat Transfer, 62-HT-46, 1962  I. Frank, An Application of Least Squares Method to the Solution of The Inverse Problem of Heat Conduction, J. Heat Transfer, 85C, pp. 378-379, 1963.  E. M. Sparrow, A. Haji-Sheikh and T.S Lundgren, The Inverse Problem inTransient Heat Conduction, J. Appl. Mech., 86e, pp. 369-375, 1964  J. V. Beck, Nonlinear Estimation Applied to the Nonlinear Inverse Heat Conduction Problem, Int. J. of Heat and Mass Transfer, 13, pp. 703-716, 1970.  O. M. Alifanov, Solution of an Inverse Problem of Heat Conduction by Iteration Methods,J. of Eng. Phys., Vol. 26, No. 4, pp. 471-476, 1974.  C. H. Huang and M. N. Özisik, A Direct Integration Approach for Simultaneously Estimating Spatially Varying Thermal Conductivity and Heat Capacity, J. Heat Fluid Flow 11, pp.262-268, 1990.  C. H. Huang and M. N. Özisik, Conjugate Gradient Method for Determining Unknown Contact Conductance During Metal Casting, Int. J. Heat Transfer A 21, pp.55-70, 1992.  C. H. Huang, M. N. Özisik and B. Sawaf, Conjugate Gradient Method for Determining Unknown Contact Conductance during Metal Casting, Int. J. Heat Mass Transfer, 35, No. 7, pp.779-786, 1992.  A. J. Silva Neto and M. N. Özisik, The Estimation of Space and Time Dependent Strength of a Volumetric Heat Source in a One Dimensional Plate, Int. J. Heat mass Transfer, 17, No.6, pp.909-915, 1994.  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.  C. L. Niliot and F. Lefèevre, A Parameter Estimation Approach to Solve the Inverse Problem of Point Heat Sources Identification, Int. J. of Heat and Mass Transfer, 47, pp.827–841, 2004.  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.  H. T. Chen and S. M. Chang, Application of the Hybrid Method to Inverse Heat Conduction Problems, Int. J. Heat mass Transfer, 33, No.4, pp. 621-628, 1990. . 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.  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.  W. K. Yeung and T. T. Lam, Second-order Finite Difference Approximation for Inverse Determination of Thermal Conductivity, Int. J. Heat Mass Transf. 39 (17), pp. 3685-3693, 1996.  A. Savitsky, and M. J. E. Gollay, 1964. Smoothing and differentiation of data by simplified least squares procedure. Anal. Chem. 36:1627-1638.||摘要:||
在物理上來說，熱傳導問題可以偏微分方程式來表示，如用已知初始條件與邊界條件及相關物理性質，則可得溫度分布狀況，此稱為直接問題(direct problem)，直接問題的解，依求解方程可分為：解析解(analytic solution)，數值解(numerical solution)，或近似解(approximation solution)。若藉由測量內部溫度逆算熱傳導係數(thermal conductivity)，內部熱源(heat source)，邊界條件(boundary condition)，熱通量(heat flux)等，則稱為逆問題(inverse problem)。在一般的熱傳問題中，其邊界條件通常都會事先給定或假設為已知，但實際上在很多熱環境下，邊界條件很難事先給定，或藉由直接量測表面溫度獲得，所以需要量測內部溫度，只需要少量的量測點，再搭配逆算法即可預估邊界條件。
本文係利用二階有限差分(second order finite difference)來反算一細長平板之時變逆熱傳導問題(inverse heat transfer problems，IHTP)時，文中探討了三個時變，不同邊界條件的例子，在逆算熱邊界條件之前，吾人結合Savitzky－Gollay digital filter數位濾波法，將量測含噪之溫度值予以平滑化，最後再配合二階有限差分法來反算邊界條件。文中探討了量測間距，量測誤差大小，平滑視窗寬度等因素對預測熱邊界的影響，其結果顯示平滑後的誤差均會變小，因此利用有限差分法搭配Savitzky－Gollay digital filter數位濾波法逆算邊界條件，是一個有效的方法。
In Physics, the heat transform problems can be illustrated with partial differential equations. For instance, the variation of the temperature of the plate would be obtained by applying known initial and boundary conditions as well as related physical characteristics. This is called direct problem. The solutions to the direct problems are analytic, numerical or approximation solution. Calculating thermal conductivity, heat source, boundary conditions and heat flux inversely by measuring interior temperature of the plate is called inverse problems. The boundary conditions of general thermal problems are given in advance or hypothetically known. However, under many thermal circumstances, the boundary conditions are difficult to be known beforehand or gained by measuring the surface temperature directly. As a result, it is necessary to measure the interior temperature. With only a few measuring points and inverse calculation, the boundary conditions would be estimated.
The study utilizes second order finite difference method to calculate inversely the inverse heat transfer problem (IHTP) of a slender slab, with inclusive of three different timewise variations of thermal boundary conditions. Before calculating thermal boundary conditions inversely, the Savitzky-Gollay digital filter is employed to smooth the measured temperature which contains noise. Then the second order finite difference method is applied to calculate inversely the boundary conditions. The research discusses the influences of measurement space, measurement errors, and smoothed window breadth on predicting thermal boundary. The results show that the smoothed errors would be lower. Therefore, combining second order finite difference method and Savitzky-Gollay digital filter is an effective method of calculating thermal boundary conditions inversely.
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