本文探討解析具時變性邊界條件之正逆向熱傳導問題。首先針對具時間邊界條件之非均勻介質暫態正向熱傳導問題，發展出一種新的解析解，無需積分變換，經由合適的轉換函數，具變量係數及時間邊界條件之二階微分方程式可轉換成具有均質邊界條件之微分方程式。若介質的物理性質以多項式形式呈現，則可以得到該系統的解析解。最後，以案例來說明分析方法。特殊情況案例與現有的文獻研究做比較。物性參數對該系統溫度分佈的影響將顯現出來。
針對具時間邊界條件之均勻介質暫態逆向熱傳導問題，則結合位移函數及、正交函數展開及最小平方法之混合逆算法搭配試件內部之量測溫度，無需積分變換來預測線性逆向熱傳導問題之未知表面條件；本文之逆算法求解時，試件之未知表面條件的函數形式不需事先已知，並針對整個時間區域進行逆運算，或將時間區域分成多個小時間區域，而後整個時間區域或每個小時間區域的預測值便可以利用本文之逆算法求得。為了驗證本文混合逆算法的精確度與可靠度，預測結果將與正確值、參考文獻之估算結果及實驗數據比較，同時也將討論量測位置對預測結果的影響；結果顯示應用本文之逆算法，可以求得良好且穩定的預測結果。

This paper discusses the analytical solution and inverse analysis of heat conduction problems with time-dependent boundary conditions. First, the non-uniform medium heat conduction problem is studied. A new analytic solution method is developed, without integral transform, to find the exact solution for the transient heat conduction in non-uniform medium with general time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. If the physic properties of the medium are in polynomial forms, the exact solution of the system can be developed. Then, examples are given to illustrate the analysis. Limiting cases are studied and compared with those in the existing literature. The influence of physic parameters on the temperature distribution of the system is revealed.
Secondly, a hybrid inverse scheme involving the shifting functions, eigenfunction expansion and least-square methods in conjunction with experimental data inside the test material, without integral transform, is proposed to estimate the unknown surface conditions for the linear inverse heat conduction problems with uniform medium. The functional form of the surface conditions is unknown a priori. We can analyze the whole time domain or divide it into several sub-time intervals for analysis and then estimates the unknown surface conditions on each sub-time interval. In order to show the accuracy and reliability of the present inverse scheme, comparisons among the present estimates, exact solution, previous results and experimental data are made. The effects of the measurement locations on the estimated results are also investigated. The results show that good estimation of the surface conditions can be obtained.

[1] M.N. Őzisik, Boundary Value Problems of Heat Conduction, first ed., International Textbook Company, Scranton, 1968.
[2] H.T. Chen, S.L. Sun, H.C. Huang, and S.Y. Lee, Analytical Closed Solution for the Heat Conduction with Time-dependent Heat Convection Coefficient at one Boundary, CMES: Compu. Modeling Eng. & Sci. 59 (2010) 107-126.
[3] S.Y. Lee, and S.M. Lin, Dynamic Analysis of Nonuniform Beams with Time-Dependent Elastic Boundary Conditions, J. Appl. Mech. 63 (1996) 474-478.
[4] S.Y. Lee, S.M. Lin, C.S. Lee, S.Y. Lu, and Y.T. Liu, Exact Large Deflection of Beams with Nonlinear Boundary Conditions, CMES: Compu. Modeling Eng. & Sci. 30 (2008) 17-26.
[5] B.T. Johansson, and D. Lesnic, A Method of Fundamental Solutions for Transient Heat Conduction, Eng. Anal. Boundary Elem. 32 (2008) 697-703.
[6] S. Chantasiriwan, Method of Fundamental Solutions for Time-Dependent Heat Conduction Problems, Int. J. Numer. Methods Eng. 66 (2006) 147-165.
[7] D.L. Young, C.C. Tsai, K. Murugesan, C.M. Fan, and C.W. Chen, Time-Dependent Fundamental Solutions for Homogeneous Diffusion Problems, Eng. Anal. Boundary Elem. 28 (2004) 1463-1473.
[8] S.P. Zhu, H.W. Liu, and X.P. Lu, A Combination of LTDRM and ATPS in Solving Diffusion Problems, Eng. Anal. Boundary Elem. 21 (1998) 285-289.
[9] J.M. Amado, M.J. Tobar, A. Ramil, and A. Ya´n˜ez, Application of the Laplace Transform Dual Reciprocity Boundary Element Method in the Modelling of Laser Heat Treatments, Eng. Anal. Boundary Elem. 29 (2005) 126–135.
[10] A. Sutradhar, G.H. Paulino, and L.J. Gray, Transient Heat Conduction in Homogeneous and Non-Homogeneous Materials by the Laplace Transform Galerkin Boundary Element Method, Eng. Anal. Boundary Elem. 26 (2002) 119-132.
[11] V. Bulgakov, B. Sarler, and G. Kuhn, Iterative Solution of Systems of Equations in the Dual Reciprocity Boundary Element Method for the Diffusion Equation, Int. J. Numer. Method Eng. 43 (1998) 713-732.
[12] A.Z. Sahin, Analytical Solutions of Transient Heat Conduction in Semi-Infinite Solid with Time Varying Boundary Conditions by Means of Similarity Transformation, Int. Commu. Heat Mass Transfer 22 (1995) 89-97.
[13] A. Barletta, E. Zanchini, S. Lazzari, and A. Terenzi, Numerical Study of Heat Transfer from an Offshore Buried Pipeline under Steady-Periodic Thermal Boundary Conditions, Appl. Thermal Eng. 28 (2008) 1168–1176.
[14] S.M. Hamza-Cherif, A. Houmat, and A. Hadjoui, Transient Heat Conduction in Functionally Graded Materials, Int. J. Comput. Methods 4 (2007) 603-619.
[15] V. Sladek, J. Sladek, M. Tanaka, and C. Zhang, Transient Heat Conduction in Anisotropic and Functionally Graded Media by Local Integral Equations, Eng. Anal. Boundary Elem. 29 (2005) 1047-1065.
[16] X. Lü, T. Lü, and M. Viljanen, A New Analytical Method to Simulate Heat Transfer Process in Buildings, Appl. Thermal Eng. 26 (2006) 1901–1909.
[17] N. Daouas, Z. Hassen, and H.B. Aissia, Analytical Periodic Solution for the Study of Thermal Performance and Optimum Insulation Thickness of Building Walls in Tunisia, Appl. Thermal Eng. 30 (2010) 319–326.
[18] M.C. Renato, P.C. Bianca, P. N-C. Carolina, and C-P. Gerson, , Hybrid Integral Transforms Analysis of the Bioheat Equation with Variable Properties, Int. J. Thermal Sci. 49 (2010) 1510-1516.
[19] X. Lü, P. Tervola, and M. Viljanen, Transient Analytical Solution to Heat Conduction in Composite Circular Cylinder, Int. J. Heat Mass Transfer 49 (2006) 341–348.
[20] H. Wang, Q.H. Qin, and Y.L. Kang, A Meshless Model for Transient Heat Conduction in Functionally Graded Materials, Compu. Mech. 38 (2006) 51-60.
[21] J.V. Beck, Calculation of surface heat flux from an integral temperature history, ASME paper 62-HT-46 (1962).
[22] J.V. Beck, Surface heat determination using an integral method, Nucl. Eng. Des. (1968) 170-178.
[23] J.V. Beck, Nonlinear estimation applied the nonlinear inverse heat conduction problem, Int. J. Heat Mass Transfer 13 (1970) 703-715.
[24] J.V. Beck, B. Litkouhi, C.R.St. Clair, Jr, Efficient sequential solution of nonlinear inverse heat conduction problem, Numer. Heat Transfer 5 (1982) 275-286.
[25] A.M. Osman, J.V. Beck, Investigation of transient heat transfer coefficients in quenching experiments, ASME J. Heat Transfer 112 (1990) 843-848.
[26] J.V. Beck, B. Blackwell, A. Haji-Sheikh, Comparison of some inverse heat conduction method using experimental data, Int. J. Heat Mass Transfer 39 (1996) 3649-3657.
[27] H.Y. Li, M.N. Özisik, Identification of the temperature profile in an absorbing, emitting and isotropically scattering medium by inverse analysis, ASME J. Heat Transfer 114 (1992) 1060-1063.
[28] C.H. Huang, M.N. Özisik, Conjudate gradient method for determining unknown contact conductance during metal casting, Int. J. Heat Mass Transfer 35 (1992) 1779-1786.
[29] C.H. Huang, M.N. Özisik, Inverse problem of determining unknown wall heat flux in laminar flow through a parallel duct, Numer. Heat Transfer A 21 (1992) 55-70.
[30] C.H. Huang, C.C. Tsai, An inverse heat conduction problem of estimating boundary fluxes in an irregular domain with conjugate gradient method, Heat Mass Transfer 34 (1998) 47-54
[31] C.H. Huang, M.N. Özisik, A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method, Int. J. Heat Mass Transfer 43 (2000) 3171-3181.
[32] H.T. Chen, S.M. Chang, Application of the hybrid method to inverse heat conduction problems, Int. J. Heat Mass Transfer 33 (1990) 621-628.
[33] H.T. Chen, S.Y. Lin, L.C. Fang, Estimation of surface temperature in two-dimensional inverse heat conduction problems, Int. J. Heat Mass Transfer 44 (2001) 1455-1463.
[34] H.T. Chen, S.Y. Lin, H.R. Wang, L.C. Fang, Estimation of two-sided boundary conditions for two-dimensional inverse heat conduction problems, Int. J. Heat Mass Transfer 45 (2002) 15-23.
[35] H.T. Chen, X.Y. Wu, Application of the hybrid method to the estimation of surface conditions from experimental data, Trends in Heat, Mass and Momentum Transfer 8 (2002) 103-114.
[36] H.T. Chen, X.Y. Wu, Y.S. Hsiao, Estimation of surface condition from the theory of dynamic thermal stresses, Int. J. Thermal Sci. 43 (2004) 95-104.
[37] H.T. Chen, J.P. Song, Y.T. Wang, Prediction of heat transfer coefficient on the fin inside one-tube plate finned-tube heat exchangers, Int. J. Heat Mass Transfer 48 (2005) 2697-2707.
[38] H.T. Chen, J.C. Chou, Investigation of natural-convection heat transfer coefficient on a vertical square fin of finned-tube heat exchangers, Int. J. Heat Mass Transfer 49 (2006) 3034-3044.
[39] H.T. Chen, J.C. Chou, H.C. Wang, Estimation of heat transfer coefficient on the vertical plate fin of finned-tube heat exchangers for various air speeds and fin spacings, Int. J. Heat Mass Transfer 50 (2007) 45–57
[40] C.Y. Yang, A sequential method to estimate the strength of the heat source based on symbol computation, Int. J. Heat Mass Transfer 41 (1998) 2245-2252.
[41] C.C. Ji, P.C. Tuan, H.Y. Jang, A recursive least-squares algorithm for on-line 1-D inverse heat conduction estimation, Int. J. Heat Mass Transfer 40 (1997) 2081-2096.
[42] C. C. Ji, H. Y. Jang, Experimental inverse investigation in inverse heat conduction problem, Numer. Heat Transfer A 34 (1998) 75-91.
[43] J.T. Wang, C.I. Weng, J.G. Chang, The influence of temperature and surface conditions on surface absorptivity in laser surface treatment, J. Appl. Phy. 87 (2000) 3245-3253.
[44] T.J. Martin, G.S. Dulikravich, Inverse determination of steady heat convection coefficient distributions, ASME J. Heat Transfer 120 (1998) 328-334.
[45] J. Taler, W. Zima, Solution of inverse heat conduction problems using control volume approach, Int. J. Heat Mass Transfer 42 (1999) 1123-1140.
[46] Y.S. Sun, C.I. Weng, T.C. Chen, W.L. Li, Estimation of surface absorptivity and surface temperature in laser surface hardening process, Jpn. J. Appl. Phys. 35 (2000) 3658-3664.
[47] O.M. Alifanov, E.A. Artyukhin, Regularized numerical solution of nonlinear inverse heat conduction problem, J. Eng. Phy. 29 (1975) 934-938.
[48] G.A. Dorai, D.A. Tortorelli, Transient inverse heat conduction problem solution via Newton’s method, Int. J. Heat Mass Transfer 40 (1997) 4115-4127.
[49] R.A. Khachfe, Y. Jarny, Numerical solution of 2-D nonlinear inverse heat conduction problems using finite-element techniques, Numer. Heat Transfer B 37 (2000) 45-67.
[50] C.Y. Yang, Estimation of boundary conditions in nonlinear inverse heat conduction problems, J. Thermophy. Heat Transfer 17 (2003) 389-395.
[51] N. Daouas, M.S. Radhouani, A new approach of the Kalman filter using future temperature measurements for nonlinear inverse heat conduction problems, Numer. Heat Transfer B 45 (2004) 565-585.
[52] S. Chantasiriwan, Inverse heat conduction problem of determining time-dependent heat transfer coefficient, Int. J. Heat Mass Transfer 42 (2000) 4275-4285.
[53] S. Chantasiriwan, Inverse determination of steady-state heat transfer coefficient, Int. Comm. Heat Mass Transfer 27 (2000) 1155-1164.
[54] P. Duda, J. Taler, Numerical method for the solution of nonlinear two-dimensional inverse heat conduction problem using unstructured meshes, Int. J. Numer. Meth. Engng. 48 (2000) 881-899.
[55] J.H. Lin, C.K. Chen, Y.T. Yang, The inverse estimation of the boundary behavior of a heated cylinder normal to a laminar air stream, Int. J. Heat Mass Transfer 43 (2000) 3991-4001.
[56] J.H. Lin, C.K. Chen, Y.T. Yang, An inverse method for simultaneous estimation of the center and surface thermal behavior of a heated cylinder normal to a turbulent air stream, ASME J. Heat Transfer 124 (2002) 601-608.
[57] P.L. Masson, T. Loulou, E. Artioukhine, Estimation of a 2D convection heat transfer coefficient during a test: comparison between two methods and experimental validation, Inverse Problems in Sci. Eng. 12 (2004) 595-617.
[58] P. Duda, J. Taler, Experimental verification of space marching methods for solving inverse heat conduction problems, Heat and Mass Transfer 36 (2000) 325-331.
[59] C. W. Chang, C. S. Liu, J. R. Chang, A group preserving scheme for inverse heat conduction problems, CMES 10 (2005) 13-38.
[60] G. Honig, U. Hirdes, A method for the numerical inversion of laplace transforms, J. Comput. Appl. Math. 10 (1984) 113-132.
[61] D. Lesnic, L. Elliott, The Decomposition Approach to Inverse Heat Conduction, J. Math. Anal. Appl. 232 (1999) 82–98.
[62] 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.
[63] 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.
[64] 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.
[65] Y.C. Hon, T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004) 489–495.
[66] 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.
[67] 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.
[68] J.T. Wang, C.I. Weng, J.G. Chang, C.C. Hwang, The influence of temperature and surface conditions on surface absorptivity in laser surface treatment, J. Appl. Phys. 87 (2000) 3245–3253.
[69] 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.
[70] H. T. Chen, J. H. Lin, X. J. Xu, Analytical estimates of surface conditions in laser surface heating process with experimental data, Int. J. Heat Mass Transf. 67 (2013) 936–943.
[71] Y.M. Qiao, S. Chandra, Spray cooling enhancements by addition of a surfactant, ASME J. Heat Transf. 120 (1998) 92–98.
[72] 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.
[73] S.S. Hsieh, T.C. Fan, H.H. Tsai, Spray cooling characteristics of water and R-134a. Part II: Transient cooling, Int. J. Heat Mass Transf. 47 (2004) 5713–5724.
[74] 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.