Effect of Oxygen and Sulfur in Molten Steel on the Agglomeration Property of Alumina Inclusions in Molten Steel

Abstract

The agglomeration force that acts between alumina cylinders in molten steel has been measured at different concentrations of oxygen and sulfur. Both oxygen and sulfur in molten steel reduce the agglomeration force between the alumina cylinders in molten steel and act as interfacial active elements. However, oxygen reduces the agglomeration force more significantly than sulfur. The agglomeration force that was obtained empirically has been analyzed by combining the formulated surface tension concerning molten steel containing oxygen and sulfur with a model representing the alumina interparticle interaction due to a cavity bridge force. Thus, this analysis enables the effect of oxygen and sulfur in the molten steel on the agglomeration property of the alumina inclusions in the molten steel to be evaluated. The agglomeration property of the alumina inclusions reduces with the increase in the concentration of oxygen and sulfur in molten steel. The reduction due to oxygen is much greater than that due to sulfur. Moreover, when the concentration of sulfur in molten steel increases, the alumina inclusions remain in an adhesion state due to the strong agglomeration force based on the cavity bridge force. However, because the agglomeration force markedly decreases when the concentration of oxygen in the molten steel increases, the alumina inclusions that have agglomerated once are likely to separate again due to the molten steel flow.

1. Introduction

In the manufacturing of high-cleanliness steel, it is useful to control appropriately the stirring and flow of molten steel in order to facilitate the agglomeration and coalescence of alumina inclusions and in order to remove the alumina inclusions as coarse inclusions which are favorable for floating separation while preventing the contamination of molten steel caused by slag, air, and refractories as much as possible during the refining process. However, in the present situation where the required level for high-cleanliness steel becomes increasingly severe and it is necessary to remove extremely small inclusions, it is difficult to meet these demands only by the process improvement by fluid dynamic action. For this reason, the agglomeration mechanism of Al₂O₃ inclusions in molten steel, including the consequences of the interfacial chemical interaction, should be investigated. Consequently, the agglomeration property of the inclusions in molten steel can be highly controlled.

Regarding the agglomeration behavior of the inclusions in molten steel, studies have been performed on the basis of the size distribution evaluation¹⁾ of inclusions in rapidly quenched steel samples, water model experiments²⁾ and molten metal model experiments^3,4) using quasi-inclusion particles, and numerical simulations⁵⁾ that considers various collision and agglomeration mechanisms. However, it is difficult to perform an experiment that replicates the actual conditions of the phenomenon that occur at the interface between high-temperature molten steel and fine oxides. As a result, very little detailed research concerning the interfacial chemical interaction that acts between inclusions in molten steel is performed, and unclear points regarding the agglomeration mechanism of Al₂O₃ inclusions in molten steel still remain. The author established a new experimental method for the direct measurement of the agglomeration force that acts between Al₂O₃ cylinders in molten steel and also analyzed the actual measured values in consideration of the interfacial properties. The results indicated that the agglomeration force between Al₂O₃ particles in molten steel is caused not by the van der Waals force but by the cavity bridge force due to molten steel, which is unlikely to wet Al₂O₃ particles.⁶⁾ Subsequently, the author made observations of the cavity bridge formation and extinction behavior between the cylinders simulating inclusions that were moved alternatively toward each other and away from each other in molten steel and in mercury. As a result, the effect of the interparticle surface distance on the agglomeration force was evaluated, and it was clarified that the agglomeration force due to the cavity bridge force is a strong long-range force that also acts between particles that are relatively distant from each other.⁷⁾

This series of basic research was targeted on aluminum deoxidized molten steel containing almost no oxygen (O) or sulfur (S), which are interfacial active elements. However, in an actual steelmaking process, interfacial active elements exist in molten steel, complicating the agglomeration behavior of inclusions in molten steel and making it difficult to determine the essence of the agglomeration behavior of the inclusions in molten steel. In this research, to understand the agglomeration mechanism of Al₂O₃ inclusions in molten steel from the viewpoint of the interfacial chemical interaction, the author uses an agglomeration force measurement experiment established in a previous report.⁶⁾ With change in concentration of O and S in the molten steel, the agglomeration force acting between Al₂O₃ cylinders is measured. Moreover, from the measured agglomeration force, the effect of O and S in the molten steel on the contact angle between the molten steel and Al₂O₃ is quantitatively analyzed. In addition, studies are performed concerning the dependence of the concentrations of O and S in molten steel on the agglomeration property of the Al₂O₃ inclusions in molten steel, based on an interaction model between two isospherical Al₂O₃ particles in the molten steel that allows the effect of the interparticle surface distance to be taken into consideration.

2. Experimental

2.1. Experimental Apparatus

Figure 1 shows the experimental apparatus for measuring the agglomeration force acting between Al₂O₃ cylinders in molten steel. In the experiment, a resistance heating furnace comprising a high-frequency induction-heated graphite cylinder was used for the purpose of restraining the molten steel flow as far as possible. In order to measure the agglomeration force, an Al₂O₃ cylinder of the specified diameter was fixed perpendicularly on the inside wall of an alumina crucible with an inner diameter of 40 mm and a height of 150 mm. Another Al₂O₃ cylinder with the length of 30 mm and the same diameter as the one at the alumina crucible was installed at the bottom end of an alumina protective tube with an outer diameter of 8 mm and a length of 380–440 mm for measuring the agglomeration force. The top end of the alumina protective tube was connected to an aluminum rod at the top of the melting furnace with the Al₂O₃ cylinders arranged in contact with each other in parallel in the molten steel inside the crucible. The apparatus has an axis of rotation that is located at a point 40 mm below the top end of the aluminum rod, allowing smooth rotation of the alumina protective tube around the axis. A force gauge that was fixed to the moving stage at the top of the guide rail was connected to the driving motor by a wire. Furthermore, the force gauge was connected horizontally to a point 30 mm above the axis of rotation of the aluminum rod so that when the aluminum rod was pulled by the driving motor, the traction force generated against the agglomeration force between the Al₂O₃ cylinders in the molten steel was output from the force gauge to a chart recorder. In this apparatus, the moving direction of the force gauge is perpendicular to the aluminum rod. In addition, the distance from the rotation axis to the mounting position of the Al₂O₃ cylinder on the lower side is maintained greater than the distance from the rotation axis to the mounting position of the force gauge above. As a result, the minute agglomeration force generated between the Al₂O₃ cylinders in molten steel can be measured by amplifying the force by using the principle of leverage.

Fig. 1.

Experimental apparatus for agglomeration force measurement.

2.2. Experimental Procedure

Here 600 g of electrolytic iron (C concentration = 0.001 mass%, S concentration = 0.0001 mass%, O concentration = 0.005 mass%) was placed in the alumina crucible with Al₂O₃ cylinders fixed to the inner wall and then melted in an Ar gas atmosphere. The temperature of the molten steel was maintained at 1600°C. When the effect of the O concentration in the molten steel was examined, the specified quantity of Al was added aiming at an Al concentration of less than 0.02 mass% in the low O concentration region, while the specified quantity of Fe₂O₃ was added in the high O concentration region, thereby the O concentration in the molten steel was adjusted over the range of 0.001 to 0.038 mass%. When the effect of the concentration of S in the molten steel was examined, Al deoxidized molten steel with an Al concentration of at least 0.02 mass% that results in a constant agglomeration force was used to remove the effect of the concentration of O in the molten steel. Subsequently, the specified quantity of S was added to vary the concentration of S in the molten steel over the range of trace to 0.104 mass%. Note that, as can be predicted from the electrolytic iron composition as well, the concentration of S in the molten steel to which S was not added was extremely low and was assumed to be 0 mass%. The Al₂O₃ cylinder installed on the bottom end of the alumina protective tube was immersed in the molten steel after the composition adjustment. Moreover, the Al₂O₃ cylinder was brought into contact with the Al₂O₃ cylinder on the inside wall of the crucible in parallel at a point that was 10 mm from the bottom of the crucible. These Al₂O₃ cylinders used to measure the agglomeration force were all made of high purity 99.6 mass% Al₂O₃ and had a diameter of 8 mm. While the moving stage was moved at a speed of 0.16 mm·s⁻¹ to pull the aluminum rod, the traction force output from the force gauge was recorded on the chart recorder. The maximum traction force F_T,Max (N) at the instant the two Al₂O₃ cylinders are separated from each other is equivalent to the agglomeration force between the Al₂O₃ cylinders in the molten steel with the corresponding composition. Consequently, the true agglomeration force F_A (N·m⁻¹) is obtained from Eq. (1) by using the principle of leverage.⁶⁾   

$${\mathrm{F}}_{\mathrm{A}}={\mathrm{L}}_{\mathrm{U}}/\left\{\left({\mathrm{L}}_{\mathrm{D}}-\mathrm{L}/2\right)\cdot \mathrm{L}\right\}\cdot {\mathrm{F}}_{\mathrm{T},\mathrm{M}\mathrm{a}\mathrm{x}}$$(1)

Here L_U is the distance from the axis of rotation to the mounting position of the force gauge (m), L_D is the distance from the axis of rotation to the bottom end of the Al₂O₃ cylinder (m), and L is the length of the Al₂O₃ cylinder (m). In order to grasp precisely the molten steel composition during the experiments, the molten steel samples were obtained before and after the agglomeration force measurement by a transparent quartz tube with an inner diameter of 6 mm and were used to analyze the Al concentration, the total oxygen concentration, and the S concentration. In the experiment with Al addition, the average value of the equilibrium O concentration calculated using the thermodynamic reevaluation value⁸⁾ of Al deoxidation equilibrium presented by Itoh et al. from the Al concentration before and after the experiment was employed as the concentration of O in the molten steel during the experiment. Also, in the experiment with only Fe₂O₃ addition, the average value of the total oxygen concentration before and after the experiment was used as the concentration of O in the molten steel during the experiment. For the Al concentration and the S concentration during the experiment, the average values obtained from the analytical concentration before and after the experiment were used.

3. Experimental Results

Figure 2 shows the effect of the Al concentration [Al] in the molten steel on the agglomeration force between the two Al₂O₃ cylinders with the same diameter and composition with the results for an Al concentration of 0.02 mass% or greater that have been discussed in an existing report.⁶⁾ Note that d_CY indicates the diameter of the Al₂O₃ cylinders. When the Al concentration in the molten steel becomes 0.02 mass% or greater, the agglomeration force between the Al₂O₃ cylinders in the molten steel becomes a roughly constant value of 14.86 N·m⁻¹. However, in the region where the Al concentration is less than 0.02 mass%, the agglomeration force falls to 5.96 N·m⁻¹ with the reduction of the Al concentration in the molten steel. In the region where the Al concentration in the molten steel is 0.02 mass% or higher, O in the molten steel is adequately deoxidized, and the effect of O, which is an interfacial active element, can be ignored. Consequently, a certain agglomeration force is indicated in this region. In the region where the Al concentration in the molten steel is less than 0.02 mass%, it is considered that the agglomeration force decreases along with the increase of the equilibrium O concentration accompanying the reduction of the Al concentration.

Fig. 2.

Effect of the concentration of A1 in molten steel on the agglomeration force between two Al₂O₃ cylinders with the same diameter and composition in molten steel.

Figure 3 shows the effect of the concentration of O [O] and also that of S [S] in the molten steel on the agglomeration force between the two Al₂O₃ cylinders with the same diameter and composition in the molten steel. The O concentration in equilibrium with the molten steel with an Al concentration of 0.02 mass%, which corresponds to the point where the agglomeration force of Fig. 2 starts to become a fixed value of 14.86 N·m⁻¹, is 0.0009 mass%. Consequently, the agglomeration force in an O concentration of 0.0009 mass% and also that in an S concentration of 0 mass% ([Al]≥0.02 mass%) are deemed to be 14.86 N·m⁻¹. The agglomeration force between Al₂O₃ cylinders in molten steel decreases with the increase in the concentration of O and S in the molten steel. However, the reduction of the agglomeration force along with the increase in the O concentration is very noticeable, which demonstrates that the action of O as an interfacial active element is stronger than that of S. This relationship is similar to the effect of the concentration of O and S in the molten steel on the surface tension of the molten steel.⁹⁾ As was clarified from previous reports,^6,7) the agglomeration force that acts between Al₂O₃ cylinders in the molten steel originates in the cavity bridge force between the Al₂O₃ cylinders, which are unlikely to be wet with molten steel. Consequently, the fact that this agglomeration force is affected by O and S, which are interfacial active elements, in the molten steel is a reasonable result.

Fig. 3.

Effect of the concentration of O and S in molten steel on the agglomeration force between two Al₂O₃ cylinders with the same diameter and composition in molten steel.

4. Discussion

4.1. Effect of O and S in Molten Steel on the Wettability of Molten Steel to Al₂O₃

4.1.1. Method of Calculating the Contact Angle between Molten Steel and Al₂O₃

The agglomeration force that acts between the Al₂O₃ cylinders in the molten steel is generated on the basis of the cavity bridge that is formed between the Al₂O₃ cylinders with low molten steel wettability.^6,7) For this reason, the agglomeration force that occurs between the two Al₂O₃ cylinders with the same diameter and composition in the molten steel is expressed as the total of the forces caused by the pressure difference ΔP_Fe (Pa) between the cavity bridge and the molten steel and the surface tension σ_Fe (N·m⁻¹) of the molten steel, as shown in Eq. (2).   

$${\mathrm{F}}_{\mathrm{A}}=2{\mathrm{X}}_4\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}+2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}$$(2)

where X₄ is the half-width of the cavity bridge neck (m). Regarding X₄, Eq. (3) is obtained from the geometrical conditions and Laplace relation.   

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{X}}_4{}^2+2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{X}}_4+2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{r}}_{\mathrm{C}\mathrm{Y}}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}=0$$(3)

where r_CY is the radius of the Al₂O₃ cylinder (m), and θ_Al2O3-Fe is the contact angle between the molten steel and Al₂O₃ (°). By using Eq. (3) to find X₄, Eq. (4) is obtained.   

$${\mathrm{X}}_4=\left\{-{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}+{({\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}{}^2-2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{r}}_{\mathrm{C}\mathrm{Y}}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}})}^{0.5}\right\}/\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}$$(4)

By substituting Eq. (4) into Eq. (2) to find θ_Al2O3-Fe, Eq. (5) is obtained.   

$${\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left((4{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}{}^2-{\mathrm{F}}_{\mathrm{A}}{}^2)/(8{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{r}}_{\mathrm{C}\mathrm{Y}})\right)$$(5)

ΔP_Fe is 3.86×10³ Pa from a previous report.⁶⁾ Therefore, if σ_Fe is already known, θ_Al2O3-Fe can be estimated from the measured value of F_A by using Eq. (5).

4.1.2. Formulation of the Effect of O Concentration in Molten Steel on Surface Tension of Molten Steel

Ogino et al.,^9,10) Takiuchi et al.,^11,12) and Nakashima et al.¹³⁾ investigated the effect of the O concentration in molten steel on the surface tension of molten steel. Table 1 shows the measurement results arranged on the basis of Szyszkowski’s Eq. (6),¹⁴⁾ which can take into consideration the reduction of the surface tension by the oxygen adsorption.   

$${\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}={\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}^{\mathrm{P}}-\mathrm{R}\cdot \mathrm{T}\cdot {\mathrm{\Gamma }}_{\mathrm{O},\mathrm{S}}^{\mathrm{S}}\cdot \mathrm{l}\mathrm{n}(1+{\mathrm{K}}_{\mathrm{O},\mathrm{S}}\cdot {\mathrm{a}}_{\mathrm{O}})$$(6)

where ${\sigma }_{\text{Fe}}^{\text{P}}$ is the surface tension of molten pure iron (N·m⁻¹), R is the gas constant (N·m·K⁻¹·mol⁻¹), T is the absolute temperature (K), ${\mathrm{\Gamma }}_{\text{O},\text{S}}^{\text{S}}$ is the saturated excess of oxygen at the surface of molten steel (mol·m⁻²), K_O,S is the adsorption coefficient of oxygen at the surface of the molten steel, and a_O is the activity of O in the molten steel. In addition, the experimental equation by Nakashima et al. was formulated by the author based on their measurement results¹³⁾ according to the procedure similar to that of the average value of the surface tension described below. However, it is unclear which of the surface tension values in Table 1 should be used; thus, the author will formulate the effect of the O concentration on the surface tension of molten steel using the average of these surface tension values.

Table 1. Effects of the concentration of O in molten steel on the surface tension of molten Fe–Al–O alloy.

σFe=1.91−0.358·ln(1+210aO) (N·m−1) at 1873 K, Ogino et al.9)

σFe=1.97−0.318·ln(1+200aO) (N·m−1) at 1823 K, Takiuchi et al.11)

σFe=1.90−0.327·ln(1+96aO) (N·m−1) at 1873 K, Takiuchi et al.12)

σFe=1.97−0.288·ln(1+280aO) (N·m−1) at 1873 K, Nakashima et al.13)

In the case where the O in the molten steel is adsorbed on the surface of the molten steel, the surface excess of oxygen Γ_O,S (mol·m⁻²) is expressed using the Gibbs’s isothermal adsorption equation of Eq. (7).   

$${\mathrm{\Gamma }}_{\mathrm{O},\mathrm{S}}=-1/(\mathrm{R}\cdot \mathrm{T})\cdot \mathrm{d}{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}/\mathrm{d}\left(\mathrm{l}\mathrm{n}{\mathrm{a}}_{\mathrm{O}}\right)$$(7)

Figure 4 shows the relation between the average value of the surface tension found from each equation in Table 1 and the logarithm of the O activity. In the region where the O activity is 0.03 and above, a constant value of dσ_Fe/d(ln a_O) (=−0.291) is acquired. Therefore, ${\mathrm{\Gamma }}_{\text{O},\text{S}}^{\text{S}}$ is 1.87×10⁻⁵ mol·m⁻² when calculated using Eq. (7). In each equation in Table 1, the surface tension obtained by assuming the O activity of 0 is at least 1.90 N·m⁻¹, which is an extremely high value. Consequently, the average value, 1.94 N·m⁻¹, of these values is deemed as the value for ${\mathrm{\sigma }}_{\text{Fe}}^{\text{P}}$, which is the surface tension of molten pure iron containing neither O nor S. The value of K_{O, S} is determined so that the average value of the surface tension in Table 1 is well fitted with Eq. (6) into which ${\mathrm{\Gamma }}_{\text{O},\text{S}}^{\text{S}}$ and ${\mathrm{\sigma }}_{\text{Fe}}^{\text{P}}$ found in the aforementioned manner are substituted, resulting in the value 237. Note that the O concentration in the agglomeration force measurement experiment is 0.04 mass% or less, and the activity coefficient of O estimated using the equilibrium values¹⁵⁾ recommended by the Japan Society for the Promotion of Science is approximately between 1 and 0.98. From this, the O concentration is used as the O activity. From the above-mentioned examination, the effect of the O activity on the surface tension of molten steel is indicated using Eq. (8).   

$${\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}=1.94-0.291\cdot \mathrm{l}\mathrm{n}\left(1+237{\mathrm{a}}_{\mathrm{O}}\right)$$(8)

Fig. 4.

Relation between the average value of the surface tension of molten steel and the logarithm of the O activity.

4.1.3. Formulation Concerning the Surface Tension of Molten Steel Containing Both O and S

In this study, to evaluate the effect of O and S on the agglomeration force between the two Al₂O₃ cylinders with the same diameter and composition in molten steel, it is necessary to formulate the effect of both O and S on the surface tension of molten steel. Accordingly, as Ogino et al.,⁹⁾ it is decided to express the surface tension of the molten steel that includes O and S as two types of surface active elements, using Eq. (9).   

$$\begin{array}{l}{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}={\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}^{\mathrm{P}}-\mathrm{R}\cdot \mathrm{T}\cdot {\mathrm{\Gamma }}_{\mathrm{O},\mathrm{S}}^{\mathrm{S}}\cdot \mathrm{l}\mathrm{n}(1+{\mathrm{K}}_{\mathrm{O},\mathrm{S}}\cdot {\mathrm{a}}_{\mathrm{O}})\\ \hspace{1em}\hspace{1em}-\mathrm{R}\cdot \mathrm{T}\cdot {\mathrm{\Gamma }}_{\mathrm{S},\mathrm{S}}^{\mathrm{S}}\cdot \mathrm{l}\mathrm{n}(1+{\mathrm{K}}_{\mathrm{S},\mathrm{S}}\cdot {\mathrm{a}}_{\mathrm{S}})\end{array}$$(9)

where ${\mathrm{\Gamma }}_{\text{S},\text{S}}^{\text{S}}$ is the saturated excess of sulfur at the surface of molten steel (mol·m⁻²), K_S,S is the adsorption coefficient of sulfur at the surface of the molten steel, and a_S is the activity of S in the molten steel. Ogino et al.⁹⁾ investigated the effect of S alone on the surface tension of the molten steel, including 0.0025–0.0034 mass% of the impurity O, at a temperature of 1600°C. As a result, Eq. (10) is obtained.   

$${\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}=1.76-0.235\cdot \mathrm{l}\mathrm{n}\left(1+185{\mathrm{a}}_{\mathrm{S}}\right)$$(10)

When the values of Eq. (8) are utilized as R·T·${\mathrm{\Gamma }}_{\text{O},\text{S}}^{\text{S}}$ and K_O,S in Eq. (9), the values of Eq. (10) are utilized as R·T·${\mathrm{\Gamma }}_{\text{S},\text{S}}^{\text{S}}$ and K_S,S in Eq. (9), and the value of Eq. (8) is utilized as the surface tension ${\sigma }_{\text{Fe}}^{\text{P}}$ of molten pure steel that does not contain O or S, the surface tension of the molten steel that contains both O and S can be represented by Eq. (11).   

$${\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}=1.94-0.291\cdot \mathrm{l}\mathrm{n}\left(1+237{\mathrm{a}}_{\mathrm{O}}\right)-0.235\cdot \mathrm{l}\mathrm{n}\left(1+185{\mathrm{a}}_{\mathrm{S}}\right)$$(11)

From Eq. (11), when the surface tension of molten steel that does not contain S but contains a maximum of 0.0034 mass% of O alone is calculated, the resulting value is 1.77 N·m⁻¹. This almost agrees with the surface tension of 1.76 N·m⁻¹ obtained by making the S activity in Eq. (10) equal to 0. From this, it can be seen that Eq. (11) contains not only Eq. (8) but also Eq. (10), which expresses the dependence of the S concentration on the surface tension of molten steel containing 0.0034 mass% O. Therefore, Eq. (11) expresses the effect of O and S on the surface tension of the molten steel universally. Note that as the S concentration used in the agglomeration force measurement experiment is 0.104 mass% or less and the estimated value of the activity coefficient of S by the equilibrium value¹⁵⁾ recommended by the Japan Society for the Promotion of Science is between 1 and 0.99, the S concentration can be used as the S activity in this research.

4.1.4. Evaluating the Effect of O and S in Molten Steel on Contact Angle between Molten Steel and Al₂O₃

In the experiment for evaluating the effect of the O concentration on the agglomeration force between Al₂O₃ cylinders in molten steel, the contact angle between molten steel and Al₂O₃ is calculated using Eq. (5) from the measured agglomeration force between the Al₂O₃ cylinders and the surface tension of the molten steel obtained from Eq. (11) in which the S concentration is assumed to be 0 mass%. Figure 5 shows a comparison of the effect of the O concentration in molten steel on the contact angle between the molten steel and the Al₂O₃ obtained from the measured agglomeration force with the measured values^10,11,12,13) by other researchers. As also mentioned in a previous report,⁶⁾ in this research the target contact angle is that between a molten Fe–Al–O alloy and Al₂O₃. Consequently, regarding other researchers’ data, only the measurement values over the O concentration range from 0.005 to 0.058 mass% without Al₂O₃ dissociation or the thermodynamically stable FeO·Al₂O₃ (hercynite) formation are utilized. The contact angles obtained by substituting the interfacial tension⁶⁾ in Table 2 and the surface tension in Table 1, which are formulated using Szyszkowski’s equation over this O concentration range, into Eq. (12) of Young below are indicated in Fig. 5 as various curves pertaining to the results obtained by each researcher.   

$$\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}=({\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}-{\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}})/{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}$$(12)

where σ_Al2O3 is the surface tension of Al₂O₃ given as 0.75 N·m⁻¹ at 1600°C,¹⁰⁾ and σ_Al2O3-Fe is the interfacial tension between molten steel and Al₂O₃. The contact angle between molten steel and Al₂O₃ calculated from the agglomeration force drastically decreases until the O concentration in the molten steel increases to approximately 0.01 mass% and subsequently the angle decreases gradually and approaches 90° asymptotically. The reduction ratio of the contact angle in the O concentration region of 0.01 mass% or less is considerably greater than those of the contact angles measured by the other researchers using the sessile drop method. At present this reason remains to be clarified because the action mechanism of the interfacial active elements on the dynamic contact angle obtained in this study is not clear. Moreover, even in the high O concentration region of 0.01 mass% or more, the contact angle based on the measurement of agglomeration force does not fall below 90°, which is the judgment criterion of wettability. Apart from the data of Ogino et al., the results of measurement performed by other researchers show the same trend. The reason why the contact angle is maintained at a value of 90° or more is explained as follows. According to McLean et al.,¹⁶⁾ the O concentration when Al₂O₃ and FeO·Al₂O₃ coexist is 0.058 mass% at 1600°C. Consequently, in the O concentration region lower than this, the interface between the molten steel and Al₂O₃ is mainly formed; conversely in a higher O concentration region (however, less than 0.23 mass%, which is the oxygen saturation concentration of molten steel), the interface between the molten steel and FeO·Al₂O₃ is mainly formed. For this reason, in the O concentration region that is lower than at least 0.058 mass%, the nature of Al₂O₃, which is not readily wet with molten steel, is strongly reflected in the interface phenomenon.

Fig. 5.

Effect of the concentration of O in molten steel on the contact angle between molten steel and Al₂O₃ obtained from the agglomeration force.

Table 2. Effects of the concentration of O in molten steel on the interfacial tension between molten Fe–Al–O alloy and Al₂O₃.

σAl2O3-Fe=2.60−1.049·ln(1+176aO) (N·m−1) at 1873 K, Ogino et al.

σAl2O3-Fe=2.60−0.660·ln(1+208aO) (N·m−1) at 1823 K, Takiuchi et al.

σAl2O3-Fe=2.60−0.834·ln(1+121aO) (N·m−1) at 1873 K, Takiuchi et al.

σAl2O3-Fe=2.20−0.275·ln(1+635aO) (N·m−1) at 1873 K, Nakashima et al.

In the experiment for evaluating the relation between the agglomeration force acting between the Al₂O₃ cylinders in the molten steel and the S concentration, the surface tension of Al deoxidized molten steel corresponding to the S concentration is calculated from Eq. (11) based on the assumption that an O concentration in the molten steel is 0.0009 mass% in equilibrium with molten steel containing the Al concentration of 0.02 mass%, where the agglomeration force becomes constant. The contact angle between the molten steel and the Al₂O₃ is calculated by means of Eq. (5) from this calculated surface tension and the measured agglomeration force acting between the Al₂O₃ cylinders in the molten steel. Figure 6 shows a comparison of the effect of the S concentration in the molten steel on the contact angle between the molten steel and the Al₂O₃ obtained from the agglomeration force with the measurement values¹⁷⁾ by Ogino et al. Note that the dotted lines in the diagram indicate the contact angle obtained by substituting the surface tension obtained from Eq. (11) assuming the O concentration to be 0.0034 mass% and the interfacial tension in Eq. (13) which is formulated with Szyszkowski’s equation in order to satisfy the measurement values of Ogino et al. into Eq. (12) of Young.   

$${\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}=1.96-0.358\cdot \mathrm{l}\mathrm{n}\left(1+30{\mathrm{a}}_{\mathrm{S}}\right)$$(13)

Whereas the contact angle by Ogino et al. is roughly constant at 137° regardless of the concentration of S in molten steel, the contact angle obtained from the agglomeration force first decreases slightly with the increase in the S concentration. Subsequently, when the S concentration becomes 0.01 mass% or more, the contact angle agrees with the measurement results of Ogino et al. Moreover, when the contact angles obtained from the agglomeration forces of Figs. 5 and 6 are compared with each other, the reduction ratio of the contact angle to the O concentration increase is much greater than that to the S concentration increase. This tendency can also be confirmed from the comparison of the data by Ogino et al. in Figs. 5 and 6. Therefore, it is clear that though at the contact angle obtained from the agglomeration force, O and S in the molten steel function as interfacial active elements, the interfacial active effect of O is higher than that of S.

Fig. 6.

Effect of the concentration of S in molten steel on the contact angle between molten steel and Al₂O₃ obtained from the agglomeration force.

In the sessile drop method, small drops of molten steel are placed on an Al₂O₃ substrate, and shape measurement is carried out to evaluate the contact angle. Compared with this sessile drop method, the contact angle evaluation method that uses the agglomeration force has the following excellent features: ① as the quantity of molten steel is large and the specific surface area to the volume is small, variations in the molten steel composition due to reaction with the gas phase are small, ② as Al₂O₃ cylinders are immersed in the molten steel, the boundary conditions that are similar to the actual Al₂O₃ inclusions are met, and ③ it is possible to evaluate the contact angle in a dynamic condition in which an external force acts between the Al₂O₃ cylinders in the molten steel. In addition to the feature ①, it is considered that as the average concentration of quenched samples obtained from the molten steel immediately before and after agglomeration force measurement is used in the contact angle evaluation by the agglomeration force measurement, the interfacial active effect of O and S in the molten steel is evaluated accurately, even when compared with the measurement results by the conventional sessile drop method. In addition, when features ② and ③ are considered in this contact angle evaluation, use of the contact angle between the molten steel and Al₂O₃ acquired in this research allows for appropriate analysis of the dynamic agglomeration behavior of the Al₂O₃ inclusions in the molten steel in the light of the effect of the interfacial active element.

4.2. Effect of O and S in Molten Steel on the Agglomeration Property of Al₂O₃ Inclusions in Molten Steel

4.2.1. Calculation Method of the Agglomeration Force between Spherical Al₂O₃ Inclusions in Molten Steel and Maximum Cavity Bridge Length

It is considered that the larger the agglomeration force acting between the inclusions in molten steel and the longer the distance over which the force acts, the easier it is for the inclusions to agglomerate and coalesce with each other to form large clusters. Accordingly, as an index of agglomeration property, the agglomeration force due to the cavity bridge force and also the maximum cavity bridge length (the largest surface distance between the particles in which cavity bridge exists.) equivalent to the distance over which this force acts are calculated from the experimental data regarding the spherical Al₂O₃ inclusions in the molten steel.

From an examination in a previous report,⁷⁾ the agglomeration force F_A,S (N) that acts between two isospherical Al₂O₃ inclusions with a cavity bridge that are away from each other by the interparticle surface distance a (m) is expressed by Eq. (14) as the sum of the forces resulting from the pressure difference between a cavity bridge and molten steel and the surface tension of the molten steel as is the case with Eq. (2).   

$${\mathrm{F}}_{\mathrm{A},\mathrm{S}}=\pi \cdot {\mathrm{R}}_4{}^2\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}+2\pi \cdot {\mathrm{R}}_4\cdot {\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}$$(14)

R₄ is the radius (m) of the cavity bridge neck. Regarding R₄, Eq. (16) is obtained from the geometrical conditions applicable to the case of two isospherical Al₂O₃ inclusions and the Laplace relation of Eq. (15).   

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}={\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\left(1/{\mathrm{R}}_3-1/{\mathrm{R}}_4\right)$$(15)

  

$${\mathrm{R}}_4{}^3+{\mathrm{A}}_1\cdot {\mathrm{R}}_4{}^2+{\mathrm{A}}_2\cdot {\mathrm{R}}_4+{\mathrm{A}}_3=0$$(16)

Here R₃ is the curvature radius (m) of cavity bridges. A₁, A₂ and A₃ are expressed by the following Eqs. (17), (18), and (19), respectively,   

$${\mathrm{A}}_1=3{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}/\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}$$(17)

  

$${\mathrm{A}}_2={\mathrm{a}}^2/4+\mathrm{a}\cdot \mathrm{r}+2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{r}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}/\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}$$(18)

  

$${\mathrm{A}}_3=\left(\mathrm{a}/4+\mathrm{r}\right)\cdot \mathrm{a}\cdot {\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}/\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}$$(19)

where r is the radius (m) of the Al₂O₃ inclusions. The cubic equation of Eq. (16) has three solutions. At least, the physical conditions under which the radius of the cavity bridge neck becomes a positive real number are satisfied when two isospherical Al₂O₃ inclusions are in complete contact with each other, which leads to R₄ between the two isospherical Al₂O₃ inclusions expressed as Eq. (20).   

$$\begin{array}{l}{\mathrm{R}}_4={\left\{-{\mathrm{B}}_2/2+{\left(-{\mathrm{B}}_3/108\right)}^{1/2}\right\}}^{1/3}\\ \hspace{1em}\hspace{0.17em}+{\left\{-{\mathrm{B}}_2/2-{\left(-{\mathrm{B}}_3/108\right)}^{1/2}\right\}}^{1/3}-{\mathrm{A}}_1/3\end{array}$$(20)

B₁, B₂, and B₃ are given by Eqs. (21), (22), and (23), respectively.   

$${\mathrm{B}}_1=-{\mathrm{A}}_1{}^2/3+{\mathrm{A}}_2$$(21)

  

$${\mathrm{B}}_2=2/27{\mathrm{A}}_1{}^3-{\mathrm{A}}_1\cdot {\mathrm{A}}_2/3+{\mathrm{A}}_3$$(22)

  

$${\mathrm{B}}_3=-108\left({\mathrm{B}}_2{}^2/4+{\mathrm{B}}_1{}^3/27\right)$$(23)

Consequently, by substituting R₄ obtained from Eq. (20) into Eq. (14), it is possible to calculate the agglomeration force acting between the two isospherical Al₂O₃ inclusions that are away from each other by a certain interparticle surface distance and form a cavity bridge. Moreover, the further the two isospherical inclusions separate from each other, the smaller R₄ becomes. However, as opposed to the case of two cylindrical particles with the same diameter, R₃ also becomes smaller as a result of the Laplace limiting conditions in Eq. (15) (where ΔP_Fe is a constant value.). Consequently, before R₄ becomes 0, the cavity bridge disappears due to the failure of the geometrical conditions of the two isospherical inclusions. In other words, the maximum cavity bridge length ${\text{D}}_{\text{CB},\text{Max}}^{\text{S}}$ when the two isospherical inclusions are gradually separated from each other means the maximum interparticle surface distance over which both the geometrical conditions and Laplace relation are satisfied, and the length is achieved when R₄ becomes the minimum positive value. Note that, from the results of a previous report⁷⁾ the maximum cavity bridge length ${\text{D}}_{\text{CB},\text{Max}}^{\text{A}}$ when the inclusions are approaching each other is made 58% of the maximum cavity bridge length ${\text{D}}_{\text{CB},\text{Max}}^{\text{S}}$ when the inclusions are separated from each other.

4.2.2. Evaluating the Effect of O and S in Molten Steel on Agglomeration Property of Spherical Al₂O₃ Inclusions in Molten Steel

The surface tension calculated from Eq. (11), the contact angle between molten steel and Al₂O₃ (Figs. 5 and 6) obtained from the agglomeration force, and also the pressure difference of 3.86 × 10³ Pa between the cavity bridge and the molten steel are used to obtain ${\text{D}}_{\text{CB},\text{Max}}^{\text{S}}$ and ${\text{D}}_{\text{CB},\text{Max}}^{\text{A}}$ by giving “a” to the Al₂O₃ inclusions having diameters between 1 and 10 μm while increasing “a” in small steps by trial and error so that R₄ in Eq. (20) becomes the minimum positive value. At the same time, by substituting R₄ into Eq. (14) when a = 0, F_A,S is also calculated under the condition in which the Al₂O₃ inclusions are in contact with each other.

The effect of the concentration of O and S in molten steel on the maximum cavity bridge length ${\text{D}}_{\text{CB},\text{Max}}^{\text{S}}\cdot {\text{d}}^{-1}$ and ${\text{D}}_{\text{CB},\text{Max}}^{\text{A}}\cdot {\text{d}}^{-1}$ is shown in Figs. 7 and 8, respectively. Note that d is the diameter of the Al₂O₃ inclusions. The maximum cavity bridge length when inclusions are separated away and when they approach each other decreases along with the increase of the concentration of O and S in the molten steel. The reduction ratio is greater in the case of the O concentration increase compared to that of the S concentration increase, and the maximum cavity length for 0.02 mass% O concentration falls to the vicinity of 0. The effect of the concentration of O and S in molten steel on the agglomeration force F_A,S·(σ_Fe·d)⁻¹ that acts between the two isospherical Al₂O₃ inclusions in contact with each other is shown in Figs. 9 and 10, respectively. As is the case with the dependence of the concentration of O and S in the molten steel on the maximum cavity bridge length, the agglomeration force between the two isospherical Al₂O₃ inclusions decreases along with the increase of the concentration of O and S in the molten steel. However, compared with the effect of the concentration of S, the effect of the concentration of O is extremely large. From these facts, it can be seen that the agglomeration property of the Al₂O₃ inclusions in the molten steel decreases along with the increase of either the O concentration or S concentration, but decreases more in the case of O compared with the case of S.

Fig. 7.

Effect of the concentration of O in molten steel on the maximum cavity bridge length.

Fig. 8.

Effect of the concentration of S in molten steel on the maximum cavity bridge length.

Fig. 9.

Effect of the concentration of O in molten steel on the agglomeration force between two isospherical Al₂O₃ inclusions in a contacting condition.

Fig. 10.

Effect of the concentration of S in molten steel on the agglomeration force between two isospherical Al₂O₃ inclusions in a contacting condition.

To understand the effect of the concentration of O and S on the agglomeration force that acts between the two isospherical Al₂O₃ inclusions in molten steel acquired from Figs. 9 and 10 by a relative comparison with the external forces that act on the Al₂O₃ inclusions, the buoyant force and drag force that act on the Al₂O₃ inclusions in the molten steel are estimated as below. The buoyant force F_B (N) that acts on the Al₂O₃ inclusions in the molten steel can be evaluated by means of Eq. (24) as follows,   

$${\mathrm{F}}_{\mathrm{B}}=4\pi \cdot {\mathrm{r}}^3\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3})\cdot \mathrm{g}/3$$(24)

where ρ_Fe is the density of the molten steel (7000 kg·m⁻³)_,ρ_Al2O3 is the density of the Al₂O₃ inclusions (3970 kg·m⁻³), and g is the acceleration of gravity (m·s⁻²). A buoyant force of 1.56×10⁻¹⁴ ~ 1.56×10⁻¹¹ N acts on the spherical Al₂O₃ inclusions with a particle diameter of 1–10 μm. Moreover, the drag force F_D (N) that acts on the Al₂O₃ inclusions due to a relative movement to molten steel is shown in Eq. (25) as follows,   

$${\mathrm{F}}_{\mathrm{D}}={\mathrm{C}}_{\mathrm{D}}\cdot {\rho }_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{v}}^2\cdot \mathrm{S}/2$$(25)

where C_D is the drag coefficient, v is the flow velocity of the molten steel (m·s⁻¹), and S is the projected area of the inclusion particle toward the flow direction (m²) and is given by π·r² for spherical inclusions. The drag coefficient is estimated by using Eq. (26) that agrees well with the experimental value when the Reynolds number Re_P (=2r·v/ν) of the particles is 1000 and less.   

$${\mathrm{C}}_{\mathrm{D}}=24\left(1+0.158\mathrm{R}{\mathrm{e}}_{\mathrm{P}}{}^{2/3}\right)/\mathrm{R}{\mathrm{e}}_{\mathrm{P}}$$(26)

Note that the kinematic viscosity coefficient ν of molten steel is 7.14×10⁻⁷ m²·s⁻¹. When the flow velocity of the molten steel through the immersion nozzle that is considered to be the fastest part in the continuous casting process is assumed to be 2m·s⁻¹, the drag force that acts on the spherical Al₂O₃ inclusions with a particle diameter of 1–10 μm ranges from 1.24×10⁻⁷ ~ 2.32×10⁻⁶ N, which is much greater than the previously obtained buoyant force. Consequently, the drag force resulting from the flow of molten steel as the main external force acting on the Al₂O₃ inclusions in the molten steel is taken up and is compared with the agglomeration force acting between the Al₂O₃ inclusions in the molten steel due to the cavity bridge force. The drag force F_D·(d·σ_Fe)⁻¹ obtained from Eq. (25) for each particle diameter between 1 and 10 μm is indicated by the dotted line in each of Figs. 9 and 10. As can be seen from Fig. 9, the agglomeration force that acts between the two isospherical Al₂O₃ inclusions in molten steel is larger than the drag force caused by the flow of the molten steel in the low O concentration region. Conversely, in the high O concentration region of 0.026 mass% and more, the agglomeration force is smaller than the drag force that acts on Al₂O₃ inclusions with a diameter of 10 μm. In addition, the O concentration at which the agglomeration force exerted between the Al₂O₃ inclusions in the molten steel starts to become smaller than the drag force increases with the decrease in the diameter of the Al₂O₃ inclusions, and the concentration is about 0.045 mass% for very fine Al₂O₃ inclusions with a diameter of 1 μm. On the other hand, from Fig. 10, the reduction of the agglomeration force along with the increase of the S concentration in molten steel is relatively small, and even in the high S concentration region of 0.02 mass% or higher, a constant strong agglomeration force of 1.56d·σ_Fe is maintained. Furthermore, the agglomeration force acting between the Al₂O₃ inclusions in the molten steel is much greater than the drag force generated by the flow of the molten steel. As discussed above, from a relative comparison with the external force that acted on the Al₂O₃ inclusions, it is found that while the Al₂O₃ inclusions maintains their adhesion state due to a strong agglomeration force based on the cavity bridge force even when the concentration of S in the molten steel increases, the agglomeration force is markedly reduced when the concentration of O in the molten steel increases; therefore, even Al₂O₃ inclusions that have agglomerated to each other are more likely to separate once again by the flow of the molten steel. Note that these results are based on the agglomeration force measured in the O concentration range lower than 0.058 mass%, where the thermodynamically stable FeO·Al₂O₃ does not readily form.

5. Conclusions

The agglomeration force that acts between the Al₂O₃ cylinders in molten steel was measured while changing the concentration of O and S in the molten steel and was analyzed in combination with the surface tension formulated concerning the molten steel in which O and S coexisted and an Al₂O₃ interparticle interaction model based on the cavity bridge force. Through these examinations, the author has studied the effect of O and S in molten steel on the contact angle between the molten steel and the Al₂O₃ and also on the agglomeration property of the Al₂O₃ inclusions in the molten steel. The conclusions obtained are as follows:

(1) Both O and S in molten steel act as interfacial active elements and reduce the agglomeration force acting between the Al₂O₃ cylinders in the molten steel. However, the agglomeration force decreases more in the case of O compared with that of S.

(2) The contact angle between the Al₂O₃ and the molten steel obtained from the agglomeration force acting between the Al₂O₃ cylinders in the molten steel decreases as the amount of both O and S increases. However, the reduction ratio of the contact angle accompanying the increase in the concentration of O is much greater than that accompanying the increase in the concentration of S.

(3) Considering the advantage in the evaluation of the contact angle in this research, the interfacial active effect of O and S in molten steel on the contact angle between the molten steel and the Al₂O₃ is evaluated accurately as compared with the results measured by the conventional sessile drop method. Therefore, use of the contact angle obtained through the evaluation method in this research allows for an appropriate analysis of the dynamic agglomeration behavior of the Al₂O₃ inclusions in the molten steel while considering the effect of the interfacial active elements.

(4) Along with the increase in the concentration of O and S in the molten steel, the agglomeration property of the Al₂O₃ inclusions in the molten steel decreases, but the effect of O is very large compared with S.

(5) Even when the concentration of S in the molten steel increases, the Al₂O₃ inclusions remain adhered to each other due to the strong agglomeration force based on the cavity bridge force. However, as the agglomeration force decreases markedly when the concentration of O in the molten steel increases, even Al₂O₃ inclusions that have agglomerated once are likely to separate once again due to the molten steel flow.

Nomenclature

a: Interparticle surface distance (m)

a_O: Activity of O in molten steel (–)

a_S: Activity of S in molten steel (–)

C_D: Drag coefficient (–)

d: Diameter of the Al₂O₃ inclusion (m)

d_CY: Diameter of the Al₂O₃ cylinder (m)

${\text{D}}_{\text{CB},\text{Max}}^{\text{A}}$: Maximum cavity bridge length when the inclusions are approaching (m)

${\text{D}}_{\text{CB},\text{Max}}^{\text{S}}$: Maximum cavity bridge length when the inclusions are separated (m)

F_A: True agglomeration force between Al₂O₃ cylinders in molten steel (N·m⁻¹)

F_A,S: Agglomeration force that acts between two isospherical Al₂O₃ inclusions (N)

F_B: Buoyant force that acts on the Al₂O₃ inclusion in molten steel (N)

F_D: Drag force that acts on the Al₂O₃ inclusion in molten steel (N)

F_T,Max: Maximum traction force at the instant Al₂O₃ cylinders are separated from (N)

g: Acceleration of gravity (m·s⁻²)

K_O,S: Adsorption coefficient of oxygen at the surface of molten steel (–)

K_S,S: Adsorption coefficient of sulfur at the surface of molten steel (–)

L: Length of the Al₂O₃ cylinder (m)

L_D: Distance from the rotation axis to the bottom end of the Al₂O₃ cylinder (m)

L_U: Distance from the rotation axis to the mounting position of the force gauge (m)

r: Radius of the Al₂O₃ inclusion (m)

r_CY: Radius of the Al₂O₃ cylinder (m)

R: Gas constant (N·m·K⁻¹·mol⁻¹)

R₃: Curvature radius of cavity bridge (m)

R₄: Radius of the cavity bridge neck (m)

Re_P: Reynolds number of the particle (–)

S: The projected area of the inclusion particle toward the flow direction (m²)

T: Absolute temperature (K)

v: Flow velocity of molten steel (m·s⁻¹)

X₄: Half-width of the cavity bridge neck (m)

ΔP_Fe: Pressure difference between the cavity bridge and molten steel (Pa)

Γ_O,S: Surface excess of oxygen (mol·m⁻²)

${\mathrm{\Gamma }}_{\text{O},\text{S}}^{\text{S}}$: The saturated excess of oxygen at the surface of molten steel (mol·m⁻²)

${\mathrm{\Gamma }}_{\text{S},\text{S}}^{\text{S}}$: The saturated excess of sulfur at the surface of molten steel (mol·m⁻²)

ν: Kinematic viscosity coefficient of molten steel (m²·s⁻¹)

θ_Al2O3-Fe: Contact angle between molten steel and Al₂O₃ (°)

ρ_Fe: Density of molten steel (kg·m⁻³)

ρ_Al2O3: Density of the Al₂O₃ inclusion (kg·m⁻³)

σ_Al2O3: Surface tension of Al₂O₃ (N·m⁻¹)

σ_Al2O3-Fe: Interfacial tension between molten steel and Al₂O₃ (N·m⁻¹)

σ_Fe: Surface tension of molten steel (N·m⁻¹)

${\sigma }_{\text{Fe}}^{\text{P}}$: Surface tension of molten pure iron (N·m⁻¹)

References

• 1)   N.  Sano,  S.  Shiomi and  Y.  Matsushita: Tetsu-to-Hagané, 51 (1965), 19.
• 2)   S.  Taniguchi,  A.  Kikuchi,  T.  Ise and  N.  Shoji: ISIJ Int., 36 (1996), S117.
• 3)   S.  Taniguchi,  T.  Nakaoka and  K.  Matsumoto: Proc. Japan-Korea Workshop on Science and Technology in Ironmaking and Steelmaking, ISIJ, Tokyo, (2003), 61.
• 4)   K.  Okumura,  M.  Hirasawa,  M.  Sano,  K.  Mori,  N.  Hakamada and  M.  Kitazawa: Tetsu-to-Hagané, 80 (1994), 107.
• 5)   S.  Taniguchi and  A.  Kikuchi: Tetsu-to-Hagané, 78 (1992), 527.
• 6)   K.  Sasai: ISIJ Int., 54 (2014), 2780.
• 7)   K.  Sasai: ISIJ Int., 56 (2016), 1012.
• 8)   H.  Itoh,  M.  Hino and  S.  Ban-ya: Tetsu-to-Hagané, 83 (1997), 773.
• 9)   K.  Ogino,  K.  Nogi and  C.  Hosoi: Tetsu-to-Hagané, 69 (1983), 1989.
• 10)   K.  Ogino,  K.  Nogi and  Y.  Koshida: Tetsu-to-Hagané, 59 (1973), 1380.
• 11)   N.  Takiuchi,  T.  Taniguchi,  N.  Shinozaki and  K.  Mukai: J. Jpn. Inst. Met., 55 (1991), 44.
• 12)   N.  Takiuchi,  T.  Taniguchi,  Y.  Tanaka,  N.  Shinozaki and  K.  Mukai: J. Jpn. Inst. Met., 55 (1991), 180.
• 13)   K.  Nakashima,  K.  Takihira,  K.  Mori and  N.  Shinozaki: J. Jpn. Inst. Met., 55 (1991), 1199.
• 14)   B.  von Szyszkowski: Z. Phys. Chem., 64 (1908), 385.
• 15)  Steelmaking Data Sourcebook, Revised Ed., The 19th Committee on Steelmaking, The Japan Society for the Promotion of Science, Tokyo, (1984), 99, 33.
• 16)   A.  McLean and  R. G.  Ward: J. Iron Steel Inst., 204 (1966), 8.
• 17)   K.  Nogi and  K.  Ogino: Can. Metall. Q., 22 (1983), 19.