本研究以數學模型預測熱軋低碳鋼經冷卻後相變態顯微組織，再與實驗所獲得相比例以及晶粒大小結果比對。將數學模型分為兩部分,第一為加熱過程的沃斯田鐵晶粒大小之預測，及第二為冷卻過程中相變態之預測，其中考慮加工應變量即差排密度之因素。
加熱過程的沃斯田鐵晶粒大小之預測，其晶粒大小的數學模型預測採用與時間和溫度有關的恆溫成長公式。而冷卻過程中相變態之預測，相比例的模擬透過熱力學機制以OE(Ortho-Equilibrium)、PE(Para- Equilibrium)兩種平衡條件計算相轉換溫度，再經由動力學機制進行相變態率的累加，而獲得相變態後相比例；晶粒大小的模擬則利用半經驗式進行預測。
實驗則分別由光學顯微鏡及電子背向散射(EBSD)觀察相變態後的沃斯田鐵、變韌鐵、波來鐵和麻田散鐵。以光學顯微鏡可以觀察及計算肥粒鐵的相比例和晶粒大小，但對於沃斯田鐵、變韌鐵、波來鐵和麻田散鐵的比例則無法明確辨識。藉由EBSD的相鑑定分辨出肥粒鐵、沃斯田鐵和麻田散鐵，而變韌鐵和波來鐵則可透過IQ(Image Quality)方式與肥粒鐵區別出來。
模擬結果與實驗比較，沃斯田鐵晶粒成長公式以成長指數n為2最接近實驗結果；相比例的預測結果PE比OE更符合實驗結果；晶粒大小預測式以Suehiro公式最符合低碳鋼的預測；應變對低冷卻速率的相變態率影響不大。

In this study a mathematical model was developed to predict phase transformation of hot rolled low-carbon steels after cooling. The mathematical model is used to predict the grain size of austenite during heating and the phase transformation under cooling. The estimation of grain size is calculated using a semi-empirical equation. The computation of phase transformation is based on ortho-equilibrium (OE) and para- equilibrium (PE). The results predicted by the mathematical model were compared with those results of experiments.
The microstructures of austenite, ferrite pearlite, bainite and martensite were characterized using optical microscope (OM) and electron backscatter diffraction (EBSD). The image quality (IQ) in EBSD was used to distinguish between bainite and ferrite.
The index n of 2 was employed for calculating the grain size of austenite. In comparison with experimental results the results of PE model is better than these of the OE model for predicting phase fractions. The mathematical model proposed in this study was successfully to predict phase transformation during cooling for hot rolled low-carbon steels. In addition the image quality (IQ) in EBSD was employed to identify ferrite, pearlite and bainite.

1.C. Ouchi and T. Okita, Dynamic recrystallization behavior of austenite in Nb-bearing high-strength low-alloy steels and stainless steel, Transactions of the Iron and Steel Institute of Japan, 22(1982), pp.543-551.
2.A. Yoshie, H. Mirikawa, and Y. Onoe, Formulation of Static Recrystallization of Austenite in Hot Rolling Process of Steel Plate, Transactions of the Iron and Steel Institute of Japan, 27(1987), pp.425-431.
3.I. Kozasu, Application of TMCP steels. Historical progress of TMCP, Transactions of the Iron and Steel Institute of Japan, 12(1972), pp.241.
4.J. Yanagimoto, Numerical Analysis for the Prediction of Microstructure after Hot Forming of. Structural Metals, Materials Transactions, 50(2009), pp.1620-1625.
5.P.A. Beck, J.C. Kremer, L.J. Demer and M.L. Holzworth, Grain Growth in High-Purity Aluminum and in an Aluminum-Magnesium Alloy, Trans.AIME, 175(1948), pp.372-394.
6.C.M. Sellars, J.A. Whiteman, Recrystallization and grain growth in hot rolling, Metal Science, 13(1979), pp.187-194.
7.S. Nanba, M. Kitamura, M. Shimada, M. Katsumata, T. Inque, H. Imamura, Y. Maeda and S. Hattori, Prediction of Microstructure Distribution in the Through-thickness Direction dfuring and after Hot Rolling in Carbon Steels, ISIJ International, 32(1992), pp.377-386.
8.T. Senuma and H. Yada, Annealing Processes,Recovery Recrystallization and Grain Growth, Proceedings of 7th Riso International Symposium on Metallurgy and Materials Science, S.S. Hansen, D. Juul Jensen, T. Lefjers, and B. Ralph, eds., Riso National Laboratory, Roshilde, Denmark, Sept. 8-12, (1986), pp.547-552.
9.M. Suehiro, K. Sato,Y. Tsukano, H. Yada, T. Senuma and Y. Matsumura, Computer modeling of microstructural change and strength of low carbon steel in hot strip rolling, Transactions of the Iron and Steel Institute of Japan, 27(1987), pp.439-445.
10.C.M. Sellars, J.A. Whiteman, Recrystallization and Grain Growth in Hot Rolling, Metal Science, 13(1979), pp.187.
11.W. Roberts, A. Sandberg, T. Siwecki and T. Warlefors, Int. Conf. on Technology and Applications of HSLA Steels, ASM, Philadelphia, (1983), pp.67.
12.P. Choquet, P. Fabregue, J. Giusti, B. Chamont, International Symposium on Mathematical Modelling of Hot Rolling Steel, in: S. Yue (Ed.), CIM Quebec, (1990), pp.34.
13.P.D. Hodgson and R.K. Gibbs, A Mathematical. Model to Predict the Mechanical Properties of Hot Rolled C-Mn and Microalloyed Steels, ISIJ International, 32(1992), pp.1329-1338.
14.H. Yada, Proc. Int. Symposium on Accelerated Cooling of Rolled Steel, G.E. Ruddle and A.F. Grawley (Eds.), Canada, (1987), pp.105.
15.T. Senuma, H. Yada, Y. Matsumura and T. Futamura, Structure of
austenite of carbon steels in high speed hot working processes, Tetsu to Hagane, 70(1984), pp.2112-2119.
16.J. Yanagimoto and J.S. Liu, Incremental Formulation for the Prediction of Microstructural Change in Multi-pass Hot Forming, ISIJ International, 39(1999), pp.171-175.
17.J.S. Liu, A. Yanagida, S. Sugiyama, and J. Yanagimoto, The Analysis of Phase Transformation for the Prediction of Microstructure Change after Hot Forming, ISIJ International, 41(2001), pp.1510-1516.
18.M. Avrami, Kinetics of Phase Change. I. General Theory, Journal of Chemical Physics, 7(1939), pp.1103–1112.
19.M. Avrami, Kinetics of Phase Change. II. Transformation-Time Relations for Random Distribution of Nuclei, Journal of Chemical Physics, 8(1940), pp.212–224.
20.M. Avrami, Kinetics of Phase Change. III. Granulation,Phase Change, and Microstructure, Journal of Chemical Physics, 9(1941), pp.177–184.
21.J.W. Cahn, The kinetics of grain boundary nucleated reactions, Acta Metallurgica, 4(1956), pp.449~459.
22.M. Suehiro, T. Senuma, H. Yada, Y. Matsumura and T. Ariyoshi, A Kinetic Model for Phase Transformations of Low Carbon Steels during Continuous Cooling, Testu-to-Hagan, 73(1987), pp.1026–1033.
23.A. Hultgren, Isotherm Omvandling av Austenit,Jernkontorets Annaler, 135(1951), pp.403-494.
24.M. Hillert, Använding av isoaktiva linjer i ternära till- stånds-diagram, Jernkontorets Annaler, 136(1952), pp.25-37.
25.H.I. Aaronson, H.A. Domain and G.M. Pound, Thermodynamics of the Austenite → Proeutectoid Ferrite Transformation. II Fe-C-X Alloys, Trans AIME, 236(1966), pp.768-781.
26.K. Hashiguchi, J.S. Kirkaldy, T. Fukuzumi and V. Pavaskar, Prediction of the Equilibrium, Paraequilibrium and No-Partition Local Equilibrium Phase Diagrams for Multicomponent Fe-C Base Alloys, CALPHAD, 8(1984), pp.173-186.
27.T.T. Pham, E.B. Hawbolt and J.K. Brimacombe, Predicting the onset of transformation under noncontinuous cooling conditions: Part I. Theory, Metallurgical and Materials Transactions A, 26(1995), pp.1987-1992.
28.J. Majta, R. Kuziak, M. Pietrzyk and H. Krzton, Use of the computer simulation to predict mechanical properties of C–Mn steel, after thermomechanical processing, Journal of Materials Processing Technology, 60(1996), pp.581-588.
29.T. Gladman, I.D. McIvor and F.B. Pickering, Journal of iron and steel institute, London, 210(1972), pp.916.
30.P. Choquet, P. Fabregue, J. Giusti, B. Chamont, J.N. Penzant and F. Blanchet In: S. Yue, Editor, Mathematical Modeling of Hot Rolling of Steel, CIM, Montreal, Canada， (1990), pp.34.
31.P.D. Hodgson and R.K. Gibbs In: S. Yue, Editor, Mathematical Modeling of Hot Rolling of Steel, (1990), pp.76–85.
32.T. Gladman, D. Dulieu and I.D. Mclvor, Microalloying '75, Union Carbide Corp., New York, (1976), pp.32.
33.T. Gladman and F.B. Pickering In: T.N. Baker, Editor, Yield, Flow and Fracture of Polycrystals, Applied Science, London, (1983), pp.141.
34.Y. Sawada, R.P. Foley, S.W. Thompson and G. Krauss, Microstructure-Property Relationships in Plain-Carbon, and V and V+ NB Microalloyed Medium Carbon Steels, ISS-AIME, Warrendale, Pennsylvania, (1994), pp.263-286.
35.H. Yada: Proc. Int. Symp. on Accelerated Cooling of Rolled Steel, ed. G.E. Ruddle and A.F.. Crawley, Pergamon, (1988), pp.105.
36.K. Irvine, F. Pickering, and I. Garstone, Low-carbon bainitic steels, Journal Iron and Steel Institute, 187(1957), pp.292-309.
37.M. Suehiro, K. Sato, Y. Tsukano, H. Yada, T. Senuma, Y. Matsumura, Computer modeling of microstructural change and strength of low carbon steel in Hot Strip Rolling, 27(1987). pp.439-445.
38.M. Umemoto, H. Ohtsuka, I. Tamura, Grain size estimation from transformation kinetics, Acta Metallurgica, 34(1986), pp.1377-1385.
39.M. Hillert, L.I. Staffansson, The Regular Solution Model for Stoichiometric Phases and Ionic Melts, Acta Chemical Scandinavica, 24(1970), pp.3618-3626.
40.H.I. Aaronson, H.A. Domain, and G.M. Pound, Thermodynamics of the Austenite to Proeutectoid Ferrite Transformation II Fe-CX Alloys, TMS-AIME, 236(1966), pp.768-781.
41.G.J. Shiflet, J.R. Bradley, and H.I. Aaronson, A re-examination of the thermodynamics of the proeutectoid ferrite transformation in Fe-C alloys, Metallurgical and Materials Transactions A, 9(1978), pp.999-1008.
42.H.K.D.H. Bhadeshia and D.V. Edmonds, The Mechanism of Bainite Formation in Steels, Acta Metallurgica, 28(1980), pp.1265-1273.
43.B. Uhrenius, A compendium of ternary iron-base phase diagrams ,Hardenability Concepts with Application to steel, ed. by D.V. Doane and J.S. Kirkaldy, TMS-AIME, (1987), pp.28-81.
44.E.A. Guggenheim, Mixture, Oxford University Press, London, (1952), pp29.
45.C.H.P. Lupis, Chemical Thermodynamics of Materials, Elservier science publishing Co., NY, (1983).
46.J.S. Kirkaldy and E.A. Baganis, Thermodynamic prediction of the ae3 temperature of steels with additions of Mn, Si, Ni, Cr, Mo, Cu, Metallurgical and Materials Transactions A, 9(1978), pp.495-501.
47.S. Yayami and T. Ferukawa, A family of high strength, cold rolled steels, Proc. Int. Symposium on High-Strength Low-Alloyed Steels，Microalloying – 75, New York, Oct. 1–3, (1975), pp.311–333.
48.廖瑞聰，二次相效應混合型牛頓迭代法之研究， (2002).
49.A. Hultgren, Isothermal transformation of Austenite, Transaction of the ASM 39, 39(1947), pp.915-1005.
50.M. Hillert, Nuclear Composition- A factor of Interest in Nucleation, Acta Metallurgical, 1(1953), pp.764-766.
51.J.B. Gilmour, G.R. Purdy, and J.S. Kirkaldy, Thermodynamics controlling the proeutectoid ferrite transformations in Fe-C-Mn alloys, Metallurgical and Materials Transactions B, 3(1972), pp.1455-1464.
52.J.S. Kirkaldy, B.A. Thomson and E.A. Baganis, Prediction of Multicomponent Equilibrium and Transformation Diagrams for Low Alloy Steels, Hardenability Concepts with Applocations to Steel, D.V. Doane and J.S. Kirkaldy, eds.,AIME, (1978) , pp.82-125
53.M. Suehiro, T. Senuma, H. Yada, Y. Matsumura and T. Ariyoshi,A Kinetic Model for Phase Transformations of Low Carbon Steels during Continuous Cooling, Tetsu-to-Hagane, 73(1987), pp.1026-1033
54.J. Liu, A. Yanagida, S. Sugiyama and J. Yanagimoto, The Analysis of Phase Transformation for the Prediction of Microstructure Change after Hot Forming, ISIJ International, 41(2001), pp.1510–1516.
55.C. Zener, Kinetics of the decomposition of austenite, Trans. AIME, 167(1946), pp.550-583.
56.M. Hillert, K. Nilsson, and L.E. Torndahl, Effect of alloying elements on the formation of austenite and dissolution of cementite, Journal of The Iron and Steel Institute, 209(1971), pp.49-66.
57.G.T. Eldis, A Critical Review of Data Sources for Isothermal Transformation and Continuous Cooling Transformation Diagrams, Hardenability Concepts with Application to steel, ed. by D.V. Doane and J.S. Kirkaldy, TMS-AIME, (1978), pp.126-148.
58.P. Choquet, P. Fabregue, J. Giusti, B. Chamont, International Symposium on Mathematical Modelling of Hot Rolling Steel, in: S. Yue (Ed.), CIM Quebec, (1990), pp.34.
59.C.M. Sellars and J.H. Beynon In: D.P Dunne and T. Chandra, Editors, Process Conference on High Strength Low Alloy Steels, Wollongong, Australia, (1985), pp.142-150.
60.L. Ryde, Application of EBSD to analysis of microstructures in commercial steels, Materials. Science and Technology, 22(2006), pp.1297-1306.
61.J. Wu, P.J. Wray, C.I. Garcia, M. Hua and A.J. Deardo, Image Quality Analysis: A New Method of Characterizing Microstructures, ISIJ International, 45(2005), pp.254.
62.S. Zaefferer, P. Romano and F. Friedel, EBSD as a tool to identify and quantify bainite and ferrite in low-alloyed Al-TRIP steels, Journal of Microscopy, 230(2008), pp.499-508.
63.J.Y. Kang, D.H. Kim, S.I. Baik, T.H. Ahn, Y.W. Kim, H.N. Han, K. H. Oh, H.C. Lee and S.H. Han, Phase Analysis of Steels by Grain-averaged EBSD Functions, ISIJ International, 51(2011), pp.130-136.
64.M. Suehiro, K. Sato, Y. Tsukano, H. Yada, T. Senuma and Y. Matsumura, Computer modeling of microstructural change and strength of low carbon steel in hot strip rolling, Transactions of the Iron and Steel Institute of Japan, 27 (1987), pp.439-445.