本文乃以拉氏轉換法( method of Laplace transform )及有限差分法( finite difference method )，並搭配最小平方法( least-squares methods )，三次仿樣曲線( cubic spline )及試件內部之量測溫度來預測二維暫態逆向熱傳導問題之邊界上的暫態熱傳係數( transient heat transfer coefficient )。本文於逆算法求解時，試件之未知熱傳係數的函數形式不須事先已知，在進行逆運算前先將整個空間域區分成p個小區域，而後以多個連續的位置三次多項式函數與時間之線性函數來模擬未知表面熱通量隨位置與時間的變化情形。

本文將探討p值、未知係數之起始猜測值、量測位置及量測誤差對預測結果的影響。結果顯示，若量測溫度無誤差時，以本文之逆算法可求得良好的表面熱通量與熱傳係數之預測值。p值、量測位置與起始猜測值對預測結果的影響並不顯著。然而，除了長時間之預測值外，對於量測溫度具有誤差時，本文之預測結果甚吻合正確結果，此意味著本文之逆算法具有良好的正確性。

The study applies the method of Laplace transform and the finite difference method in conjunction with the least-squares methods, the cubic spline and the measured temperature inside the test material to predict the transient heat transfer coefficient on the boundary for the two-dimensional transient inverse heat conduction problems. For the inverse algorithm of the study, the functional form of the heat transfer coefficient is unknown a priori. The whole spatial domain is first divided into p sub-intervals. A series of connected cubic polynomial functions in space and a linear function in time are then introduced to simulate the distribution of the unknown surface heat flux over space and time for the transient inverse heat conduction problem.

The study investigates into the effects of p value, the initial guesses of the unknown coefficient, the measurement locations and the measurement errors on the estimated results. The results show that when there is no temperature measurement error, a good estimation on the surface heat flux and the heat transfer coefficient can be derived with the inverse algorithm. The estimated results seem to be not very sensitive to the initial guesses, the measurement locations and the p value. Nevertheless, the predictions agree with the correct results perfectly even if there exist measurement errors, except for the long time estimations. It means that the inverse algorithm of the study presents a good accuracy.

1. 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, 771-781, 1957.
2. G. Stolz, Jr, Numerical solution to an inverse problem of heat condition for simple shapes, ASME J. Heat Transfer 82, 20-26, 1960.
3. J.V. Beck, Calculation of surface heat flux from an integral temperature history, ASME paper 62-HT-46, 1962.
4. J.V. Beck, Surface heat determination using an integral method, Nucl. Eng. Des. , 170-178, 1968.
5. J.V. Beck, Nonlinear estimation applied the nonlinear inverse heat conduction problem, Int. J. Heat Mass Transfer 13, 703-715, 1970.
6. J.V. Beck, B. Litkouhi, C.R.St. Clair, Jr, Efficient sequential solution of nonlinear inverse heat conduction problem, Numer. Heat Transfer 5, 275-286, 1982.
7. J.V.Beck, B.Blackwell, C.R.St.Clair, Jr, Inverse Heat Conduction: Ill-posed Problems, Wiley, New York, 1985.
8. A.M. Osman, J.V. Beck, Investigation of transient heat transfer coefficients in quenching experiments, ASME J. Heat Transfer 112 843-848, 1990.
9. J.V. Beck, B. Blackwell, A. Haji-Sheikh, Comparison of some inverseheat conduction method using experimental data, Int. J. Heat Mass Transfer 39, 3649-3657, 1996.
10. A.M. Osman, J.V. Beck, Nonlinear inverse problem for the estimation of time-and-space-dependent heat-transfer coefficients, J. Thermophysics, vol. 3, 146-152, 1987.
11. E.M. Sparrow, Haji-Sheikh A. and Lundgern, T. S., The inverse problem in transient heat conduction, J, Appl. Mech., Vol. 86e, pp. 369-375, 1964.
12. 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.
13. M.N. Özisik, H.R.B. Orlande, Inverse Heat Transfer: Fundamentala and Applications, Taylor & Francis, New York, 1990.
14. H.C. Huang, M.N. Özisik, Conjudate gradient method for determining unknown contact conductance during metal casting, Int. J. Heat Mass Transfer 35, 1779-1786, 1992.
15. H.C. Huang, M.N. Özisik, Inverse problem of determining unknown wall heat flux in laminar flow through a parallel duct, Numer. Heat Transfer A, 21, 55-70, 1992
16. 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, 1060-1063, 1992.
17. H.T. Chen, S.M. Chang, Application of the hybrid method to inverse heat conduction problems, Int. J. Heat Mass Transfer 33, 621-628, 1990.
18. 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, 1455-1463, 2001.
19. 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, 15-23, 2002.
20. 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, 103-114, 2002.
21. 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, 95-104, 2004.
22. D. Naylor, P. H. Oosthuizen, Evaluation of an inverse heat conduction procedure for determining local convective heat transfer rates, ASME, Inverse Problems in Engineering: Theory and Practice, 125-131,1993.
23. T.J. Martin, G.S. Dulikravich, Inverse determination of steady heat convection coefficient distributions, ASME J. Heat Transfer 120, 328-334, 1998.
24. J. Taler, W. Zima, Solution of inverse heat conduction problems using control volume approach, Int. J. Heat Mass Transfer 42, 1123-1140, 1999.
25. Jian Su, Geoffrey F. Hewitt, Inverse heat conduction problem of estimating time-varying heat transfer coefficient. Numerical Heat Transfer. Part A, 45, 777-789, 2004.
26. H.T. Chen, X.Y. Wu, Estimation of heat transfer coefficient in two- dimensional inverse heat conduction problems, Numerical Heat Transfer. Part B,50, 375-394, 2006.
27. J.P. Holman, Experimental Methods for Engineers, McGraw-Hill, New York, 48-143, 2001.