Transformation of Inclusions in Solid GCr15 Bearing Steels During Heat Treatment

Laboratory heating experiments with a varied holding time of GCr15 bearing steels at 1498 K were performed to study the transformation of inclusions in solid GCr15 bearing steels during high temperature diffusion processes. Heating experiments at 1573 and 1648 K were also carried out to study the effect of these heating temperatures. Experimental results showed that inclusions transformed from Al2O3-CaO-(MgO) to Al2O3-CaS-(MgO-CaO) when the heat treatment was in the range of 1498 to 1648 K due to reactions between Al and S in the steel matrix and CaO in the inclusions. This is in good agreement with thermodynamic calculations. Moreover, the size of the inclusions hardly changed after heat treatment. The transformation rate of the inclusions depended strongly on both the heating temperature and the size of the inclusions. Kinetic analyses on the transformation of inclusions during heat treatment were performed based on a simplified analytical model. The mass transfer coefficients of CaO and CaS in inclusions were calculated, which ranged from 0.73 × 10−10 to 4.48 × 10−10 m/s.


Introduction
Bearing steels are widely applied in machinery manufacturing, railway transportation, automobile manufacturing, national defense industries, and other fields [1]. It is well known that the mass fraction, composition, size, morphology, and distribution of non-metallic inclusions have an important influence on the rolling contact fatigue (RCF) life of bearing steels [2][3][4][5]. The demand for the RCF life of bearing steels becomes more stringent, which requires a better control of steel cleanliness. The RCF life of Al-killed bearing steels can be increased 30 times by decreasing the oxygen content from 30 ppm to 5 ppm [6]. Besides, the RCF life is also closely related to the composition of inclusions in bearing steels. It is found that titanium nitride inclusions are more harmful than oxide inclusions with the same size. The detrimental effect of a 7 µm TiN inclusion is equivalent to that of a 25 µm oxide inclusion. However, large globular calcium aluminate inclusions are the most detrimental to bearing steels [2].
Extensive research on the formation, removal, and modification of non-metallic inclusions in the molten steel at steelmaking temperatures have been conducted, such as the calcium treatment [7][8][9][10][11][12][13][14][15], the rare earth element treatment [16,17], slag refining [18][19][20][21][22], and reoxidation [23,24]. However, less attention has been paid to the variation of oxide inclusions in the solid steel during heat treatment. It was reported that MnO-SiO 2 oxide inclusions in Type 304 stainless steels may transform to MnO-Cr 2 O 3 during heat treatment in the range of 1273 to 1623 K [25][26][27] The contents of Mn, Si, and Cr in the steel had a strong influence on the transformation [28,29]. It was found that Al-Ca-O-S complex inclusions were the predominating particles in EH36 shipbuilding steels, but the TiN ones were profusely populated after being heated at 1473 K [30,31]. CaO-Al 2 O 3 type of inclusions in pipeline The cylindrical steel sample with a 34 mm diameter was taken from the tundish and quenched in water. The quenched steel sample was machined into sectors (each measuring approximately 17 mm × 10 mm with an angle of 60 • ), as shown in Figure 1. Next, the machined samples were heated under an argon atmosphere at 1498, 1573, or 1648 K for various periods from 5 min to 12 h, respectively, in a Si-Mo resistance furnace (SHIMADEN FP21, Tokyo, Japan), followed by water quenching. The schematic of experimental procedures is shown in Figure 2.
Metals 2019, 9, x FOR PEER REVIEW 2 of 21 steels transformed into Al2O3-CaS type after heat treatment in the range of 1273 to 1673 K [27,32]. Yang et al also investigated the transformation of inclusions in pipeline steels during solidification and cooling [33]. For a high temperature diffusion process of high carbon chromium bearing steels to reduce the segregation of bloom and precipitation of ribbon type carbide as well as aliquation carbide, 1473 K is usually utilized [34]. However, the transformation of inclusions in Al-killed GCr15 bearing steels during high temperature diffusion processes has rarely been studied. Thermodynamic models [35,36] and kinetic models [37][38][39][40] have been established to predict the compositions of steel, slag, and inclusions in the molten steel, as well as the precipitation of sulfides [41,42] and nitrides [43,44] during cooling and solidification. In the current work, the transformation of inclusions in a GCr15 bearing steel during high temperature diffusion processes was investigated through laboratory experiments, thermodynamic analyses and kinetic analyses. Isothermal heat treatment of the Al-killed GCr15 bearing steel was performed at varied temperatures that ranged from 1498 to 1648 K under an argon atmosphere.

Experimental Procedure
Industrial trails for the production of a GCr15 bearing steel by a route of "EAF-LF refiningvacuum degassing-bloom CC-hot rolling" were performed. The molten steel was casted into 320 mm × 480 mm blooms. Before hot rolling, the bloom was heated for 3 h, 7 h, or 12 h in the range of 1493 to 1503 K as a high temperature diffusion process. The chemical composition of the bearing steel used in this study is listed in Table 1. The cylindrical steel sample with a 34 mm diameter was taken from the tundish and quenched in water. The quenched steel sample was machined into sectors (each measuring approximately 17 mm × 10 mm with an angle of 60°), as shown in Figure 1. Next, the machined samples were heated under an argon atmosphere at 1498, 1573, or 1648 K for various periods from 5 min to 12 h, respectively, in a Si-Mo resistance furnace (SHIMADEN FP21, Tokyo, Japan), followed by water quenching. The schematic of experimental procedures is shown in Figure 2.   Quenched steel samples were mounted in epoxy resin, ground and polished. The size, composition, morphology and amount of inclusions in steel samples were analyzed by an automatic SEM-EDS equipment (FEI, Pittsburgh, PA, USA). The scanned area was over 60 mm 2 for each sample.

Effect of Holding Time on the Transformation of Inclusions in the Steel During Heating
The composition and size of the inclusions in the steel before and after the heat treatment at 1498 K with varied holding time are shown in Figure 3. As shown in Figure 3a, the inclusions were mainly MgO·Al2O3 and Al2O3-CaO-(MgO) which represents Al2O3-CaO with limited MgO content, before heat treatment. A typical inclusion with mapping and line scanning is shown in Figure 4. The average composition of inclusions was 15.42 wt % MgO-63.76 wt % Al2O3-17.18 wt % CaO-3.64 wt % CaS. According to Figure 3b, after being annealed at 1498 K for 5 min, the average content of CaO in the inclusions decreased while the CaS content slightly increased. After 20 min, the average content of CaO in the inclusions decreased from 17.18 to 9.36 wt % while the CaS content increased from 3.64 to 13.06 wt %, as shown in Figure 3c. Figure 5 shows a typical inclusion in the steel after being held at 1498 K for 1 hour, which was the MgO-Al2O3 core enveloped by CaS. The average composition of inclusions in the steel had no obvious change when the holding time was longer than 3 h, according to Figure 3e-g. The average composition of the inclusions in the steel after being held at 1498 K for 12 h was 13.20 wt % MgO-69.94 wt % Al2O3-4.12 wt % CaO-12.73 wt % CaS. An irregular shaped MgO-Al2O3-CaS inclusion was observed after the steel was held at 1498 K for 12 h, as shown Figure  6, which was MgO-Al2O3 in the core and CaS in the periphery. Wang et al. investigated the effect of CaS on the evolution of CaO-Al2O3 inclusions in low-carbon Al-killed and Ca-treated steels during the solidification process [7]. It was found that the inclusions were originally liquid and spherical, and transformed to an irregular state during solidification due to the reaction between CaO-Al2O3 inclusions with the dissolved sulfur and aluminum in the steel, resulting in the formation of dense CaS shells around the inclusions. The effect of the holding time on the average composition of inclusions at 1498 K is shown in Figure 7. It indicates that inclusions are transformed from Al2O3-CaO-(MgO) to Al2O3-CaS-(MgO-CaO). Error bars represented the 95% confidence interval of the mean composition. Quenched steel samples were mounted in epoxy resin, ground and polished. The size, composition, morphology and amount of inclusions in steel samples were analyzed by an automatic SEM-EDS equipment (FEI, Pittsburgh, PA, USA). The scanned area was over 60 mm 2 for each sample.

Effect of Holding Time on the Transformation of Inclusions in the Steel During Heating
The composition and size of the inclusions in the steel before and after the heat treatment at 1498 K with varied holding time are shown in Figure 3. As shown in Figure 3a, the inclusions were mainly MgO·Al 2 O 3 and Al 2 O 3 -CaO-(MgO) which represents Al 2 O 3 -CaO with limited MgO content, before heat treatment. A typical inclusion with mapping and line scanning is shown in Figure 4. The average composition of inclusions was 15.42 wt % MgO-63.76 wt % Al 2 O 3 -17.18 wt % CaO-3.64 wt % CaS. According to Figure 3b, after being annealed at 1498 K for 5 min, the average content of CaO in the inclusions decreased while the CaS content slightly increased. After 20 min, the average content of CaO in the inclusions decreased from 17.18 to 9.36 wt % while the CaS content increased from 3.64 to 13.06 wt %, as shown in Figure 3c. Figure 5 shows a typical inclusion in the steel after being held at 1498 K for 1 h, which was the MgO-Al 2 O 3 core enveloped by CaS. The average composition of inclusions in the steel had no obvious change when the holding time was longer than 3 h, according to Figure 3e-g. The average composition of the inclusions in the steel after being held at 1498 K for 12 h was 13.20 wt % MgO-69.94 wt % Al 2 O 3 -4.12 wt % CaO-12.73 wt % CaS. An irregular shaped MgO-Al 2 O 3 -CaS inclusion was observed after the steel was held at 1498 K for 12 h, as shown Figure 6, which was MgO-Al 2 O 3 in the core and CaS in the periphery. Wang et al. investigated the effect of CaS on the evolution of CaO-Al 2 O 3 inclusions in low-carbon Al-killed and Ca-treated steels during the solidification process [7]. It was found that the inclusions were originally liquid and spherical, and transformed to an irregular state during solidification due to the reaction between CaO-Al 2 O 3 inclusions with the dissolved sulfur and aluminum in the steel, resulting in the formation of dense CaS shells around the inclusions. The effect of the holding time on the average composition of inclusions at 1498 K is shown in Figure 7. It indicates that inclusions are transformed from Al 2 O 3 -CaO-(MgO) to Al 2 O 3 -CaS-(MgO-CaO). Error bars represented the 95% confidence interval of the mean composition.       The evolution of the average content of each component in the inclusions as a function of the inclusion diameter is shown in Figure 8. According to Figure 8a, before heat treatment, inclusions larger than 3 μm were mainly calcium aluminate. As for inclusions smaller than 3 μm, the CaO content increased with the increase of the diameter, and the contents of Al2O3 and MgO decreased.    The evolution of the average content of each component in the inclusions as a function of the inclusion diameter is shown in Figure 8. According to Figure 8a, before heat treatment, inclusions larger than 3 µm were mainly calcium aluminate. As for inclusions smaller than 3 µm, the CaO content increased with the increase of the diameter, and the contents of Al 2 O 3 and MgO decreased.
As shown in Figure 8b, after being held at 1498 K for 1 h, the CaO content in inclusions with the size of 1-4 µm decreased while the CaS content increased. As for the inclusions less than 3 µm, the average composition of the inclusions is almost the same regardless of the size. The CaO content of the inclusions decreased to 6 wt % after the steel was heated for 3 h, as shown in Figure 8c. The composition of inclusions with different sizes varied slightly with a longer holding time, as shown in Figure 8d. All the above experimental results indicated that the composition of small inclusions varied more severely than large ones. According to Figure 8, the CaS and Al 2 O 3 content of the inclusions increased, and the CaO content decreased with the increase of the holding time, especially for large inclusions. The main reaction that occurred in the steel during heat treatment at 1498 K is as follows: The variation in the number density and average diameter of the inclusions in the steel are shown in Figures 9 and 10, respectively. The number density of inclusions varied slightly with time when the steel was held at 1498 K. The average diameter of the inclusions in the steel decreased after heating, especially after 12 h heat treatment at 1498 K. As shown in Figure 8b, after being held at 1498 K for 1 h, the CaO content in inclusions with the size of 1-4 μm decreased while the CaS content increased. As for the inclusions less than 3 μm, the average composition of the inclusions is almost the same regardless of the size. The CaO content of the inclusions decreased to 6 wt % after the steel was heated for 3 h, as shown in Figure 8c. The composition of inclusions with different sizes varied slightly with a longer holding time, as shown in Figure 8d. All the above experimental results indicated that the composition of small inclusions varied more severely than large ones. According to Figure 8, the CaS and Al2O3 content of the inclusions increased, and the CaO content decreased with the increase of the holding time, especially for large inclusions. The main reaction that occurred in the steel during heat treatment at 1498 K is as follows: The variation in the number density and average diameter of the inclusions in the steel are shown in Figures 9 and 10, respectively. The number density of inclusions varied slightly with time when the steel was held at 1498 K. The average diameter of the inclusions in the steel decreased after heating, especially after 12 h heat treatment at 1498 K.   The variation in the size distribution of the inclusions in the steel against the holding time is shown in Figure 11. The number of small inclusions increased after heat treatment at 1498 K, especially The variation in the size distribution of the inclusions in the steel against the holding time is shown in Figure 11. The number of small inclusions increased after heat treatment at 1498 K, especially heat treatment of 7 and 12 h. Less variation in the area fraction of the inclusions in the steel with holding time was observed. The results are shown in Figure 12.    The variation in the size distribution of the inclusions in the steel against the holding time is shown in Figure 11. The number of small inclusions increased after heat treatment at 1498 K, especially heat treatment of 7 and 12 h. Less variation in the area fraction of the inclusions in the steel with holding time was observed. The results are shown in Figure 12.

Effect of Temperature on the Transformation of Inclusions in the Steel During Heating
The effect of holding time on the average composition of the inclusions in the steel during heat treatment at 1573 K is shown in Figure 13. After heating for 5 min at 1573 K, the average content of CaS in inclusions increased while the CaO content slightly decreased. The average content of CaO in

Effect of Temperature on the Transformation of Inclusions in the Steel During Heating
The effect of holding time on the average composition of the inclusions in the steel during heat treatment at 1573 K is shown in Figure 13. After heating for 5 min at 1573 K, the average content of CaS in inclusions increased while the CaO content slightly decreased. The average content of CaO in inclusions decreased from 17.18 wt % to 5.33 wt % and the CaS content increased from 3.64 wt % to 19.41 wt % after heating for 20 min at 1573 K. The average composition of inclusions slightly varied after heating for 1 h or longer at 1573 K. The average composition of inclusions in the steel after heating for 7 h at 1573 K was 17.10 wt % MgO-58.77 wt % Al 2 O 3 -2.60 wt % CaO-21.53 wt % CaS. The CaS content of inclusions increased and the CaO content decreased with time at 1573 K.

Effect of Temperature on the Transformation of Inclusions in the Steel During Heating
The effect of holding time on the average composition of the inclusions in the steel during heat treatment at 1573 K is shown in Figure 13  The effect of holding time on the average composition of inclusions in the steel heated at 1648 K is shown in Figure 14. After heating for 20 min at 1648 K, the content of CaS in inclusions increased from 3.64 wt % to 7.59 wt %, while the CaO content decreased from 17.18 wt % to 11.95 wt %. The content of CaO in inclusions slightly decreased and that of CaS further increased after the steel was heated for 1 hour at 1648 K. After heating for 3 h at 1648 K, the CaO content decreased to 7.28 wt % and the CaS content increased to 18.13 wt %. The average composition of inclusions varied little after The effect of holding time on the average composition of inclusions in the steel heated at 1648 K is shown in Figure 14. After heating for 20 min at 1648 K, the content of CaS in inclusions increased from 3.64 wt % to 7.59 wt %, while the CaO content decreased from 17.18 wt % to 11.95 wt %. The content of CaO in inclusions slightly decreased and that of CaS further increased after the steel was heated for 1 h at 1648 K. After heating for 3 h at 1648 K, the CaO content decreased to 7.28 wt % and the CaS content increased to 18.13 wt %. The average composition of inclusions varied little after the steel was heated for 7 h. The CaS content in the inclusions increased and the CaO content decreased with time at 1648 K. Figure 15 shows the composition of inclusions against the annealing temperature at a fixed holding time of 7 h. With the increase of temperature, the CaS content in inclusions increased, while the CaO content decreased significantly, and the Al 2 O 3 and MgO contents slightly varied. According to the above results, the main reaction that occurred in the steel during heat treatment at 1573 and 1648 K is:

(CaO)+[S] =(CaS)+[O]
(2) Figure 15 shows the composition of inclusions against the annealing temperature at a fixed holding time of 7 h. With the increase of temperature, the CaS content in inclusions increased, while the CaO content decreased significantly, and the Al2O3 and MgO contents slightly varied. According to the above results, the main reaction that occurred in the steel during heat treatment at 1573 and 1648 K is:  the CaO content decreased significantly, and the Al2O3 and MgO contents slightly varied. According to the above results, the main reaction that occurred in the steel during heat treatment at 1573 and 1648 K is:    Figure 16 shows the effect of the holding time and temperature on the size of the inclusions in the steel. The average size of inclusions decreased slightly with the increase of temperature and holding time, implying that small inclusions were generated, especially CaS inclusions. Heating time (h)

Thermodynamic Considerations on the Transformation of Inclusions in the Steel During Heat Treatment
The comprehensive transformation of inclusions in the bearing steel consisted of three parts: Self-precipitation of inclusions, new precipitates from the steel, and the transformation of the inclusions from the reaction between steel and inclusions. In order to understand the transformation of the inclusions in the steel during heat treatment, thermodynamic calculations were carried out.

Self-Precipitation of Inclusions with the Decrease of Temperature only Considering Inclusions
The precipitation of phases in the inclusions with the decrease of temperature only considering the inclusions was calculated using FactSage 7.0 (Thermfact/CRCT & GTT-Technologies, Aachen, Germany) with the Ftoxid database. The result is shown in Figure 17. The initial composition of inclusions was 15.42 wt % MgO-63.76 wt % Al2O3-17.18 wt % CaO-3.64 wt % CaS, which was the average composition of inclusions in the bearing steel before heat treatment. According to Figure 17, at 1773 K, the liquid and solid phase of MgO·Al2O3 exist in inclusions. As the temperature decreases, solid CaO·Al2O3 precipitates accompanied with the decrease in the liquid phase, and the MgO·Al2O3 ones slightly increase. The liquid phase disappears at 1611 K, along with the precipitation of solid 3CaO·2MgO·Al2O3. The calculated results show that the precipitation of phases in the inclusions at heat treatment temperature did not result from the formation of CaS inclusions, and the CaO content in the inclusions hardly decreased, indicating that the variation of the composition of inclusions in the bearing steel was not caused by self-precipitation of inclusions during heat treatment.

Thermodynamic Considerations on the Transformation of Inclusions in the Steel During Heat Treatment
The comprehensive transformation of inclusions in the bearing steel consisted of three parts: Self-precipitation of inclusions, new precipitates from the steel, and the transformation of the inclusions from the reaction between steel and inclusions. In order to understand the transformation of the inclusions in the steel during heat treatment, thermodynamic calculations were carried out.

Self-Precipitation of Inclusions with the Decrease of Temperature only Considering Inclusions
The precipitation of phases in the inclusions with the decrease of temperature only considering the inclusions was calculated using

Precipitation of Inclusions in the Bearing Steel at Various Temperature with Consideration of Steel
For the steel with 7 ppm T.O. (total oxygen content), which was used in the current experiments, the precipitation of inclusions in the bearing steel at various temperatures with consideration of steel was calculated using FactSage 7.0 with the FactPS, FToxid, and Fsstel databases. The result is shown in Figure 18. Due to the similarity of transformations under different conditions according to Figure  18, only the comprehensive transformation of inclusions in the bearing steel is illustrated in detail. As shown in Figure 18a, at the initial decreasing of temperature, inclusions in the bearing steel are liquid oxides and solid MgO·Al2O3 ones. As the temperature decreases, the amount of spinel phase increases and a small amount of solid MgO inclusions forms. At 1737 K, solid CaS inclusions start to precipitate and the content of solid MgO·Al2O3 further increases with the decrease of liquid phase. The liquid phase disappears at 1719 K. Trace amount of solid MgO and MgS precipitates in bearing steels with the further decrease of temperature.
During heat treatment in the temperature range of 1498 to 1648 K, the main equilibrium inclusion phases are solid CaS and MgO·Al2O3. By combining the content of each phase, the average composition of inclusions during heat treatment can be obtained, as shown in Figure 19. The CaO content of inclusions decreases to zero when the temperature decreases to 1719 K, and the CaS content in inclusions increases significantly. Moreover, the Al2O3 and MgO contents of inclusions vary slightly, indicating that the transformation from CaO into CaS in inclusions mainly occurs during heat treatment.  During heat treatment in the temperature range of 1498 to 1648 K, the main equilibrium inclusion phases are solid CaS and MgO·Al 2 O 3 . By combining the content of each phase, the average composition of inclusions during heat treatment can be obtained, as shown in Figure 19. The CaO content of inclusions decreases to zero when the temperature decreases to 1719 K, and the CaS content in inclusions increases significantly. Moreover, the Al 2 O 3 and MgO contents of inclusions vary slightly, indicating that the transformation from CaO into CaS in inclusions mainly occurs during heat treatment.      Figure 20. The amount of MgO·Al2O3 inclusions increases with the decrease of temperature. Solid CaS inclusions begin to precipitate at 1717 K. Solid CaO·Al2O3 inclusions generate and liquid phase disappears at 1698 K. As the temperature further decreases, solid CaO·2Al2O3 and CaO·2MgO·8Al2O3 precipitate successively, along with the disappearance of CaO·Al2O3 and CaO·2Al2O3 in succession. According to the calculated results above, it can be concluded that the steel composition has an important influence on the transformation of inclusions in the bearing steel during heat treatment.    The calculated and measured results are in good agreement with each other and show that the transformation of inclusions happens during heat treatment. However, minor deviations between experimental and calculated values were also observed. Thermodynamic calculation results were obtained based on the equilibrium state, which means a sufficient reaction time and no consideration of the reaction between the slag and refractories. However, in heat treatment experiments in the current work, the reaction time is insufficient to reach the equilibrium, resulting in the minor deviation between the experimental and calculated results.

Kinetic Analyses on the Transformation of Inclusions During Heat Treatment
As shown in Figure 22

Kinetic Analyses on the Transformation of Inclusions During Heat Treatment
As shown in Figure 22, the initial inclusions in the steel were mainly Al 2 O 3 -CaO-(MgO) and [Ca]+[S] =(CaS) x(MgO) + y(Al 2 O 3 ) = xMgO · yAl 2 O 3 (6) According to thermodynamic calculation results in Section 4, the CaS content, as well as the CaO content, in the inclusions at the equilibrium state was a little distinct at different temperatures. Thus, the transformation of the inclusions could be treated as the same from the thermodynamic perspective. However, under the experimental conditions, according to Figures 7, 13, and 14, the evolution of the composition of inclusions as a function of holding time at different temperatures was diverse, which should be attributed to the transformation kinetics. It is assumed that the transfer process during heat treatment is a stationary diffusion process and the equilibrium state of reactions are held at a steel-inclusions interface, indicating that the concentration of substances at the interface are the equilibrium concentration. Therefore, Fick's first law, as shown Equation (7), can be used to describe the transformation.
where X ,r J is the mass flux of X , kg/s; X D is the diffusion coefficient of X , m 2 /s; A is the area of the interface, m 2 ; X c is the concentration of X , kg/m 3 ; r is the distance from the interface, m; δ is the boundary layer thickness, m. In this paper, superscript * represents the interface between the steel matrix and the inclusions, and b denotes the steel matrix.
where X m is the mass of X , kg; V is the volume of the inclusion or steel matrix, m 3 ; t is the diffusion time, s. Equation (7) is equal to Equations (8) and (9) can be obtained.
Equation (10) derived from Equation (9) is usually utilized to describe the kinetic reaction of the steel matrix. According to thermodynamic calculation results in Section 4, the CaS content, as well as the CaO content, in the inclusions at the equilibrium state was a little distinct at different temperatures. Thus, the transformation of the inclusions could be treated as the same from the thermodynamic perspective. However, under the experimental conditions, according to Figure 7, Figure 13, and Figure 14, the evolution of the composition of inclusions as a function of holding time at different temperatures was diverse, which should be attributed to the transformation kinetics. It is assumed that the transfer process during heat treatment is a stationary diffusion process and the equilibrium state of reactions are held at a steel-inclusions interface, indicating that the concentration of substances at the interface are the equilibrium concentration. Therefore, Fick's first law, as shown Equation (7), can be used to describe the transformation.
where J X,r is the mass flux of X, kg/s; D X is the diffusion coefficient of X, m 2 /s; A is the area of the interface, m 2 ; c X is the concentration of X, kg/m 3 ; r is the distance from the interface, m; δ is the boundary layer thickness, m. In this paper, superscript * represents the interface between the steel matrix and the inclusions, and b denotes the steel matrix.
where m X is the mass of X, kg; V is the volume of the inclusion or steel matrix, m 3 ; t is the diffusion time, s. Equation (7) is equal to Equations (8) and (9) can be obtained.
Equation (10) derived from Equation (9) is usually utilized to describe the kinetic reaction of the steel matrix.
where [%M] is the mass percentage of M element; n is the number of inclusions; A steel−inc is the contact area between the steel matrix and inclusions, m 2 ; V steel is the volume of the steel matrix, m 3 ; k m_inc , defined as D M δ , is the mass transfer coefficient of M element in the boundary layer, m/s. However, the variation of the element content in the steel matrix, such as the dissolved oxygen, sulfur, and calcium cannot be measured during heat treatment due to the limitation of current detection methods. Besides, it is difficult to determine the contact area between the steel matrix and inclusions. The situation could be simplified when kinetic calculations were performed assuming that mass transfer takes place as the molecules of inclusion components between the steel matrix and inclusions, because the composition of inclusions can be obtained after heat treatment. Moreover, the inclusion diffusion was assumed as a rate-limiting step of reactions in current work. Consequently, Equation (11) can be applied to describe the kinetic reaction of inclusions. Equations (12) and (13) are derived from Equation (11).
where (%MO n ) inc is the mass percentage of MO n in inclusions; k inc , defined as D MOn δ , is the mass transfer coefficient of MO n in the boundary layer, m/s; d p is the diameter of the inclusions; C is the constant.
The mass percentage of MO n in inclusions after a certain time of heat treatment and that at equilibrium state (%MO n ) * inc can be acquired from Figure 7, Figure 13, and Figure 14. As stated previously, the diameter of the inclusions d p can be regarded as the constant value, according to Figure 16. Thus, k CaS and k CaO at different temperatures are obtained as empirical parameters shown in Figure 23. At 1498 K and 1573 K, the mass transfer coefficient of CaS in inclusions is larger than that of CaO, and they are almost the same at 1648 K. In addition, the mass transfer coefficient of CaS (or CaO) in inclusions at 1573 K is larger than that of CaS (or CaO) at 1498 and 1648 K.