本文應用混合拉氏轉換法（Laplace transform）和有限差分法（finite difference method）的數值方法，並配合最小平方法（least squares method）來預測冷激鐵(chill)與鑄件(casting)間之接觸熱阻。首先利用拉氏轉換法處理系統之時間域，再以有限差分法處理系統轉換後之空間域，最後以數值逆拉氏轉換法求得於兩接觸面處之溫度和熱通量。本文之數值反算法乃假設於冷激鐵與鑄件之交界處的熱通量相等以預測其之間的接觸熱阻。為求得較精確的交界處熱通量估算值，將整個時間域分割成數個小時間域，而後再利用量測溫度配合反算法求出小時間區間的估算值。本文列舉各種不同的實例來驗證本文反算法之精確性。計算結果顯示接觸熱傳係數(contact heat transfer coefficient)預測值與前人的預測值差異不大。為了更進一步證實本文之預測值的可靠性，利用已獲得之預測值所求得於其他量測位置之溫度值與實驗值也甚吻合。另外，文中也將探討量測位置對於接觸熱傳導係數的影響。

The present study introduces a hybrid numerical method to predict the contact heat transfer coefficients between casting and metal mold. This algorithm combines the Laplace transform and the finite-difference method in conjunction with the least-squares scheme. Time-dependent terms in the governing equation are removed by using the Laplace transform, and then the resulting differential equation is discretized by the finite-difference method. Finally, the temperature and heat flux of the contact surface are obtained by using the numerical inversion of Laplace transform. It is assumed that the heat flux of the casting and mold interface are equal. In order to predict the heat flux of the interface accurately, the whole time domain is divided into several sub-time intervals, and then using the inverse scheme to estimate the heat flux in the individual sub-time domain. In this thesis, various examples are illustrated to show the accuracy of the present hybrid inverse numerical method. The numerical results show that the contact heat transfer coefficients are predicted well in comparison with the previous results of the other research. Also, the estimated heat flux and the temperature distribution are coincided with the experimental data. The effects of thermocouple location on the accuracy of estimated contact heat transfer coefficient are also investigated.

1. C. V. Madhusudana, “Thermal Contact Resistance”, Springer-Verlag, New York. pp.1-8, 1996.

2. T. N. Cetinkale and M. Fishenden, “Thermal Conductance of metallic surfaces in contact”, General Discussion on Heat Transfer, Proc. Inst. Mech. Engrs., London Sept. pp.271-275,1951.

3. A. M. Clausing, “Some influence of macroscopic constrictions on the thermal contact conductance”, NASA Report No. ME-TN-242-2, 1966.

4. A. M. Clausing and B. T. Chao, “Thermal Contact Resistance in a vacuum environment”, ASME J. Heat Transfer, 87 No.3, pp.243-251,1965.

5. M. G. Cooper, B. B. Mikic and M. M. Yovanovich, “Thermal Contact Conductance”, Int. J. Heat Transfer, 12, January, pp.279-300, 1969.

6. W. M. Moses, Jr., “An experimental investigation of the heat transfer across periodically contacting surface”, Ph. D. Dissertation, North Carolina State University, Raleigh, NC.,1985.

7. W. M. Moses and R. R. Johnson., “An experimental study of the transient behavior of the Thermal Contact Conductance and temperature in periodically contact conductance”, AIAA Paper, No. AI-AA-86-1244, 1986.

8. W. M. Moses and R. R. Johnson, “Experimental study of the transient heat transfer across periodically contacting surfaces”, Journal of Thermophysics and Heat Transfer, Vol. 2, pp.37-42, 1988.

9. C. H. Huang and T. M. Ju, “An inverse problem of simultaneously estimating contact conductance and heat transfer coefficient of exhaust gases between engine’s exhaust valve and seat”, Int. J. for Numerical Methods in Engineering, Vol. 38, pp.735-754, 1995.

10. J. R. Howard, “An experimental study of heat transfer through periodically contacting surfaces”, Int. J. Heat and Mass Transfer vol. 19, pp.376-372, 1976.

11. K. Ho, and R.D. Pehlke, “Metal-mold interfacial heat transfer”, Metallurgical Transactions B, Vol.16B, pp.585-594, 1985.

12. A. K. Das, S. S. Sadhal, “Analytical Solution for Constriction Resistance with Interstitial Fluid in the Gap”, Heat and Mass Transfer, pp. 111–119, 1998.

13. T. R. Thomas, and S. D. Probert, 1970, “Thermal Contact Resistance: The Directional Effect and Other Problems”, Int. J. Heat Mass Transf., Vol. 13, pp. 789–807.

14. 黃志準,王如竹, 一種接觸熱阻的預測方法,低溫工程,第六期, pp.40-46, 2000.

15. 趙蘭萍,徐烈,用輪廓離散法研究粗糙表面間的接觸導熱,低溫工程,第六期, pp.52-57, 2000.

16. W.D. Griffiths, “The Heat-Transfer Coefficient during the Unidirectional Solidification of an Al-Si Alloy casting”, Metall. Trans. Vol. 30B, pp. 473-482, 1999.

17. M. Trovant, S. Argyropoulos, “Finding Boundary Conditions: A Coupling Strategy for the Modeling of Metal Casting Processes: Part І. Experimental Study and Correlation Development”, Metall. Trans. 31B, pp. 75-86, 2000.

18. A. A. Rostami, A. Y. Hassan, P. C. Lim, “Parametric Study of Thermal Constriction Resistance”, Heat Mass Transfer, Vol. 37, pp. 5-10, 2000

19. C. A. Santos, J. M. Quaresma and A. Garcia, “Determination of transient interfacial heat transfer coefficients in chill mold castings”. Journal of Alloys and Compounds, Vol.319 p.174-186, 2001.

20. T. S. Prasanna and K. N. Prabhu, “Heat Flux Transients at the Casting/Chill Interface during Solidification of Aluminum Base Alloys”, Metall. Trans. Vol. 22B, pp. 717, 1991.

21. M.A. Gafur, M. N. Haque and K. N. Prabhu, “Effect of chill thickness and superheat on casting/chill interfacial heat transfer during solidification of commercially pure aluminum”, Journal of Material Processing Technology. Vol. 133, p. 257-265, 2003.

22. L. S. Fletcher, “A Review of Thermal Enhancement Techniques for Electronic Systems”, IEEE Trans on Components, Hybrids. and Manufacturing Technology. Vol. 13, No 4, 1990.

23. W. S. Childres and G. P. Peterson. “Quantification of Thermal Contact Conductance in Electronic Packages”, IEEE Trans on Components, Hybrids and Manufacturing Technology. Vol. 12 No 4, pp. 717~723, 1989.

24. Y. C. Lee, H. T. Ghaffari and J. M. Segelken, “Internal Thermal Resistance of a Multi-Chip Packaging Design for VLSI Based Systems”, IEEE Trans on Components, Hybrids. and Manufacturing Technology. Vol. 12, No 2, pp.163-169, 1989.

25. Lasance. C. and Lacaze. A Transient Method for the Accurate Measurement of Interface Resistance, Proc 12th IEEE SEMI THERM, Austin (1996)

26. K. Mizuhara. and N. Ozawa., “Estimation of Thermal Contact Resistance Based on Electrical Contact Resistance Measurements”, Int. Journal Japan Soc. Prec. Eng., Vol. 33, No 1, pp.59-61, 1990.

27. M. Krishnan and D. G. R. Sharma, “Determination of the Interfacial Heat Transfer Coefficient h in Unidirectional Heat Flow by Beck’s Non Linear Estimation Procedure”, Int. Comm. Heat Mass Transfer, Vol. 23, pp. 203-214. 1996.

28. Y. Nishida, W. Droste, and S. Engler, 1986, “The Air-Gap Formation Process at the Casting-Mold Interface and the Heat Transfer Mechanism through the Gap”, Metall. Trans. B, Vol. 17B, pp. 833–844. 1986.

29. C. V. Madhusudana, L. S. Fletcher, “Contact Heat Transfer-The Last Decade”, AIAA Journal, Vol. 24, No.3, pp. 510–523. 1984.

30. L. S. Fletcher, “Recent Developments in Contact Conductance Heat Transfer”, ASME J. Heat Transfer, Vol. 110, pp. 1059–1070, 1988.

31. E. G. Wolff and D.A. Schneider, “Prediction of thermal contact resistance between polished surfaces”, Int. J. Mass Transfer, Vol. 41, pp. 3469-3482, 1998.

32. H. T. Chen and J. Y. Lin “Application of the Laplace Transform to Nonlinear Transient Problems”, Appl. Math. Modeling, vol. 15, pp. 144-151, 1991.

33. H. T. Chen and J. Y. Lin “Hybrid Laplace transform technique for non-linear transient thermal problems”, Int. J. Heat Mass Transfer, Vol. 34, pp. 1301-1308, 1991.

34. H. T. Chen, S. Y. Lin and L. C Fang, “Estimation of Surface Temperature in Two-Dimensional Inverse Heat Conduction Problems”, Int. J. Heat Mass Transfer, Vol. 44, pp. 1455-1463. 2000.

35. H. T. Chen, S. Y. Peng, P. C. Yang and L. C. Fang, “Numerical Method for Hyperbolic Inverse Heat Conduction Problems”, Int. Comm. Heat Mass Transfer, Vol. 28, pp.847-856, 2001.

36. 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, Vol. 45, pp. 15-23 , 2002.

37. H. T. Chen, X. Y. Wu, Y. S. Hsiao, L. C Fang, “Estimate of surface condition from the theory of dynamic thermal stresses”, Int. J. Thermal Sciences , Vol. 43, 2003.