本文提出了一種解決長時間熱傳導問題的逆向分析方法，包含了雷射表面加熱問題及噴灑散熱問題。藉由最小均方誤差法將在物體內得到的實驗量測數據與經由時變性邊界的熱傳導問題之解析解得到的估算數據之誤差最小化，可以得到其未知邊界的溫度。我們可以求得所有位置與時間的溫度分佈和熱通量。此方法無需使用積分轉換及繁雜的數值運算，亦無需使用將時間區域分成數個微小時間區域的技巧。在數學和實驗上的範例更是說明了此方法為一個簡單、有效率及準確的方法。

This paper proposes a solution method for inverse analysis of long time heat conduction problems, include of laser surface heating problem and spray cooling problem. By minimizing the mean square error between the experimental data obtained from inside the body and estimated data from the derived analytical solution of a heat conduction problem with time-dependent boundary conductions, the temperature at the unknown boundary can be determined. Consequently, the temperature distribution and the heat flux over the entire time and space domains can also be obtained. The integral transform and tedious numerical operations are not required in the proposed solution method. In addition, the technique of dividing time into serval sub-time intervals is not required in long time heating treatment analysis. Mathematical and experimental examples are given to illustrate the simplicity, efficiency, and accuracy of the proposed method.

[1] B. Jin, Y. Zheng, A Meshless Method for Some Inverse Problems Associated with The Helmholtz equation, Comput. Meth. Appl. Mech. Eng. 195 (2006) 2270-2288.
[2] D. Lesnic, L. Elliott, The Decomposition Approach to Inverse Heat Conduction, J. Math. Anal. Appl. 232 (1999) 82-98.
[3] H.T. Chen, X.Y. Wu, Estimation of Surface Absorptivity in Laser Surface Heating Process with Experimental Data, J . Phys. D: Appl. Phys. 39 (2006) 1141-1148.
[4] H.T. Chen, H.C. Lee, Estimation of Spray Cooling Characteristics on a Hot Surface Using the Hybrid Inverse Scheme, Int. J. Heat Mass Transf. 50 (2007) 2503-2513.
[5] H.T. Chen, SL. Sun, H.C. Huang, S.Y. Lee, Analytical Closed Solution for The Heat Conduction with Time Dependent Heat Convection Coefficient at One Boundary, CMSE 59 (2010) 107-126.
[6] J.T. Wang, C.I. Weng, J.G. Chang, C.C. Huang, The Influence of Temperature and Surface Conditions on Surface Absorptivity in Laser Surface Treatment, J. Appl. Phys. 87 (2000) 3245-3253.
[7] L. Yan, C.L. Fu, F.L. Yang, The Method of Fundamental Solutions for The Inverse Heat Source Problem, Eng. Anal. Bound. Elem. 32 (2008) 216-222.
[8] M. Monde, Y. Mitsutake, A New Estimation Method of Thermal Diffusivity Using Analytical Inverse Solution For One-dimensional Heat Conduction, Int. J. Heat Mass Transf. 44 (2001) 3169-3177.
[9] M. Monde, H. Arima, Y. Mitsutake, Estimation of Surface Temperature and Heat Flux Using Inverse Solution for One-Dimensional Heat Conduction, J. Heat Transf. 125 (2003) 213-223.
[10] P.L. Woodfield, M. Monde, Y. Mitsutake, Improved Analytical Solution for Inverse Heat Conduction Problems on Thermally thick and semi-infinite solids, Int. J. Heat Mass Transf. 49 (2006) 2864-2876.
[11] Q. Cui, S. Chandra, S. McCahan, The effect of Dissolving Salts in Water Sprays Used for Quenching a Hot Surface: Part 2 – Spray cooling, ASME J. Heat Transf. 125 (2003) 333-338.
[12] S.Y. Lee, S.M. Lin, Dynamic Analysis of Nonuniform Beams with Time-dependent Elastic Boundary Conditions, J. Appl. Mech. 63 (1996) 474-478.
[13] S.S. Hsieh, T.C. Fan, H.H. Tsai, Spray Cooling Characteristics of Water and R-134a. Part 2: Transient cooling, Int. J. Heat Mass Transf. 47 (2004) 5713-5724.
[14] S.Y. Lee, S.M. Lin, C.S. Lee, S.Y. Lu, Y.T. Liu, Exact Large Deflection of Beams with Nonlinear boundary conditons, CMES 30 (2008) 17-26.
[15] S.Y. Lee, T.W. Huang, A Method for Inverse Analysis of Laser Surface Heating with Experimental Data, Int. J. Heat Mass Transf. 72 (2014) 299-307.
[16] S.Y. Lee, T.W. Huang, Inverse Analysis of Spray Cooling on a hot surface with Experimental Data, 國立成功大學機械工程學系博士論文83-111(2014).
[17] Y.M. Qiao, S. Chandra, Spray Cooling Enhancements by Addition of a Surfactant, ASME J. Heat Transf. 120 (1998) 92-98.
[18] Y.C. Hon, T. Wei, A Fundamental Solution Method for Inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004) 489-495.
[19] 黃得晉, 洪炎星, 金屬二次加工技術roadmap-鑄造、熱處理, 金屬中心5-14 (2005).