Effect of Na 2 O and B 2 O 3 on the Distribution of P 2 O 5 between Solid Solution and Liquid Phases Slag

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1 , pp Effect of Na 2 O and B 2 O 3 on the Distribution of P 2 O 5 between Solid Solution and Liquid Phases Slag Senlin XIE, 1) Wanlin WANG, 1) * Yongzhen LIU 1) and Hiroyuki MATSUURA 2) 1) School of Metallurgy and Environment, Central South University, Changsha, P. R. China. 2) Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba, Japan. (Received on September 21, 2013; accepted on December 18, 2013) With the development of multiphase slag system for hot-metal treatment to achieve better dephosphorization efficiency, it is very important to improve the distribution ratio of P 2O 5 between the solid solution (2CaO SiO 2 3CaO P 2O 5) and liquid phase slag. This study was carried out to investigate the effects of Na 2O and B 2O 3 on P 2O 5 distribution ratio and morphologies of corresponding solid solutions in CaO SiO 2 Fe 2O 3 P 2O 5 slag system. The results indicated that the distribution ratio of P 2O 5 would be improved with the increase of Na 2O content due to the formation of (2CaO SiO 2 Na 2O 2CaO P 2O 5) solid solution with a similar morphology as that of reference solid solution (2CaO SiO 2 3CaO P 2O 5) in the reference slag. While B 2O 3 plays an opposite role, it would not only reduce the phosphorus distribution ratio, but also change the morphology of its corresponding solid solution due to the formation of complex solid solution (2CaO SiO 2 Ca 9.93(P 5.84B 0.16O 24) (B 0.67O 1.79)). Besides, the effect of cooling rate on the size of the solid solution was also studied. It would provide an instructive way for the design of multiphase slag for hot metal treatment. KEY WORDS: multiphase slag; dephosphorization; distribution ratio; solid solution; morphology. 1. Introduction In order to meet the increasing demand for high quality grade steels such as low-phosphorus steel, it requires a high efficient hot metal dephosphorization process. The current slag system for hot metal treatment contains a large amount of solid CaO dispersed in liquid slag, which introduce significant CaO consumption due to its low solubility; especially when CaF 2 as fluxing agent was restricted due to the environmental concerns. Therefore, new refining method to improve the utilization efficiency of CaO by using a solid/ liquid coexisting multiphase slag was proposed, and the studies regarding to the solid solution formation mechanism, mass transfer behavior of phosphorus and the phase relationship for multi-phase slag have been conducted. 1 6) Normally, hot metal dephosphorization slag consists of CaO SiO 2 FeO P 2O 5, and it has been approved that 2CaO SiO 2 forms a solid solution with 3CaO P 2O 5 at the hot metal treatment temperature over a wide range of compositions. 7) Kitamura et al., 8) pointed out that the solid solution plays an important role in achieving efficient dephosphorization, and (2CaO SiO 2 3CaO P 2O 5) solid solution has been regarded as a absorbent for phosphorus to lower the phosphorus content in liquid phase, which makes it more capable for future dephosphorization. 9) In addition, the multiphase slag system has the potential to be recycled as a source of phosphorus, if (2CaO SiO 2 * Corresponding author: wanlin.wang@gmail.com DOI: 3CaO P 2O 5) solid solution could contain high phosphorus content. Yokoyama et al., 10,11) found that the concentrated phosphorus phase can be separated from Fe to matrix phase due to the large difference of their magnetic properties under the strong magnetic field. Therefore, it is essential to increase the distribution ratio of phosphorous between the solid solution and liquid slag as well as the phosphorus content in the solid solution. In order to increase the efficiency of dephosphorization and to use the slag as an alternative source of phosphate ores, a series of studies regarding to the distribution ratio of phosphorous between the liquid slag and the solid solution have been carried out 8,9,12 14). Kitamura et al., 8) studied the effect of P 2O 5 content and oxide additives such as MgO and MnO on the distribution ratio of phosphorus for CaO SiO 2 Fe 2O 3 P 2O 5 slag system, and the results show that the phosphorus distribution ratio would increase with the increase P 2O 5 content in the slag, while the influence of MgO and MnO was not significant. Shimauchi et al., 14) indicated that the similar results regarding to the influence of MgO and MnO on the distribution P 2O 5 ratio was obtained. Recently Pahlevani et al., 9) made a detailed study based on the previous results, it was found that, in the CaO SiO 2 P 2O 5 Fe 2O 3 system, the P 2O 5 distribution ratio decreased with the addition of MgO and MnO, but it did not change with the addition of Al 2O 3; besides, the effect of adding MgO, MnO and Al 2O 3 on the P 2O 5 distribution ratio was smaller in the case of CaO SiO 2 P 2O 5 FeO system than that in the case of CaO SiO 2 P 2O 5 Fe 2O 3 system. Therefore, studies regard ISIJ 766

2 ing to the influence of oxides additives such as MgO, MnO and Al 2O 3 on the P 2O 5 distribution in the multiphase slag would provide insight for development of industrial slag. Generally, Na 2O and B 2O 3 have been chosen as the effective additives for various slags to enhance the dissolution rate of CaO and to increase the phosphorus distribution between lime-based slag and liquid iron. Pak and Fruehan 15) observed a fivefold increase in the phosphorus partition coefficient between lime-based slag and liquid iron, for a 5- wt% addition of Na 2O to a CaO-saturated CaO FeO t SiO 2 slag at K. Hamano et al., 16,17) reported that the phosphorous partition ratio between lime-based slag and liquid iron didn t change, when SiO 2 was replaced by B 2O 3 for the Fe to CaO MgO satd. SiO 2 slag at K. Above results have shown that Na 2O and B 2O 3 would tend to influence the phosphorus partition between lime-based slag and liquid iron; however, the effect of the addition of Na 2O and B 2O 3 on the distribution ratio of P 2O 5 between solid solution and liquid slag in multi phase slag system hasn t been conducted, which may have the potential to further remove phosphorus from liquid iron to liquid slag and then into solid solution in a multiphase slag system. This paper would first investigate the effect of different oxides, i.e. Na 2O and B 2O 3 contents on the distribution ratio of P 2O 5 between the solid solution and liquid slag in the CaO SiO 2 Fe 2O 3 slag system, followed by the study of the effect of cooling rate on the distribution ratio of P 2O 5. Finally the morphology and size of the solid solution under the different conditions was also studied Preparations of Sample The design of samples was based on the CaO SiO 2 Fe 2O 3 system, the isothermal CaO SiO 2 Fe 2O 3 ternary phase diagram at K was calculated by FactSage software as shown in Fig. 2, where the composition of the reference slag (except P 2O 5) was represented by a red circle, The chemical compositions of all slag samples used in this study are listed in Table 1, where Sample O was designed as the reference sample, and others were prepared by adjusting the proportion of Na 2O and B 2O 3. All slag samples were prepared by using regent-grade SiO 2, Fe 2O 3, CaO, 3CaO P 2O 5, Na 2CO 3 and B 2O 3. After weighing, the reagents were fully mixed in various ratios to produce CaO SiO 2 Fe 2O 3 P 2O 5 (Na 2O or B 2O 3) slag system. The desired criterion for the slag systems in the study is that it should be in a homogenous liquid state at K and in a semisolid state at K Experimental Procedure The mixed regents were first placed in an corundum crucible as the effect of Al 2O 3 on distribution ratio of P 2O 5 is slight in the CaO SiO 2 P 2O 5 Fe 2O 3 system, 5,9) then the crucible was heated up to K in the electric resistance furnace, and was held for 60 min to eliminate bubbles and homogenize chemical composition for producing a homogenous slag. Then, the homogenous slag was cooled to K at a cooling rate of 5.0 K/min. After that, it was held at K for 60 min to produce a semisolid slag and ensure 2. Experiment 2.1. Experimental Apparatus A schematic of the experimental apparatus was shown in Fig. 1, where the electric resistance furnace with MoSi 2 heater was used, the accuracy of temperature control was about ± 5 K, and it could reach up to K due to its good insulation properties. The heating profile could be designed according to the experimental requirement. Besides, a peephole was mounted on the top of the heating furnace tube for the in-situ observation to check whether the mixed slag samples in resistance furnace has been melted or not. Fig. 2. The isothermal CaO SiO 2 Fe 2O 3 ternary phase diagram at 1623 K. Table 1. The chemical compositions of slag with different Na 2O and B 2O 3 (mass%). Fig. 1. Schematic drawing of experimental apparatus. Composition based on the mixing ratio (mass%) Sample CaO SiO 2 Fe 2O 3 P 2O 5 Na 2O B 2O 3 Total O A B C D E F ISIJ

3 that the equilibrium between the solid phase and the liquid phase would be obtained. 9,14) Finally, the slag sample was quenched through a rotating water bath for further analysis. The thermal profile for the experiments was shown in Fig. 3. It should be mentioned that Fe 2O 3 was used in the present work, even most of the iron oxide would be in the form of FeO in practical steelmaking; however, it has been indicated that the trend regarding to the effect of oxide additives on the distribution behavior of P 2O 5 between solid solution and liquid phase was consistent when the iron oxide was changed from FeO to Fe 2O 3. 9,12) 2.4. Experimental Method The samples selected from the core of each quenched slag were embedded in the polyester resin; then, they were polished following the standard metallographic procedure. Finally the polished samples were coated by Au evaporation for SEM observation. The composition of each phase in different regions was analyzed by SEM/EDS, as described in previous papers. 9,18) Through the element analysis, the amounts of oxides, such as Fe 2O 3, CaO, SiO 2, P 2O 5, Na 2O and Al 2O 3 in different phases can be obtained. Then, the distribution ratio of P 2O 5 (L P) between the solid solution and the liquid phase was calculated by Eq. (1). Meanwhile, some extra slag samples was crushed and ground into powders for XRD tests. XRD data were collected by using Cu Ka radiation ( Å), in a range of 2θ = 10 to 80 deg with a step size of 4 deg/s. PO SS LP = (% ) (1) (% PO ) 2 5 where the subscripts SS and L denote the solid solution and liquid phase, respectively. L 3. Experimental Results Figure 4 schematically shows the typical four stages during the experimental process of sample powders. Stage I is a period in which the different particle reacts with each other when the slag is heated. Stage II is the 60 minutes period of melting for bubbles elimination and chemical composition homogenization at K. Stage III is the time of the solid solution crystallization due to the drop of temperature with the rate of 5.0 K/min. Stage IV is the time of the growth of the solid solution when the temperature at the K holds for 60 min. The results regarding to the average compositions of the precipitated solid phases and matrix liquid phases at K are summarized in Table 2. In this Table 2. Sample Composition of the solid solution and liquid phases in slag (mass%). Phase Chemical composition of different phases CaO SiO 2 Fe 2O 3 Al 2O 3 Na 2OB 2O 3 P 2O 5 Lp O solid solution liquid phase A solid solution liquid phase B solid solution liquid phase C solid solution liquid phase D solid solution liquid phase solid solution E 3.72 liquid phase F solid solution liquid phase O(2.5) solid solution liquid phase O(10) solid solution liquid phase B(2.5) solid solution liquid phase Fig. 3. Thermal profile for slag melting, cooling and holding process. B(10) solid solution liquid phase Note: (XX) represents different cooling rate Fig. 4. Illustration of the melting, crystallization and growth processes of sample slag ISIJ 768

4 table, the distribution ratio of P 2O 5 between the solid and liquid phase for each sample is also shown. It should be mentioned that the element of Boron is not measured because its atomic number is too small to be analyzed by SEM/EDS, which might cause the slight increase of the other compounds content. Besides, the results of Sample O and Sample B at the different cooling rate of 2.5K/min and 10.0 K/min are also shown in the Table 2. Fig. 5. The distribution ratio of P 2O 5 versus mass fraction of Na 2O Distribution of P 2O 5 between the Solid Solution and Liquid Phases (1) Effect of Na 2O on the Distribution Ratio of P 2O 5 Figure 5 shows the relationship between distribution ratio of P 2O 5 and varied Na 2O content. It can be seen that the distribution ratio of P 2O 5 between the solid solution and liquid slag increases with the addition of Na 2O. As for the reference slag Sample O, the distribution ratio of P 2O 5 is 6.39; and it goes up to 6.69 and 8.16, when the Na 2O content increases from 1% to 5%. (2) Effect of B 2O 3 on the Distribution Ratio of P 2O 5 Figure 6 shows the relationship between the distribution ratio of P 2O 5 and B 2O 3 content for CaO SiO 2 Fe 2O 3 P 2O 5 slag, and it is clear that the distribution ratio of P 2O 5 decreases from 6.39 to 3.19 with the addition of B 2O 3 content. Compared with the effect of Na 2O, the results indicates that when dephosphorization is conducted by using multiphase slag, the addition of B 2O 3 might not be a good option for the enrichment of phosphorus from the liquid phase to solid solution. (3) Effect of Cooling Rate on the Distribution Ratio of P 2O 5 In order to understand the effect of the crystallization behavior of the solid solution on the distribution ratio of P 2O 5, the effect of cooling rate on crystallization of the precipitated solid solution was investigated. Figure 7 shows the phosphorus distribution ratio of both Sample O and Sample B at the different cooling rate of 2.5 K/min, 5.0 K/min and 10.0 K/min under the same holding time of 60 min. It could be observed that the cooling rate has very trivial effect on P 2O 5 distribution ratio, as it is firstly reduced and then getting increased slightly for both samples. Meanwhile, the phosphorus distribution ratio of Sample B is always higher than that of Sample O regardless of cooling rate, which is consistent with the results in Fig Morphology and Size of the Solid Solution In order to make best use of the solid solution as an alternative source for phosphate ore, it is important not only to increase phosphorus content in the solid solution, but also to understand the characteristics of the solid solution such as the morphology and size. Therefore, the morphology and size of the solid solution under the different conditions was investigated. (1) Effects of Na 2O on Morphology of the Solid Solution Figure 8 shows the SEM images for the reference Sample O along with other three Sample A, B and C. The morphology of the solid solution is appearing three kinds of structures: dendrite, strip and nummular for each sample. Compared all fours samples, it was found that addition of Na 2O would not change the morphology of the solid solution. (2) Effects of B 2O 3 on Morphology of the Solid Solution The SEM images of the reference Sample O and other three B 2O 3 containing Sample D, E and F are shown in Fig. 9. It is clear shown that the morphology of the solid solution in Samples D, E and F is different from that of Sample O, where the irregular polygonal structure was formed in solid solution, which indicated that the morphology of the solid solution was changed with the addition of B 2O 3. (3) Effects of Cooling Rate on Size of the Solid Solution The effect of cooling rate on the crystal size of the solid Fig. 6. The distribution ratio of P 2O 5 versus mass fraction of B 2O 3. Fig. 7. Effects of cooling rate on the phosphorus distribution ratio ISIJ

5 Fig. 8. SEM images of samples with different Na 2O content. Fig. 9. SEM images of samples with different B 2O 3 content. solution was investigated by comparing their SEM images with a low magnification of 100 as shown in Fig. 10. It is found that the size of the solid solution at the cooling rate of 2.5 K/min is larger than that of 10 K/min. This may because the molten slag viscosity becomes larger with the addition of cooling rate, 19,20) which makes it more difficult for the transportation of molten species. Thus it requires a larger driving force to initiate the crystallization, and results in a larger undercooling. As the size of the crystal is mainly determined by the growth rate of the crystals, and the growth rate in a larger cooling system is limited due to the increased difficulty of the transportation of molten species, the crystal size tends to become smaller with the increase of cooling rate. Similar trends are observed for Sample B at the different cooling rate. Furthermore, comparing the results of Sample O and Sample B at the same cooling rate, it is found that 2014 ISIJ 770

6 Fig. 10. SEM images of Sample O and Sample B at the different cooling rate. Fig. 11. SEM image of Sample A with 1% Na 2O. the crystal size of the solid solution in Sample O is smaller than that in Sample B, it indicates that adding Na 2O oxides is favorable for the crystallization of the solid solution, which is again in good agreement with the results in Fig. 5, the possible reason would be analyzed in Section 4.1. Fig. 12. Results of line analysis from M to N by EDS Parcel Phenomenon There is a parcel phenomenon observed in many cases. Take Sample A with 1% Na 2O for an example, it can be observed from Fig. 11 that several gray phases are embedded in the black solid phase, and its color is almost as same as the liquid phase. In order to determine the composition of the gray phase, EDS line analysis was made from point (M) to point (N). The obtained results are shown in Fig. 12. According to the line scanning results, the average content of elements in gray phase is almost equal to that in the liquid phase. Therefore, the gray phase was confirmed as the liquid phase. A possible explanation was found in another Sample B with 3% Na 2O, as shown in Fig. 13. The formation process could be considered as follows: (1) several adjacent Fig. 13. SEM image of Sample B with 3% Na 2O. solid phases gradually precipitate and grow up (Fig.13(A)); (2) the adjacent solid phases continue to grow and merge with each other (Fig.13(B)); (3) when the equilibrium ISIJ

7 between the liquid and the solid phase is obtained, there will be no more phase transformation and the retaining liquid in the solid during the crystals merge would be reserved in the solid phase during quenching (Fig.13(C)). In order to elaborate this phenomenon, further studies are needed in the future. 4. Discussion In the present work, the influence of Na 2O and B 2O 3 on both the distribution ratio of P 2O 5 and the corresponding solid solution morphology for CaO SiO 2 Fe 2O 3 P 2O 5 system was investigated. The results show that Na 2O and B 2O 3 play different roles for above system. In order to further investigate above oxides introduced variations, Sample O, Sample B with 3% Na 2O and Sample E with 3% B 2O 3 were chosen as representative examples for XRD testing, respectively. The XRD results are shown in Fig. 14 through Fig. 16, the intensity of detected peaks for each sample is not very strong due to the presence of glass phase obtained through quenching of liquid slag. According to the XRD analysis, it is found in Fig.14 that the solid solution in the reference slag sample O is (2CaO SiO 2 3CaO P 2O 5), which has been confirmed by previous paper, 14) Figure 15 shows that the major phase of solid solution in Sample B is the (2CaO SiO 2 Na 2O.2CaO P 2O 5), and (2CaO SiO 2 Ca 9.93(P 5.84B 0.16O 24) (B 0.67O 1.79)) was formed in Sample E, B 2O 3-containg slag, as shown in Fig. 16. It is necessary to understand the formation mechanism of Fig. 14. X-ray diffraction of Sample O without oxide additive. (2CaO SiO 2 3CaO P 2O 5) solid solution before the discussion of phosphorus partition. According to the ionic solution model, Silicon and Phosphorus are existed in the form of [SiO 4] 4 tetrahedral unit and [PO 4] 3 tetrahedral unit in the molten slag, respectively. As the silicon ionic (Si 4+ ) radius is nm that is close to that of phosphorus ionic (P 5+ ) radius which is nm, the [SiO 4] 4 tetrahedral unit and [PO 4] 3 tetrahedral unit could be easily replaced with each other in a crystal lattice. Therefore, the molten slag with complex silicate-phosphate structure tends to form solid solution (2CaO SiO 2 3CaO P 2O 5) during cooling. 21) 4.1. Effect of Na 2O on Both Phosphorus Partition and Morphology of the Solid Solution When adding Na 2O into CaO SiO 2 Fe 2O 3 P 2O 5 (12%) slag system, Sodium and Calcium will present in the form of ions in the molten slag as described above. As pointed by r-r Hume-Rothery rule, 22) 1 2 when < 15% (where r 1 and r 2 r1 stand for the radius of two different ions in the solute, respectively), both ions in the molten system could be replaced with each other and form the solid solution. Therefore, Calcium ion (Ca 2+ ) and Sodium ion (Na + ) could be replaced with each other in the molten multi phase slag system, as the ionic radius of Na + (0.095 nm) is very close to that of Ca 2+ (0.106 nm), such that the value of r-r = = 10.38%, is less r than 15%. For the molten slag, each ion has its electrostatic field, and the cation prefers to be together with the anion, which has the large electrostatic field. 23) However, for the Na 2O-containing slag, there are sufficient calcium ions (Ca 2+ ) in the molten slag as the composition of slag is located in the C 2S saturation zone. Therefore, the Sodium ion (Na + ) would be mainly together with [PO 4], 3 because the electrostatic field of [PO 4] 3 is larger than that of [SiO 4] 4. 23) Thus, Sodium ions (Na + ) would mainly replace the Calcium ions (Ca 2+ ) that bonded with [PO 4] 3 during the cooling and a new solid solution of (2CaO SiO 2 Na 2O 2CaO P 2O 5) was obtained. There would be more (2CaO SiO 2 Na 2O 2CaO P 2O 5) solid solution precipitated with the increase of Na 2O content. Besides, a better kinetic condition would be obtained, as Na 2O could lower the viscosity of slag that is favorable for the nucleation and growth of the crystals. Combing above two issues, the phos- Fig. 15. X-ray diffraction of Sample B with 3% Na 2O. Fig. 16. X-ray diffraction of Sample E with 3% B 2O ISIJ 772

8 phorus partition between the solid solution and liquid phase would increase with the increase of Na 2O content. Comparing (2CaO SiO 2 3CaO P 2O 5) and (2CaO SiO 2 Na 2O 2CaO P 2O 5) solid solution s structure, the change is that part of the Calcium ions (Ca 2+ ) in the lattice was replaced by the Sodium ions (Na + ) that is with almost the same radius, and this replacement does not affect the main structure of the (2CaO SiO 2 3CaO P 2O 5) solid solution. Thereby, the new solid solution of (2CaO SiO 2 Na 2O 2CaO P 2O 5) remains the similar structure as that of (2CaO SiO 2 3CaO P 2O 5) Effect of B 2O 3 on Both Phosphorus Partition and Morphology of the Solid Solution Comparing the XRD results of Figs.14 and 16, the solid solution in multiphase slag was changed to (2CaO SiO 2 Ca 9.93(P 5.84B 0.16O 24) (B 0.67O 1.79)) when B 2O 3 was added. For the B 2O 3-containg slag, Boron could form two types of structure unit as [BO 3] 3 triangular and [BO 4] 4 tetrahedral, while there would be only one type of structure formed by Phosphorus and Silicon. The structure units such as [BO 4] 4, [PO 4] 3 and [SiO 4] 4 are 3D, except for [BO 3] 3 which is a planar shape but it could link with other tetrahedral units. 24,25) [BO 4] 4 and [PO 4] 3 tetrahedral units prefer to be together in the molten slag to form complex B O P bond based network during the cooling process, 24) and a minor portion of [BO 3] 3 triangular units may connect to [BO 4] 4 or [PO 4] 3 tetrahedral units; thus, it would result in the formation of (2CaO SiO 2 Ca 9.93 (P 5.84B 0.16O 24) (B 0.67O 1.79)) solid solution during molten slag solidification. Therefore, the addition of B 2O 3 is favorable for improving the glass-forming ability and weakening the crystallization ability. 26) So the crystallization of the solid solution is restrained with the increase of B 2O 3, and the phosphorus partition between the solid solution and liquid phase would decrease with the increase of B 2O 3 content. Based on the above analysis, it was indicated that the crystal structure of the (2CaO SiO 2 Ca 9.93(P 5.84B 0.16O 24)(B 0.67O 1.79)) solid solution is different from that of the (2CaO SiO 2 3CaO P 2O 5). Therefore, the crystal structure of the new solid solution was eventually changed to irregular polygonal structure with the addition of B 2O Conclusions In order to increase the efficiency of dephosphorization and to use the slag as an alternative source of phosphate ores, the influences of Na 2O and B 2O 3 on the distribution ratio of P 2O 5 between the solid solution and the liquid phase and the morphology of the solid solution for the CaO SiO 2 Fe 2O 3 P 2O 5 system, were investigated by using the meltquenching technique. The conclusions obtained in the present study are summarized as follows: (1) The distribution ratio of P 2O 5 in the solid solution was enhanced with the increase of Na 2O content, as the new (2CaO SiO 2 Na 2O 2CaO P 2O 5) solid solution was formed due to the replacement of Na + with Ca 2+. Besides, the addition of Na 2O would tend to provide a better kinetic condition for the crystal nucleation and growth, which in turn to improve the distribution ratio. However, the morphology of the solid solution does not change significantly, as the lattice replacement of Sodium ions by Calcium ions in the phosphate tetrahedral structure is very small. (2) The addition of B 2O 3 study showed that the distribution ratio of P 2O 5 and the P 2O 5 concentration in the solid solution would reduce with the increase of B 2O 3 content, as the new formed complex (2CaO SiO 2 Ca 9.93(P 5.84B 0.16O 24) (B 0.67O 1.79)) solid solution is hard to precipitate from the liquid slag due to the complexity of the new formed network structure. The morphology of the new-formed solid solution also changed to irregular polygon with change of complex networks. (3) The parcel phenomenon was observed during the formation of solid solution, which may be due to the growth of the crystals and liquid-solid equilibrium between the solid solution and liquid slag. 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