於於鑄造過程中，相變化或凝固是相當重要的物理現象，處理相變化問題的數值模式不少，一般是直接將能量方程式以有限差分離散或有限元素積分來求解。本文提出一個不一樣的求解方法，也就是應用混合拉氏轉換法，並搭配數種潛熱效應的處理方法，來求解凝固過程之溫度場分佈。文中將介紹混合拉氏轉換法的原理，以及處理潛熱效應的數種方法，包含割線法、比熱法、熱焓法、等效比熱-熱焓法。本文以所提出的方法求解有正解的凝固問題，一維史蒂芬問題、紐曼問題和二維Rathjen問題；並將數值解與正解進行準確性和誤差比較。由分析的結果發現，在求解史蒂芬問題與紐曼問題時，混合拉氏轉換法搭配熱焓法有最好的準確性。比起有限差分法，混合拉氏轉換法搭配比熱法在相同的時間步伐與空間間隔下，明顯地有較佳的準確性，因此本文的方法可以有效地簡化數值求解的過程。

Phase change or solidification is an important physical phenomenon in the casting process. Generally, finite difference and finite element methods are frequently used to solve the governing equation of phase change problems. The present study employs a method involving the combined use of the hybrid Laplace transform and various ways to deal with the effect of latent heat to investigate nonlinear phase-change problems. In the paper, the theorems of hybrid Laplace transform method are presented. Secant method, effective specific heat method, enthalpy method and effective specific heat-enthalpy method are used to estimate the effect of latent heat in phase-change problems. The present numerical methods are utilized to solve the one-dimensional Stefan and Neumann phase-change problems and the two-dimensional Rathjen phase-change problem. The comparison results show that the numerical solutions agree well with the analytical ones. The numerical scheme incorporating the hybrid Laplace transform and enthalpy methods is the most accurate when dealing with the Stefan and Neumann problems. Combined with the effective specific heat method, the hybrid Laplace transform scheme is more accurate than finite difference method. As a result, the proposed approach applying the hybrid Laplace transform method could provide an effective way in solving phase change problems.

1. Ceylan, H. T., “Long-Time Solutions to Heat Conduction Transient with Time-Dependent Inputs,” Thesis, Mechanical Engineering Department, University of Wisconsin-Madison, WS., 1979.
2. Mikhailov, M. D., and Ozisik, M. N., “General Solution of One-Dimensional, Time Dependent Coupled Heat Flow Systems,” Int. COMM. Heat Mass Transfer, vol. 12, pp. 393-404, 1985.
3. Rizzo, F. J., and Shippy, D. J., “A method of solution for certain problems of transient heat conduction,” AIAA J, vol. 8, pp. 2004-2009, 1970.
4. Holzlohner, U., “A finite element analysis for time-dependent problems,” Int. J. Numer. Meth. Engng, Vol. 8, pp. 55-69, 1974.
5. Aral, M. M., and Gulcat, U., “A finite element Laplace transform solution technique for the wave equation,” Int. J. Numer. Meth. Engng., vol. 11, pp. 1719-1732, 1977.
6. Liggett, J. A., and Liu, P. L. F., “Unsteady flow in confined aquifers : a comparison of two boundary integral methods,” Water Resources Research, vol. 15, pp. 861-866, 1979.
7. Tamma, K. K. and Railkar, S. B., “Hybrid transfinite element methodology for nonlinear transient, thermal problems,” Numer. Heat Transfer, vol. 11, pp. 443-459, 1987.
8. Tamma, K. K., and Railkar, S. B., “Transfinite element methodology for nonlinear/linear transient thermal modeling/analysis : process and recent advances,” Int. J. Numer. Meth. Engng., vol. 25, pp. 475-494, 1988.
9. Tamma, K. K. and Railkar, S. B., “A new hybrid transfinite element computational methodology for applicability to conduction/convection/radiation heat transfer,” Int. J. Numer. Meth. Engng., vol. 26, pp. 1087-1100, 1988.
10. Tamma, K. K. and Railkar, S. B., “Specially tailored transfinite-element formulation for hyperbolic heat conduction involving non-Fourier effects,” Numer. Heat Transfer, vol. l5 (Part B)., pp. 211-226, 1989.
11. Chen, H. T., “Application of hybrid numerical method to transient heat conduction problems,” Ph. D. Thesis, Mechanical Engineering Department, National Cheng Kung University, R.O.C, 1987.
12. Chen, H. T. and Lin, J. Y., “Application of the Laplace transform to nonlinear transient problems,” Appl. Math. Modelling, vol. 15, pp. 144-151, 1991.
13. Chen, H. T. and Lin, J. Y., “Hybrid Laplace transform technique for non-linear transient thermal problems,” Int. J. Heat Mass Transfer, vol. 34, pp. 1301-1308, 1991.
14. Chen, H. T. and Lin, J. Y., “Application of the hybrid method to transient heat conduction in one-dimensional composite layers,” Comp. Struct., vol. 39, pp. 451-458, 1991.
15. Chen, H. T. and Lin, J. Y., “Application of the Laplace transform to one-dimensional non-linear transient heat conduction in hollow cylinders,” Comm. Appl. Numer. Meth., vol. 7, pp. 241-252, 1991.
16. Lin, J. Y. and Chen, H. T., “Radiar'axrsymmetric transient heat conduction in composite hollow cylinders with variable thermal conductivity,” Engng. Anal. Boundary Elements, vol. 10, pp. 27-33, 1992.
17. 林傑毓， “解析暫態熱傳問題之新數值方法，” 國立成功大學機械工程研究所博士論文， 1994.
18. Carslaw, H. S. and Jaeger, J. C., “ Conduction of Heat in Solids,” 2nd ed., Clarendon Press, London, 1959.
19. Sikarskie, D. L. and Boley, B. A., “The solution of a class of two-dimensional melting and solidification problems,” Int. J. Solids Structures, vol. 1, pp. 207-234, 1965.
20. Rathjen, K. A. and Jiji, L. M., “Heat conduction with melting or freezing in a corner,” J. Heat Transfer ASME, vol. 93, pp. 101-109.
21. Budhia, H. and Kreith, F., “Heat transfer with melting or freezing in a wdge,” Int. J. Heat Mass Transfer, vol. 16, pp. 195-211, 1973.
22. Ku, J. Y. and Chan, S. H., “A generalized Laplace transform technique for phase-change problems,” J. Heat Transfer ASME, vol.112, pp. 495-497, 1990.
23. Chen, H. T. and Lin, J. Y., “Hybrid Laplace transform technique for Stefan problems with radiation-convection boundary condition,” Int. J. Heat Mass Transfer vol.35, pp. 3345-3351, 1992.
24. 黃俊誠， “混合拉式轉換與數值分析法在傅立葉與非傅立葉熱傳問題之研究，” 國立成功大學工程科學研究所碩士論文, 2008.
25. Anderson, D. A., Tannehill, J. C. and Pletcher, R. H., “Computational Fluid Mechanics and Heat Trans,” Hemisphere, 1990.
26. Honig, G. and Hirdes, U., “A Method for the Numerical Inversion of Laplace Transforms,” J. of Computational and Applied Mathematics, vol. 10, pp. 113-133, 1984.
27. Hsiao, J. S., “An Efficient Algorithm for Finite Difference Analysis of Heat Transfer with Melting and Solidification,” Numerical Heat Transfer, Vol. 8, pp. 653-666, 1985.
28. Date, A. W., “A Strong Enthalpy Formulation for the Stefan Problem,” International Journal of Heat and Mass Transfer, vol. 34, pp. 2231-2283, 1991.
29. Crank, J., “Free and Moving Boundary Problems,” Oxford Uni. Press, Oxford, 1984.
30. Rappaz, M., “Modelling of Microstructure Formation in Solidification Processes,” Internal Materials Reviews, Vol. 34, No. 3, pp. 93-198, 1989.
31. Dantzig, J. A., “Modeling liquid–solid phase changes with melt convection, ” International Journal for Numerical Methods in Engineering, ” Vol. 28, pp. 1769–1785, 1989.