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Comparative study on structural transformation of low-melting pure Al and high-melting stainless steel under external pulsed magnetic field
Materials Letters 61 (2007) 4011 – 4014 www.elsevier.com/locate/matlet
Comparative study on structural transformation of low-melting pure Al and high-melting stainless steel under external pulsed magnetic field Yu-Lai Gao a , Qiu-Shu Li a,b , Yong-Yong Gong a , Qi-Jie Zhai a,⁎ a
b
School of Materials Science and Engineering, Shanghai University, Shanghai 200072, PR China School of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, PR China Received 23 October 2006; accepted 8 January 2007 Available online 12 January 2007
Abstract A comparative study on the structural transformation of low-melting pure Al and high-melting 1Cr18Ni9Ti stainless steel under external pulsed magnetic field was carried out. The results showed that totally equiaxed grains were produced for pure Al, however, only thin columnar grains were generated for stainless steel even treated with higher magnetic intensity. It is deemed that grain refinement can be attributed to the heterogeneous nucleus created on the mould wall as well as their falling by the oscillating resulting from the magnetic field. In contrast, a dense chilling layer was generated at the primary solidification stage of the stainless steel due to the large temperature gradient between the high temperature melt and the mould and accordingly the nucleus falling was prevented. Therefore, only dendrites refinement possibly occurred. © 2007 Elsevier B.V. All rights reserved. Keywords: Metals and alloys; Solidification; Microstructure
1. Introduction Magnetic field plays a particular role in materials processing, especially in the field of metal solidification, which, in essence, differs from the traditional incubation treatment due to its protecting the melt from pollution. It is, therefore, a new type of solidification structure refinement technique. Generally, direct current magnetic field can efficiently impair the convection during solidification and improve the solidification structure [1–3], whereas the aim of applying alternating and rotating magnetic fields is to generate electromagnetic stirring [4–6] or levitation [7,8], altering the solidification process and controlling the solidification microstructure. Different from the said types of magnetic fields, pulsed magnetic field processing has become one of the most promising new techniques to refine the solidified structure attributing to its large energy capacity and easy manipulation. However, previous studies were focused on the low-melting metals [9,10], and few attempts had been made on high-melting ⁎ Corresponding author. Tel./fax: +86 21 56331218. E-mail addresses: [email protected] (Y.-L. Gao), [email protected] (Q.-S. Li), [email protected] (Y.-Y. Gong), [email protected] (Q.-J. Zhai). 0167-577X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2007.01.007
alloys due to the high requirement for the appropriate apparatuses. As a result, uncertainties still exist for the influence of the external physical fields on solidification process because of their various solidification characteristics of low-melting metals and high-melting ones. The purpose of this paper is to conduct a comparative study on alloys with various melting temperatures, and thus to investigate the influence of the external pulsed magnetic field on their structure transformation, by which to understand its influence mechanism. 2. Experimental In order to study the difference of the effect on structure transformation of low-melting and high-melting alloys, pure Al (purity ≥ 99.7 wt.%) and 1Cr18Ni9Ti stainless steel were chosen as the raw materials. To avoid the fast liquid–solid transition and ensure enough actuation time of the pulsed magnetic field, the specimen sizes were designed Φ50 × 130 mm for pure Al and Φ40 × 130 mm for stainless steel with the consideration of the available space inside the induction coil of the magnetic field and the limitation of the melting equipment.
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Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
The experiment was carried out by using the self-designed high-voltage pulsed power source and the solidification tester. The highest value of the intensity of pulsed magnetic field can reach 6 T in the central homogeneous zone. The aluminum ingots were melted in an electrical resistance furnace yet the stainless steel ingots were melted by a medium-frequency induction furnace. In view of the temperature decrease, the liquid metal was superheated to 850 °C for pure Al and 1600 °C ± 10 °C for stainless steel and subsequently poured into the sand moulds. The etching reagents used to reveal the solidification structure were mixtures of HNO3:HF:HCl:H2O = 3:3:9:5 for pure Al and FeCl3 + HCl+ H2O = 18 g:30 ml:100 ml for 1Cr18Ni9Ti stainless steel, respectively.
to say, no divaricated structure existed in those equiaxed or columnar grains. It is, therefore, not possible to achieve grain multiplication via grain fragmentation. Thus, it is speculated that the promotion of nucleation during solidification is responsible for the grain refinement. The pulsed magnetic field was produced when unidirectional pulse current passed the induction coil. Meanwhile, the induction current was created on the side surface of the specimen inside the pulsed magnetic field. As a result, Lorentz force was generated by the coupling effects of the induction current and the magnetic field, given by
F ¼J B
ð1Þ
3. Results and discussion The liquid Al was treated as soon as it was poured into the sand mould. In light of the principle of the self-designed power source, there is a dependence between the magnetic intensity and its frequency. Fig. 1 shows the variation of the solidification structure of pure Al when different magnetic fields were employed during the solidification process. It was found that only coarse columnar grains were produced without the influence of pulsed magnetic field, however, the structure was refined under the external influence of pulsed magnetic field. In addition, totally refined equiaxed grains were gained when the magnetic intensity exceeded 0.31 T (Fig. 1(d)). Namely the solidification structure of pure Al could be evidently refined under the effect of pulsed magnetic field. In contrast, only thin columnar grains were produced even under stronger external pulsed magnetic field (as shown in Fig. 2), implying that it was more difficult to obtain refined solidification structure for high-melting alloys than that of low-melting ones. Thus, an issue was thereby put forward, namely what was the dominating factor for structure refinement. As to pure Al, it experiences nucleation and grain growth during its solidification process, without the formation of dendrites. That is
where F is the Lorentz force, B the magnetic intensity and J the induction current. According to the right-hand rule, the direction of the Lorentz force is pointed from the side surface to the specimen center, directly imposing on the liquid metal. During the pouring and solidification process of liquid Al, heterogeneous nucleation was caused by the relatively cool mould wall. Attributing to this radial force, these nuclei were possibly oscillated apart from the mould wall. As a result, continuously produced heterogeneous nucleus promoted the structure refinement of pure Al. Nevertheless, a dense solidification layer was formed due to the large temperature gradient between the liquid steel and the sand mould, and therefore no heterogeneous nucleus could be oscillated apart from the mould wall. Anyhow, the magneto-oscillating was possibly destabilizing the dendrite growth and refining the columnar grains. On the other hand, the induced Joule heat also influenced the solidification process. According to the operating principle of the magnetic field, not only the induction magnetic field but also the induction current was produced when pulse current passed through the induction coil. Thus, Joule heat was obtained attributing to the induction current, and the specimen temperature was possibly increased.
Fig. 1. Structure transformation of pure Al specimens under the influence of the pulsed magnetic field with the intensity and frequency of: (a) 0 T, (b) 0.12 T (82.7 Hz), (c) 0.22 T (45.1 Hz), (d) 0.31 T (32 Hz), (e) 0.37 T (26.8 Hz) and (f) 0.51 T (19.5 Hz).
Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
4013
Fig. 2. Structure transformation of austenitic stainless steel under the influence of pulsed magnetic field with the intensity and frequency of: (a) 0 T, (b) 0.48 T (20.7 Hz), (c) 0.72 T (13.8 Hz), (d) 0.96 T (10.3 Hz), (e) 1.10 T (9.0 Hz), and (f) 1.35 T (7.3 Hz).
In light of the electromagnetic theory, the induction current created in the metallic specimen under the influence of pulsed magnetic field was expressed by [11] B s K ¼ exp − ð2Þ l0 sm And the current density of the specimen was given by B s exp − J¼ ll0 sm
ð3Þ
where B is the magnetic intensity, T; μ0 is the vacuum permeability, H/m; τ is the discharge time, s; τm equals to 1/2μσδr; s; μ is the specimen permeability; σ is the electric conductivity of the specimen, S/m; δ is the diffusion depth of the magnetic field, m; r is the specimen radius, m; and l is the specimen length in the magnetic field zone with stable intensity, m. With respect to the aforementioned induction current, its heating effect could be calculated by the following equation: Q ¼ I 2 Rt
ð4Þ
with I ¼ JS
ð5Þ
and l q ð6Þ S where ρ is the electrical resistivity of the melt, Ω m; and S the crosssectional area of the specimen, m2. R¼
As for the utilized pulse, the time width of each pulse was 0.035 s, then the effective time of the magnetic field on the specimen could be expressed as t ¼ 0:035f s V
ð7Þ
where τ′ is the processing time, s; and f the pulse frequency, Hz. Obviously the parameter 0.035f indicates the duty ratio of the pulse. Substituting Eqs. (3), (5), (6), and (7) into Eq. (4), we get B2 2s Q ¼ 0:035J 2 Slqf s V¼ 0:035f qV 2 2 s Vexp − ð8Þ sm l l0 where V is the specimen volume. Then the temperature rise can be evaluated by the following equation Q f q B2 2s DT ¼ ð9Þ s V e xp − ¼ 0:035 cm Dc l2 l20 sm where c is the specific heat capacity of the specimen, J/kg °C; m the specimen mass, kg; and D the specimen density, kg/m3. Eq. (9) indicated that the temperature rise induced by the magnetic field was linearly increased with increasing the value of B2. Consequently, higher temperature rise happened when a stronger magnetic field was applied. In addition, the temperature rise produced in the surface was higher than that inside the specimen due to bigger induction current generated on the surface zone, flatting the temperature gradient from the surface to the center. Table 1 shows the changes of the temperature and temperature
Table 1 Changes in the temperature and temperature gradient from the surface to the center of pure Al specimen Solidification time, s Without treatment
Surface temperature, °C Center temperature, °C
Temperature gradient, °C/mm Treated under 0.51 T Surface temperature, °C Center temperature, °C Temperature gradient, °C/mm
10 656.9 667.5 − 0.43 670.2 674 − 0.15
20 655 662 − 0.28 664.5 668.1 − 0.15
50 654.1 661 − 0.28 657.1 662 − 0.20
100 652.7 659.7 − 0.28 657.4 661.1 − 0.15
200 650.1 657.1 −0.28 657.1 661.2 −0.17
300 646.4 653 − 0.27 657.3 660.1 − 0.11
400 642.1 647.8 − 0.23 656.5 657.2 − 0.03
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Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
gradient from the surface to the center of pure Al specimen during their solidification process without and with the influence of external pulsed magnetic field, in accordance with the above analysis. For pure Al, the Joule heat decreased the rapid formation of the chilling surface layer, promoting the separation of the incompacted heterogeneous nucleus. As for high-temperature stainless steel, the quantity of the induced Joule heat was not enough to essentially change the temperature gradient, so nucleus falling was ceased due to the surface rapid chilling layer. It should be noted that this study has examined only the structure transformation under given parameters of the self-developed pulsed magnetic field. Notwithstanding its limitation, this study does suggest that nucleation promotion dominates the structure refinement of metals when external pulsed magnetic field is applied.
4. Conclusions (1) Totally equiaxed grains could be obtained in pure Al under the external influence of the pulsed magnetic field, yet only thin columnar grains generated in high-melting steel even treated by higher magnetic intensity. (2) It is reasonable to attribute the grain refinement to the heterogeneous nucleus created on the mould wall as well as their falling by the oscillating which resulted from the magnetic field. For high-melting alloys, structure refinement was difficult if a surface dense chilling layer was formed at their primary solidification process.
(3) The structure refinement was influenced by the coupling effects of Lorentz force and Joule heat. In addition, more obvious effects could be obtained in low-melting metals. Acknowledgement The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant nos. 50274050 and 50674064). References [1] C. Hans, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 287 (2000) 205. [2] D.R. Uhlmann, T.P. Seward, Trans. Metall. Soc. AIME 236 (1966) 527. [3] S. Asai, Trans. ISIJ 18 (1978) 754. [4] A. Ohno, T. Motegi, J. Jpn. Inst. Met. 46 (1982) 554. [5] K. Murakami, T. Fujiyama, Acta Metall. 31 (1983) 1425. [6] M. Seki, H. Kawamura, J. Heat Transfer 101 (1979) 227. [7] R.S. Lin, M.G. Frohberg, High Temp. High Press. 24 (1992) 543. [8] M. Barth, B.B. Wei, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 226 (1997) 770. [9] C. Vives, Metall. Mater. Trans., B, Proc. Metall. Mater. Proc. Sci. 27 (1996) 445. [10] C. Vives, J. Cryst. Growth 173 (1997) 541. [11] A.H. Hermann, R.M. James, Electromagnetic Field and Energy, PrenticeHall International Editions, 1989, p. 427.
Comparative study on structural transformation of low-melting pure Al and high-melting stainless steel under external pulsed magnetic field Yu-Lai Gao a , Qiu-Shu Li a,b , Yong-Yong Gong a , Qi-Jie Zhai a,⁎ a
b
School of Materials Science and Engineering, Shanghai University, Shanghai 200072, PR China School of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, PR China Received 23 October 2006; accepted 8 January 2007 Available online 12 January 2007
Abstract A comparative study on the structural transformation of low-melting pure Al and high-melting 1Cr18Ni9Ti stainless steel under external pulsed magnetic field was carried out. The results showed that totally equiaxed grains were produced for pure Al, however, only thin columnar grains were generated for stainless steel even treated with higher magnetic intensity. It is deemed that grain refinement can be attributed to the heterogeneous nucleus created on the mould wall as well as their falling by the oscillating resulting from the magnetic field. In contrast, a dense chilling layer was generated at the primary solidification stage of the stainless steel due to the large temperature gradient between the high temperature melt and the mould and accordingly the nucleus falling was prevented. Therefore, only dendrites refinement possibly occurred. © 2007 Elsevier B.V. All rights reserved. Keywords: Metals and alloys; Solidification; Microstructure
1. Introduction Magnetic field plays a particular role in materials processing, especially in the field of metal solidification, which, in essence, differs from the traditional incubation treatment due to its protecting the melt from pollution. It is, therefore, a new type of solidification structure refinement technique. Generally, direct current magnetic field can efficiently impair the convection during solidification and improve the solidification structure [1–3], whereas the aim of applying alternating and rotating magnetic fields is to generate electromagnetic stirring [4–6] or levitation [7,8], altering the solidification process and controlling the solidification microstructure. Different from the said types of magnetic fields, pulsed magnetic field processing has become one of the most promising new techniques to refine the solidified structure attributing to its large energy capacity and easy manipulation. However, previous studies were focused on the low-melting metals [9,10], and few attempts had been made on high-melting ⁎ Corresponding author. Tel./fax: +86 21 56331218. E-mail addresses: [email protected] (Y.-L. Gao), [email protected] (Q.-S. Li), [email protected] (Y.-Y. Gong), [email protected] (Q.-J. Zhai). 0167-577X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2007.01.007
alloys due to the high requirement for the appropriate apparatuses. As a result, uncertainties still exist for the influence of the external physical fields on solidification process because of their various solidification characteristics of low-melting metals and high-melting ones. The purpose of this paper is to conduct a comparative study on alloys with various melting temperatures, and thus to investigate the influence of the external pulsed magnetic field on their structure transformation, by which to understand its influence mechanism. 2. Experimental In order to study the difference of the effect on structure transformation of low-melting and high-melting alloys, pure Al (purity ≥ 99.7 wt.%) and 1Cr18Ni9Ti stainless steel were chosen as the raw materials. To avoid the fast liquid–solid transition and ensure enough actuation time of the pulsed magnetic field, the specimen sizes were designed Φ50 × 130 mm for pure Al and Φ40 × 130 mm for stainless steel with the consideration of the available space inside the induction coil of the magnetic field and the limitation of the melting equipment.
4012
Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
The experiment was carried out by using the self-designed high-voltage pulsed power source and the solidification tester. The highest value of the intensity of pulsed magnetic field can reach 6 T in the central homogeneous zone. The aluminum ingots were melted in an electrical resistance furnace yet the stainless steel ingots were melted by a medium-frequency induction furnace. In view of the temperature decrease, the liquid metal was superheated to 850 °C for pure Al and 1600 °C ± 10 °C for stainless steel and subsequently poured into the sand moulds. The etching reagents used to reveal the solidification structure were mixtures of HNO3:HF:HCl:H2O = 3:3:9:5 for pure Al and FeCl3 + HCl+ H2O = 18 g:30 ml:100 ml for 1Cr18Ni9Ti stainless steel, respectively.
to say, no divaricated structure existed in those equiaxed or columnar grains. It is, therefore, not possible to achieve grain multiplication via grain fragmentation. Thus, it is speculated that the promotion of nucleation during solidification is responsible for the grain refinement. The pulsed magnetic field was produced when unidirectional pulse current passed the induction coil. Meanwhile, the induction current was created on the side surface of the specimen inside the pulsed magnetic field. As a result, Lorentz force was generated by the coupling effects of the induction current and the magnetic field, given by
F ¼J B
ð1Þ
3. Results and discussion The liquid Al was treated as soon as it was poured into the sand mould. In light of the principle of the self-designed power source, there is a dependence between the magnetic intensity and its frequency. Fig. 1 shows the variation of the solidification structure of pure Al when different magnetic fields were employed during the solidification process. It was found that only coarse columnar grains were produced without the influence of pulsed magnetic field, however, the structure was refined under the external influence of pulsed magnetic field. In addition, totally refined equiaxed grains were gained when the magnetic intensity exceeded 0.31 T (Fig. 1(d)). Namely the solidification structure of pure Al could be evidently refined under the effect of pulsed magnetic field. In contrast, only thin columnar grains were produced even under stronger external pulsed magnetic field (as shown in Fig. 2), implying that it was more difficult to obtain refined solidification structure for high-melting alloys than that of low-melting ones. Thus, an issue was thereby put forward, namely what was the dominating factor for structure refinement. As to pure Al, it experiences nucleation and grain growth during its solidification process, without the formation of dendrites. That is
where F is the Lorentz force, B the magnetic intensity and J the induction current. According to the right-hand rule, the direction of the Lorentz force is pointed from the side surface to the specimen center, directly imposing on the liquid metal. During the pouring and solidification process of liquid Al, heterogeneous nucleation was caused by the relatively cool mould wall. Attributing to this radial force, these nuclei were possibly oscillated apart from the mould wall. As a result, continuously produced heterogeneous nucleus promoted the structure refinement of pure Al. Nevertheless, a dense solidification layer was formed due to the large temperature gradient between the liquid steel and the sand mould, and therefore no heterogeneous nucleus could be oscillated apart from the mould wall. Anyhow, the magneto-oscillating was possibly destabilizing the dendrite growth and refining the columnar grains. On the other hand, the induced Joule heat also influenced the solidification process. According to the operating principle of the magnetic field, not only the induction magnetic field but also the induction current was produced when pulse current passed through the induction coil. Thus, Joule heat was obtained attributing to the induction current, and the specimen temperature was possibly increased.
Fig. 1. Structure transformation of pure Al specimens under the influence of the pulsed magnetic field with the intensity and frequency of: (a) 0 T, (b) 0.12 T (82.7 Hz), (c) 0.22 T (45.1 Hz), (d) 0.31 T (32 Hz), (e) 0.37 T (26.8 Hz) and (f) 0.51 T (19.5 Hz).
Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
4013
Fig. 2. Structure transformation of austenitic stainless steel under the influence of pulsed magnetic field with the intensity and frequency of: (a) 0 T, (b) 0.48 T (20.7 Hz), (c) 0.72 T (13.8 Hz), (d) 0.96 T (10.3 Hz), (e) 1.10 T (9.0 Hz), and (f) 1.35 T (7.3 Hz).
In light of the electromagnetic theory, the induction current created in the metallic specimen under the influence of pulsed magnetic field was expressed by [11] B s K ¼ exp − ð2Þ l0 sm And the current density of the specimen was given by B s exp − J¼ ll0 sm
ð3Þ
where B is the magnetic intensity, T; μ0 is the vacuum permeability, H/m; τ is the discharge time, s; τm equals to 1/2μσδr; s; μ is the specimen permeability; σ is the electric conductivity of the specimen, S/m; δ is the diffusion depth of the magnetic field, m; r is the specimen radius, m; and l is the specimen length in the magnetic field zone with stable intensity, m. With respect to the aforementioned induction current, its heating effect could be calculated by the following equation: Q ¼ I 2 Rt
ð4Þ
with I ¼ JS
ð5Þ
and l q ð6Þ S where ρ is the electrical resistivity of the melt, Ω m; and S the crosssectional area of the specimen, m2. R¼
As for the utilized pulse, the time width of each pulse was 0.035 s, then the effective time of the magnetic field on the specimen could be expressed as t ¼ 0:035f s V
ð7Þ
where τ′ is the processing time, s; and f the pulse frequency, Hz. Obviously the parameter 0.035f indicates the duty ratio of the pulse. Substituting Eqs. (3), (5), (6), and (7) into Eq. (4), we get B2 2s Q ¼ 0:035J 2 Slqf s V¼ 0:035f qV 2 2 s Vexp − ð8Þ sm l l0 where V is the specimen volume. Then the temperature rise can be evaluated by the following equation Q f q B2 2s DT ¼ ð9Þ s V e xp − ¼ 0:035 cm Dc l2 l20 sm where c is the specific heat capacity of the specimen, J/kg °C; m the specimen mass, kg; and D the specimen density, kg/m3. Eq. (9) indicated that the temperature rise induced by the magnetic field was linearly increased with increasing the value of B2. Consequently, higher temperature rise happened when a stronger magnetic field was applied. In addition, the temperature rise produced in the surface was higher than that inside the specimen due to bigger induction current generated on the surface zone, flatting the temperature gradient from the surface to the center. Table 1 shows the changes of the temperature and temperature
Table 1 Changes in the temperature and temperature gradient from the surface to the center of pure Al specimen Solidification time, s Without treatment
Surface temperature, °C Center temperature, °C
Temperature gradient, °C/mm Treated under 0.51 T Surface temperature, °C Center temperature, °C Temperature gradient, °C/mm
10 656.9 667.5 − 0.43 670.2 674 − 0.15
20 655 662 − 0.28 664.5 668.1 − 0.15
50 654.1 661 − 0.28 657.1 662 − 0.20
100 652.7 659.7 − 0.28 657.4 661.1 − 0.15
200 650.1 657.1 −0.28 657.1 661.2 −0.17
300 646.4 653 − 0.27 657.3 660.1 − 0.11
400 642.1 647.8 − 0.23 656.5 657.2 − 0.03
4014
Y.-L. Gao et al. / Materials Letters 61 (2007) 4011–4014
gradient from the surface to the center of pure Al specimen during their solidification process without and with the influence of external pulsed magnetic field, in accordance with the above analysis. For pure Al, the Joule heat decreased the rapid formation of the chilling surface layer, promoting the separation of the incompacted heterogeneous nucleus. As for high-temperature stainless steel, the quantity of the induced Joule heat was not enough to essentially change the temperature gradient, so nucleus falling was ceased due to the surface rapid chilling layer. It should be noted that this study has examined only the structure transformation under given parameters of the self-developed pulsed magnetic field. Notwithstanding its limitation, this study does suggest that nucleation promotion dominates the structure refinement of metals when external pulsed magnetic field is applied.
4. Conclusions (1) Totally equiaxed grains could be obtained in pure Al under the external influence of the pulsed magnetic field, yet only thin columnar grains generated in high-melting steel even treated by higher magnetic intensity. (2) It is reasonable to attribute the grain refinement to the heterogeneous nucleus created on the mould wall as well as their falling by the oscillating which resulted from the magnetic field. For high-melting alloys, structure refinement was difficult if a surface dense chilling layer was formed at their primary solidification process.
(3) The structure refinement was influenced by the coupling effects of Lorentz force and Joule heat. In addition, more obvious effects could be obtained in low-melting metals. Acknowledgement The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant nos. 50274050 and 50674064). References [1] C. Hans, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 287 (2000) 205. [2] D.R. Uhlmann, T.P. Seward, Trans. Metall. Soc. AIME 236 (1966) 527. [3] S. Asai, Trans. ISIJ 18 (1978) 754. [4] A. Ohno, T. Motegi, J. Jpn. Inst. Met. 46 (1982) 554. [5] K. Murakami, T. Fujiyama, Acta Metall. 31 (1983) 1425. [6] M. Seki, H. Kawamura, J. Heat Transfer 101 (1979) 227. [7] R.S. Lin, M.G. Frohberg, High Temp. High Press. 24 (1992) 543. [8] M. Barth, B.B. Wei, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 226 (1997) 770. [9] C. Vives, Metall. Mater. Trans., B, Proc. Metall. Mater. Proc. Sci. 27 (1996) 445. [10] C. Vives, J. Cryst. Growth 173 (1997) 541. [11] A.H. Hermann, R.M. James, Electromagnetic Field and Energy, PrenticeHall International Editions, 1989, p. 427.