A Kinetic Model for the Modification of Al2O3 Inclusions during Calcium Treatment in High-Carbon Hard Wire Steel

Laboratory-scale experiments for the modification of Al2O3 inclusions by calcium treatment in high-carbon hard wire steel were performed and the compositions and morphological evolution of inclusions were studied. The kinetics of the modification of Al2O3 inclusions by calcium treatment were studied in high-carbon hard wire steel based on the unreacted shrinking core model, considering the transfer of Ca and Al through the boundary layer and within the product layer, coupled with thermodynamic equilibrium at the interfaces. The diffusion of Al in the inclusion layer was the limiting link in the inclusion modification process. The Ca concentration in molten steel had the greatest influence on the inclusion modification time. The modification time for inclusions tended to be longer in the transformation of higher CaO-containing calcium aluminate. The modification of Al2O3 into CA6 was fastest, while the most time was needed to modify CA into C12A7. It took about six times time longer at the later stage of inclusion modification than at the early stage. The complete modification time for inclusions increased with the square of their radii. The changes of CaO contents with melting time were estimated based on a kinetic model and was consistent with experimental results.


Introduction
High-carbon hard wire steels are mainly applied in massive engineering projects such as bridges, cables, airports, power stations, and dams [1][2][3]. High-carbon hard wire products are drawn into filaments with a diameter of about 5 mm [4,5]. The hard inclusions with large sizes affect the steel yield and performance. Al 2 O 3 inclusions are brittle with high melting temperatures, which are the major factors believed to impact the performance of high-carbon hard wire steels [6]. Therefore, Ca is usually used to modify Al 2 O 3 inclusions to improve the performance of high-carbon hard wire steels.
Calcium treatment is one of the most commonly used and effective methods for modifying non-metallic inclusions into liquid ones [7][8][9][10]. The solid alumina can be converted into calcium aluminate inclusions partially or completely during calcium treatment [11,12], reducing the blockage of the immersion nozzle during continuous casting [10,13]. Since the 1990s, many studies have been conducted to understand the modification mechanism for alumina inclusions using calcium treatment [9,[14][15][16]. Research studies on the modification kinetics for Al 2 O 3 inclusions have been conducted in order to evaluate the accurate addition amounts for calcium and to understand the modification evolution process. Lu et al. [17] first established the kinetic model for oxide and sulfide inclusions in the calcium treatment process. They assumed that the internal diffusion rate of the inclusions is extremely fast and the interface reaction is rapid, and they developed an inclusion evolution model. Higuchi et al. [18] revised the kinetic model for modification of Al 2 O 3 inclusions by using a first-order reaction equation. They studied the gasification rate of calcium from the melt and the reaction rate between the melt and inclusions. In this model, they assumed that the size and number of inclusions remained constant. Visseret al. [19] divided the ladle into two reaction zones: one is a high-calcium and low-oxygen zone, while the (1) Numerical simulation has many advantages, such as operating at low temperatures, having good reproducibility, being low cost, and detailed experimental data being available. Therefore, more metallurgical workers are using numerical simulation methods to study the behavior of inclusions in steel, and then to obtain the variation law for each parameter in the process and the quantitative relationships between each parameter [27].
On the basis of the multilayer unreacted core model for alumina inclusions, a step-bystep reaction kinetic model for the modification of Al 2 O 3 inclusions by calcium treatment in high-carbon hard wire steel was established. The effects of Al, Ca, and O contents on the modification of high-carbon hard wire steel was studied during calcium treatment. The conversion ratio, radii of inclusions, calcium oxide contents in inclusions, and modification times were predicted. This work is helpful for understanding the inclusion modification process and for improving calcium treatment technology.

Experimental Procedure
The tested steels were produced based on the chemical compositions of SWRH62A steel. Two heat experiments with different amount of deoxidants (A and B) were carried out in a tubular resistance furnace, as shown in Figure 1. The corundum crucible containing about 400 g pure iron, ferrosilicon alloy, and electrolytic manganese was placed in the furnace. Table 1 lists the compositions of raw materials used in the study. The temperature was increased to 1600 • C using electric heating. The added alloys and sampling procedures used in the experiments are shown in Figure 2. The aluminum alloy and Si-Ca alloy were added to the liquid steel for deoxidation at certain times. Calcium treatment was carried out after Al deoxidization. Four samples were sucked out using a quartz tube (φ 5 mm) and then quenched by insertion into a sodium chloride solution at different times after calcium addition (60, 180, 600, and 720 s). During the steel smelting process, the argon gas flow rate was maintained at 5 min/L. different times after calcium addition (60, 180, 600, and 720 s). During the steel smelting process, the argon gas flow rate was maintained at 5 min/L.

Composition Analysis for Steels and Inclusions
The contents of calcium and aluminum in experimental steels were assessed using inductively coupled plasma-mass spectrometry (ICP-MS ,Su Zhou, China)The contents of C, Si, Mn, and S in experimental steels were assessed using a direct-reading spectrometer (Q4-TASMAN,Brooke, Germany). The O contents in experimental steels were assessed using an inorganic oxygen-hydrogen tester.
The metallographic samples were ground using abrasive papers and then polished. The two-dimensional morphologies and compositions of inclusions in a cross section of each sample were analyzed using scanning election microscopy (ZeissΣIGMA+ X-Max20,Baden-Wurttemberg, Germany) and energy-dispersive spectrometry. About 30 inclusions were detected in each sample. Table 2 lists the measured compositions of steels. A significant difference was found in the mass fractions of calcium between the two experiments due to the different amounts of Si-Ca alloy added. This showed that steel A had a high calcium content of 0.0025 mass%, while steel B had a low calcium content of 0.0017 mass%.

Composition Analysis for Steels and Inclusions
The contents of calcium and aluminum in experimental steels were assessed using inductively coupled plasma-mass spectrometry (ICP-MS, Su Zhou, China) The contents of C, Si, Mn, and S in experimental steels were assessed using a direct-reading spectrometer (Q4-TASMAN, Brooke, Germany). The O contents in experimental steels were assessed using an inorganic oxygen-hydrogen tester.
The metallographic samples were ground using abrasive papers and then polished. The two-dimensional morphologies and compositions of inclusions in a cross section of each sample were analyzed using scanning election microscopy (ZeissΣIGMA+ X-Max20, Baden-Wurttemberg, Germany) and energy-dispersive spectrometry. About 30 inclusions were detected in each sample. Table 2 lists the measured compositions of steels. A significant difference was found in the mass fractions of calcium between the two experiments due to the different amounts of Si-Ca alloy added. This showed that steel A had a high calcium content of 0.0025 mass%, while steel B had a low calcium content of 0.0017 mass%.

Compositions and Morphologies of Inclusions
The compositions of inclusions detected at 60, 180, 600, and 720 s after calcium addition are shown in Figure 3. The mass fractions of CaO in calcium aluminate inclusions were in the range of 7.75-29.23% at 60 s after calcium addition and the average mass fraction of CaO was 19.22% in steel A. This indicates that the main types of inclusions were CA 6 and CA 2 in steel A. The average mass fractions of CaO in calcium aluminate inclusions in steel A increased to 40.17%, 50.05%, and 60.43% after adding calcium for 180, 600, and 720 s, respectively. This indicates that the inclusions were modified into CA + C 12 A 7 , C 12 A 7 , and C 12 A 7 + C 3 A gradually with the prolonging of the calcium treatment time. The mass fraction of CaO in calcium aluminate inclusions for steel B was less than that in steel A. The average mass fractions of CaO in calcium aluminate inclusions in steel B increased to 9.09%, 20.04%, 32.53%, and 40.12% after adding calcium for 60, 180, 600, and 720 s, respectively. This indicates that the inclusions were modified into CA 6 , CA 6 + CA 2 , CA 2 + CA, and CA + C 12 A 7 gradually with the prolonging of the calcium treatment time.  Figure 4a shows that the typical inclusion CaO·2Al2O3 that formed after calcium treatment for 60 s was irregular in steel A. The CaO·Al2O3 that formed after calcium treatment for 180 s was similar to a hexagon, of which the edges tended to be smooth. Typical inclusions 12CaO·7Al2O3 and 3CaO·Al2O3 occurred in steel A at later stages of deoxidation and their morphologies tended to be spherical. Figure 4a shows the typical inclusions of CaO·6Al2O3, 3CaO·8Al2O3, CaO·2Al2O3, and CaO·Al2O3 formed in steel B after calcium treatment for 60, 180, 600, and 720 s, respectively. Their morphologies were irregular and their sharp corners tended to be disappeared. The compositions, morphologies, and elemental mappings of typical inclusions detected in samples after calcium addition are shown in Figure 4 (steel A) and Figure 5 (steel B). Figure 4a shows that the typical inclusion CaO·2Al 2 O 3 that formed after calcium treatment for 60 s was irregular in steel A. The CaO·Al 2 O 3 that formed after calcium treatment for 180 s was similar to a hexagon, of which the edges tended to be smooth. Typical inclusions 12CaO·7Al 2 O 3 and 3CaO·Al 2 O 3 occurred in steel A at later stages of deoxidation and their morphologies tended to be spherical. Figure 4a shows the typical inclusions of CaO·6Al 2 O 3 , 3CaO·8Al 2 O 3 , CaO·2Al 2 O 3 , and CaO·Al 2 O 3 formed in steel B after calcium treatment for 60, 180, 600, and 720 s, respectively. Their morphologies were irregular and their sharp corners tended to be disappeared.

Dynamic Model
The mass transfer process and modification of Al2O3 inclusions in high-carbon hard wire steels can be described by the unreacted core model shown in Figure 6 based on experimental results. The modification process of Al2O3 inclusions can be described as shown in Figure 6 based on experimental results. The Al2O3 inclusion wrapped in the CA6 layer formed at the start of calcium treatment and Ca transferred into the unreacted core of Al2O3 through the CA6 layer. Therefore, the unreacted core of Al2O3 gradually decreased and the CA6 layer gradually thickened, then Al2O3 inclusions transformed into CA6 completely. With the diffusion of Ca and its increasing content, a complex inclusion with a core of CA6 wrapped in a CA2 layer formed and Ca transferred from the product layer (CA6 layer) to boundary layer (the interface between CA2 and CA6), resulting in the formation of a spherical inclusion with CA2. Similarly, the CA2 was transformed into CA, C12A7, and C3A step-by-step with the transfer of Ca and the chemical reaction between calcium aluminates. The assumptions in the current model are as follows:

Dynamic Model
The mass transfer process and modification of Al 2 O 3 inclusions in high-carbon hard wire steels can be described by the unreacted core model shown in Figure 6 based on experimental results. The modification process of Al 2 O 3 inclusions can be described as shown in Figure 6 based on experimental results. The Al 2 O 3 inclusion wrapped in the CA 6 layer formed at the start of calcium treatment and Ca transferred into the unreacted core of Al 2 O 3 through the CA 6 layer. Therefore, the unreacted core of Al 2 O 3 gradually decreased and the CA 6 layer gradually thickened, then Al 2 O 3 inclusions transformed into CA 6 completely. With the diffusion of Ca and its increasing content, a complex inclusion with a core of CA 6 wrapped in a CA 2 layer formed and Ca transferred from the product layer (CA 6 layer) to boundary layer (the interface between CA 2 and CA 6 ), resulting in the formation of a spherical inclusion with CA 2 . Similarly, the CA 2 was transformed into CA, C 12 A 7 , and C 3 A step-by-step with the transfer of Ca and the chemical reaction between calcium aluminates. The assumptions in the current model are as follows: 1.
All inclusions in molten steel are spherical before and during the calcium treatment process; 2.
The temperature of molten steel is very high at 1600 • C, so the interfacial reaction is assumed to be in equilibrium; 3.
To simplify the discussion of the model, the concentrations of calcium, aluminum, and oxygen in molten steel are assumed to be constant; 4.
The diffusion of all substances in the liquid calcium aluminate layer is steady, which is in accordance with Fick's first law.  Taking the transformation of CA inclusions into C12A7 as an example, the process is divided into the following three steps.
1. Ca in molten steel diffuses to the C12A7 layer-molten steel interface, for which the reaction formula is: 2.
[Ca] passes through the C12A7 liquid phase, diffuses to the CA layer, and reacts with it: 3. At this time, the generated [Al] diffuses outward through the C12A7 liquid phase layer and enters into the molten steel. Figure 7 is a schematic of the transformation process of CaO·Al2O3 into 12CaO·7Al2O3 inclusions. In the Figure 7, r0 represents the radius of C12A7 inclusion after complete modification, r represents the radius of the unreacted CA inclusion, and l1 and l2 represent the interface between CA inclusion and C12A7 inclusions, respectively. In the refining process, argon blowing and stirring are used. (Ca) and (Al) diffuse rapidly in molten steel and in the high-temperature reaction process. Therefore, the rate control link in the modification process for inclusions is solute diffusion in the calcium aluminate layer. At 1600 °C, the diffusion coefficients of Ca and Al in the calcium aluminate layer are [24,29] DCa ≈ 10 −8.6 m 2 ·s −1 and DAl ≈ 10 −10.4 m 2 ·s −1 , respectively. In this paper, the kinetics of the Al2O3 inclusion modification in high-carbon hard wire steel were analyzed in two cases. In the first case, the diffusion process of Al in the calcium aluminate layer was the limiting link. In the second case, the diffusion of Ca in the calcium aluminate layer was the limiting link in the process of inclusion modification. Taking the transformation of CA inclusions into C 12 A 7 as an example, the process is divided into the following three steps.

1.
Ca in molten steel diffuses to the C 12 A 7 layer-molten steel interface, for which the reaction formula is: Ca passes through the C 12 A 7 liquid phase, diffuses to the CA layer, and reacts with it: 2.
At this time, the generated [Al] diffuses outward through the C 12 A 7 liquid phase layer and enters into the molten steel. Figure 7 is a schematic of the transformation process of CaO·Al 2 O 3 into 12CaO·7Al 2 O 3 inclusions. In the Figure 7, r 0 represents the radius of C 12 A 7 inclusion after complete modification, r represents the radius of the unreacted CA inclusion, and l 1 and l 2 represent the interface between CA inclusion and C 12 A 7 inclusions, respectively. In the refining process, argon blowing and stirring are used. (Ca) and (Al) diffuse rapidly in molten steel and in the high-temperature reaction process. Therefore, the rate control link in the modification process for inclusions is solute diffusion in the calcium aluminate layer. At 1600 • C, the diffusion coefficients of Ca and Al in the calcium aluminate layer are [24,29] D Ca ≈ 10 −8.6 m 2 ·s −1 and D Al ≈ 10 −10.4 m 2 ·s −1 , respectively. In this paper, the kinetics of the Al 2 O 3 inclusion modification in high-carbon hard wire steel were analyzed in two cases. In the first case, the diffusion process of Al in the calcium aluminate layer was the limiting link. In the second case, the diffusion of Ca in the calcium aluminate layer was the limiting link in the process of inclusion modification.
From Equation (6), we get: In the formula, According to Equation (7), the rate of Al consumption in the modification reaction is: Combining Equation (7) with Equation (8), we can obtain: When the diffusion of Al in the calcium aluminate layer is the limiting link in inclusion modification, the diffusion rate of Al in the C 12 A 7 layer is expressed as: where n Al represents the amount of Al, r represents the radius of unreacted nuclear CA, D Al indicates the diffusion rate of Al in molten steel, c Al is the concentration of Al in C 12 A 7 , and t represents the modification time of inclusions: Equation (5) From Equation (6), we get: In the formula, cAl,l 1 represents the Al concentration at the interface between two inclusions and cAl,l 2 represents the Al concentration at the interface between inclusion C 12 A 7 and molten steel.
According to Equation (7), the rate of Al consumption in the modification reaction is: where ρ CA represents the density of CA, ρ CA = 2.96 × 10 3 kg/m 3 , and M CA represents the molar mass of CA, M CA =158g/mol. Combining Equation (7) with Equation (8), we can obtain: The relationship between the unreacted nucleus radii of inclusions and the modification time (t) can be obtained by finishing the following: To calculate the modification time of inclusions, the activity of solute elements in steel was used instead of its concentration. Equation (10) can be expressed as: When CA inclusions are completely transformed into C 12 A 7 , when r = 0, the complete modification time (t f ) of inclusions is: where ρ CA , M CA , and D Al are all constants and the modification time for CA inclusions depends on their radii and the activity difference of Al between the interface between two inclusions and the interface between molten steel and inclusions.
When the diffusion of Ca in the calcium aluminate layer is the limiting link in inclusion modification, the diffusion rate of Ca in the C 12 A 7 layer is expressed as follows: where n Ca represents the amount of Ca, r represents the radius of unreacted nuclear CA, D Ca indicates the diffusion rate of Ca in molten steel, c Ca is the concentration of Ca in C 12 A 7 , and t represents the modification time of inclusions: Equation (14) is integrated as: From Equation (15), we get: In the formula, cCa,l 1 represents the Ca concentration at the interface between two inclusions and cCa,l 2 represents the Ca concentration at the interface between inclusion C 12 A 7 and molten steel.
The rate of Ca generated by the modification reaction is: where ρ C 12 A 7 represents the density of C 12 A 7 , ρ C 12 A 7 = 2.83 × 10 3 kg/m 3 , and M C 12 A 7 represents the molar mass of C 12 A 7 , M C 12 A 7 = 1386g/mol. Combining Equation (16) with Equation (17), we can obtain: The relationship between unreacted nucleus radii of inclusions and modification time (t) can be obtained by finishing: To calculate the modification time of the inclusions, the activity of solute elements in steel was used instead of its concentration. Equation (18) can be expressed as: When CA inclusions are completely transformed into C 12 A 7 , when r = 0, the complete modification time (t f ) of inclusions is: where ρ C 12 A 7 , M C 12 A 7 , and D Ca are all constants, and the modification time of CA inclusions depends on their radius and the activity difference of Ca between the interface between two inclusions and the interface between molten steel and inclusions.

Model and Parameter Determination
When the diffusion of Al in the calcium aluminate layer is the limiting link in inclusion modification, the concentration of [Al] in calcium aluminate inclusions is difficult to determine and can be replaced by a Al as the following formula: For interface l 1 : The Gibbs free energy of this reaction is [25,29,30]: In the formula, a Al ,l 1 represents the activity of the interface layer Al between inclusion CA and C 12 A 7 , while a where a Al ,l 2 represents the activity of Al at the interface between C 12 A 7 inclusions and molten steel. When the diffusion of Ca in the calcium aluminate layer is the limiting link in inclusion modification, the activity of the interface layer Ca between inclusion CA and C 12 A 7 can be estimated as Equation (28).
In the formula, a Ca,l 1 represents the activity of the interface layer Ca between inclusion CA and C 12 A 7 , while a The formula shows the activity of Ca at the interface between C 12 A 7 inclusions and molten steel.
According to the interface shown in Equation (13), combined with the experimental steel composition in Table 2 and the interaction coefficient between elements in molten steel at 1873 K as shown in Table 3.  Table 4).

Determination of Restrictive Links
According to Equations (12) and (21), the diffusion of Al and Ca in the C 12 A 7 layer, which treated as limiting link in the transformation of CA inclusions into C 12 A 7 in steel A, was calculated as shown in Figure 8. It can be seen that the value of t f is larger in the red line than that in the black line when the radius is same. This indicates that the diffusion of Al in the inclusion layer was the limiting link in the modification process of inclusions, which was consistent with the research results of Park et al. [24].
In the formula, The formula shows the activity of Ca at the interface between C12A7 inclusions and molten steel.
According to the interface shown in Equation (13), combined with the experimental steel composition in Table 2 and the interaction coefficient between elements in molten steel at 1873 K as shown in Table 3.  Table 4).

Determination of Restrictive Links
According to Equations (12) and (21), the diffusion of Al and Ca in the C12A7 layer, which treated as limiting link in the transformation of CA inclusions into C12A7 in steel A, was calculated as shown in Figure 8. It can be seen that the value of tf is larger in the red line than that in the black line when the radius is same. This indicates that the diffusion of Al in the inclusion layer was the limiting link in the modification process of inclusions, which was consistent with the research results of Park et al. [24].

Effects of Solute Element Content in Molten Steel on Modification Time
The modification of inclusions depends on the activity of Al, which is affected by the content of each solute element in molten steel. The effects of Ca, Al, and O contents in molten steel on the time taken to modify CA into C 12 A 7 inclusions with a radius of 1.5 µm was studied based on Equations (12), (25), and (27), as shown in Figure 9. The modification time increased linearly with increasing oxygen content and decreased with increasing calcium content in molten steel. When the content of O increased in the range of 0.0002-0.0045%, the modification time was prolonged from 0.0011 s to 118 s, increasing by approximately 118 s. When the content of Ca increased in the range of 0.0002-0.0045%, the modification time for complete transformation of Ca inclusions into C 12 A 7 was reduced from 48616 s to 247 s, which was reduced by approximately 48369 s. The change of modification time with increasing Al content was very small. It can be seen that the Ca concentration in molten steel had the greatest influence on the modification time of inclusions. The steel with high Ca content was favorable for the modification of CA inclusions into C 12 A 7 inclusions.

Effects of Solute Element Content in Molten Steel on Modification Time
The modification of inclusions depends on the activity of Al, which is affected by the content of each solute element in molten steel. The effects of Ca, Al, and O contents in molten steel on the time taken to modify CA into C12A7 inclusions with a radius of 1.5 μm was studied based on Equations (12), (25), and (27), as shown in Figure 9. The modification time increased linearly with increasing oxygen content and decreased with increasing calcium content in molten steel. When the content of O increased in the range of 0.0002-0.0045%, the modification time was prolonged from 0.0011 s to 118 s, increasing by approximately 118 s. When the content of Ca increased in the range of 0.0002-0.0045%, the modification time for complete transformation of Ca inclusions into C12A7 was reduced from 48616 s to 247 s, which was reduced by approximately 48369 s. The change of modification time with increasing Al content was very small. It can be seen that the Ca concentration in molten steel had the greatest influence on the modification time of inclusions. The steel with high Ca content was favorable for the modification of CA inclusions into C12A7 inclusions. The relationship between the Ca content in molten steel and modification time of inclusions was calculated based on Equations (12) and (25), as shown in Figure 10. The complete modification time for inclusions was significantly shortened with the increase of the Ca content in molten steel and decrease of the inclusion radius. When the Ca content in molten steel was 0.0025% and the radius of inclusion was 1.5 μm, the modification times for Al2O3 into CA6, CA6 into CA2, CA2 into CA, CA into C12A7, and C12A7 into C3A were 4.5, 16, 116, 601, and 449 s, respectively. This indicates that the modification times for inclusions tend to be longer in the transformation of higher CaO-containing calcium aluminate. The modification of Al2O3 into CA6 was fastest, while the most time was required to modify CA into C12A7. When calcium aluminate inclusions changed from solid state to liquid state, this process was the most difficult to carry out and the required modification time was the longest. The relationship between the Ca content in molten steel and modification time of inclusions was calculated based on Equations (12) and (25), as shown in Figure 10. The complete modification time for inclusions was significantly shortened with the increase of the Ca content in molten steel and decrease of the inclusion radius. When the Ca content in molten steel was 0.0025% and the radius of inclusion was 1.5 µm, the modification times for Al 2 O 3 into CA 6 , CA 6 into CA 2 , CA 2 into CA, CA into C 12 A 7 , and C 12 A 7 into C 3 A were 4.5, 16, 116, 601, and 449 s, respectively. This indicates that the modification times for inclusions tend to be longer in the transformation of higher CaO-containing calcium aluminate. The modification of Al 2 O 3 into CA 6 was fastest, while the most time was required to modify CA into C 12 A 7 . When calcium aluminate inclusions changed from solid state to liquid state, this process was the most difficult to carry out and the required modification time was the longest.
for Al2O3 into CA6, CA6 into CA2, CA2 into CA, CA into C12A7, and C12A7 into C3A were 4.5, 16, 116, 601, and 449 s, respectively. This indicates that the modification times for inclusions tend to be longer in the transformation of higher CaO-containing calcium aluminate. The modification of Al2O3 into CA6 was fastest, while the most time was required to modify CA into C12A7. When calcium aluminate inclusions changed from solid state to liquid state, this process was the most difficult to carry out and the required modification time was the longest.

Influence of Inclusion Conversion Rate in Molten Steel on Modification Time
In order to evaluate the modification rate, taking CA to C12A7 as an example, the inclusion conversion ratio is defined as: According to Equations (11) and (30), the relationship between the conversion ratio and modification time during inclusion deformation was calculated, as shown in Figure  11. The conversion ratio was affected by the melting time, inclusion type, and size. The conversion ratio increased quickly at the beginning of the modification but became slow as the reaction progressed. It can be seen that when the conversion ratio of CA to C12A7 with a 2 μm radius increased from 0% to 57.8%, this took 167 s, while the time needed to prolong the process was 902 s, with a conversion ratio increase from 57.8% to 100%; it took about six times longer time at the later stage of inclusion modification than at the early stage. Therefore, in the later stage of inclusion modification, the stirring speed should be increased to promote inclusion modification and reduce the modification time.

Influence of Inclusion Conversion Rate in Molten Steel on Modification Time
In order to evaluate the modification rate, taking CA to C 12 A 7 as an example, the inclusion conversion ratio is defined as: According to Equations (11) and (30), the relationship between the conversion ratio and modification time during inclusion deformation was calculated, as shown in Figure 11. The conversion ratio was affected by the melting time, inclusion type, and size. The conversion ratio increased quickly at the beginning of the modification but became slow as the reaction progressed. It can be seen that when the conversion ratio of CA to C 12 A 7 with a 2 µm radius increased from 0% to 57.8%, this took 167 s, while the time needed to prolong the process was 902 s, with a conversion ratio increase from 57.8% to 100%; it took about six times longer time at the later stage of inclusion modification than at the early stage. Therefore, in the later stage of inclusion modification, the stirring speed should be increased to promote inclusion modification and reduce the modification time.
11. The conversion ratio was affected by the melting time, inclusion type, and size. The conversion ratio increased quickly at the beginning of the modification but became slow as the reaction progressed. It can be seen that when the conversion ratio of CA to C12A7 with a 2 μm radius increased from 0% to 57.8%, this took 167 s, while the time needed to prolong the process was 902 s, with a conversion ratio increase from 57.8% to 100%; it took about six times longer time at the later stage of inclusion modification than at the early stage. Therefore, in the later stage of inclusion modification, the stirring speed should be increased to promote inclusion modification and reduce the modification time.  Figure 11. Modification process of inclusions with different particle sizes (a) Al2O3 is modified into CA6, (b) CA6 is modified into CA2, (c) CA2 is modified into CA, (d) CA is modified into C12A7, (e) C12A7 is modified into C3A).

Relationship Between Inclusion Radius and Modification Time
The relationship between the inclusion radius and modification time was studied based on Equation (12), as shown in Figure 12. The complete modification times for inclusions increased with the square of their radii. The complete modification time was prolonged by four times, while the radii of unmodified inclusions doubled. The complete modification time for inclusions with a 1 μm radius from CA to C12A7 was 267 s, and was 1069 s for inclusions with a 2 μm radius. During the whole modification process of solid Al2O3 inclusions to liquid calcium aluminate inclusions of the same size, the modification time for inclusions from CA to C12A7 was the longest. Figure 11. Modification process of inclusions with different particle sizes (a) Al 2 O 3 is modified into CA6, (b) CA 6 is modified into CA 2 , (c) CA 2 is modified into CA, (d) CA is modified into C 12 A 7 , (e) C 12 A 7 is modified into C 3 A).

Relationship between Inclusion Radius and Modification Time
The relationship between the inclusion radius and modification time was studied based on Equation (12), as shown in Figure 12. The complete modification times for inclusions increased with the square of their radii. The complete modification time was prolonged by four times, while the radii of unmodified inclusions doubled. The complete modification time for inclusions with a 1 µm radius from CA to C 12 A 7 was 267 s, and was 1069 s for inclusions with a 2 µm radius. During the whole modification process of solid Al 2 O 3 inclusions to liquid calcium aluminate inclusions of the same size, the modification time for inclusions from CA to C 12 A 7 was the longest.

Modeling Verification
In order to verify the calculation of the kinetic model for the modification of Al2O3 inclusions during calcium treatment, the calculated CaO content was compared with experimental results, considering the boundary conditions and unreacted core model parameters, as shown in Figure 13. The modification of inclusions with radii of 1, 2, and 3 μm was simulated based on observed results in experiments. It can be seen that the modification of Al2O3 inclusions in sample A is much faster than that in sample B, which is consistent with experimental results. The model calculation is closer to the experimental data, indicating that they are in good agreement. Additionally, it was found that the inclusions with 1 μm evolving from Al2O3 to CA6 took no longer than 1 s. It took about 1000 s for inclusions with a 3 μm radius to modify Al2O3 into liquid calcium aluminate in sample A and about 6000 s for that in sample B.

Modeling Verification
In order to verify the calculation of the kinetic model for the modification of Al 2 O 3 inclusions during calcium treatment, the calculated CaO content was compared with experimental results, considering the boundary conditions and unreacted core model parameters, as shown in Figure 13. The modification of inclusions with radii of 1, 2, and 3 µm was simulated based on observed results in experiments. It can be seen that the modification of Al 2 O 3 inclusions in sample A is much faster than that in sample B, which is consistent with experimental results. The model calculation is closer to the experimental data, indicating that they are in good agreement. Additionally, it was found that the inclusions with 1 µm evolving from Al 2 O 3 to CA 6 took no longer than 1 s. It took about 1000 s for inclusions with a 3 µm radius to modify Al 2 O 3 into liquid calcium aluminate in sample A and about 6000 s for that in sample B.

Conclusions
In the experiments with high-carbon hard wire steels treated with different contents of calcium at 1600 °C, the compositions and morphological evolution of inclusions were studied. The kinetic model for modification of Al2O3 inclusions during calcium treatment in high-carbon hard wire steels was established based on unreacted core theory. This model considers the transfer of Ca and Al through the boundary layer and within the product layer coupled with thermodynamic equilibrium at the interfaces. The calculated results based on the kinetic model were compared with experimental results. The main findings are summarized below: 1. The diffusion of Al in the inclusion layer was the limiting link in the inclusion modification process. The modification time increased linearly with increasing oxygen content and decreased with increasing calcium content in molten steel. The change of modification time with increasing Al content was very small. The Ca concentration in molten steel had the greatest influence on the modification time of inclusions; 2. The modification times for inclusions tended to be longer in the transformation of higher CaO-containing calcium aluminate. The modification of Al2O3 into CA6 was

Conclusions
In the experiments with high-carbon hard wire steels treated with different contents of calcium at 1600 • C, the compositions and morphological evolution of inclusions were studied. The kinetic model for modification of Al 2 O 3 inclusions during calcium treatment in high-carbon hard wire steels was established based on unreacted core theory. This model considers the transfer of Ca and Al through the boundary layer and within the product layer coupled with thermodynamic equilibrium at the interfaces. The calculated results based on the kinetic model were compared with experimental results. The main findings are summarized below: 1.
The diffusion of Al in the inclusion layer was the limiting link in the inclusion modification process. The modification time increased linearly with increasing oxygen content and decreased with increasing calcium content in molten steel. The change of