Agglomeration and Removal of Alumina Inclusions in Molten Steel with Controlled Concentrations of Interfacial Active Elements

Abstract

In this study, Al deoxidation experiments have been performed in a mildly stirred steel bath with controlled O and S concentrations, to investigate the effects of interfacial active elements on the agglomeration and removal of Al₂O₃ inclusions in molten steel. The decrease rate constants of total Al₂O₃ inclusions, Al₂O₃ cluster inclusions, and Al₂O₃ single inclusions as well as the maximum average diameter of Al₂O₃ cluster inclusions decrease with increasing O and S concentrations in molten steel. However, the effect of O is much greater than that of S. These experimental results have been analyzed based on the kinetics of Al₂O₃ inclusion removal and the interfacial chemical interaction between Al₂O₃ inclusions in molten steel. The following findings have been obtained on the agglomeration and removal mechanisms of Al₂O₃ inclusions in molten steel. The Al₂O₃ inclusions in molten steel are removed by a mechanism whereby large Al₂O₃ cluster inclusions, formed by Al deoxidation, float and separate while repeatedly agglomerating and coalescing with fine Al₂O₃ single inclusions suspended in molten steel. The agglomeration of Al₂O₃ inclusions during floating and separation can also be explained by a mechanism whereby the agglomeration force due to the cavity bridge force is exerted between the Al₂O₃ inclusions and the Al₂O₃ inclusions come in complete contact when the Al₂O₃ inclusions with thermodynamically agglomerating tendency are approaching each other. The effects of O and S interfacial active elements are considered in both these mechanisms.

1. Introduction

As the quality and properties required of products have advanced in recent years, the content of hitherto nonproblematic fine inclusions in products has become more strictly limited and the required level for higher cleanliness has become increasingly severe. Stirring of molten steel has been widely employed as an effective method for the separation and removal of inclusions in molten steel. However, this method has limited effectiveness and it cannot fully meet the increasingly severe inclusion requirements. Under these situations, it is essential to control inclusions consistently from the refining process to the casting process. That means promoting the agglomeration and coalescence to remove the fine inclusions from molten steel in the refining process, while restricting the further agglomeration and coalescence to prevent the inclusion building-up on the continuous casting nozzle caused the mold level fluctuations in the following casting process. Therefore, the elucidation of the agglomeration mechanism of alumina inclusions in molten steel, for not only fluid dynamic action but also interfacial chemical interaction, and the precise control of the agglomeration properties of inclusions in molten steel are expected based on the obtained scientific findings.

The author established a new experimental method for directly measuring the agglomeration force between the alumina particles in molten steel, analyzed the measured agglomeration force by considering the interfacial properties, and highlighted that a strong agglomeration force due to the cavity bridge force, which is created due to the difficulty of being wet with molten steel, acts between the alumina inclusions.^1,2) The author also quantitatively evaluated the effect of interfacial active elements on the agglomeration force using a new experimental method and clarified that both oxygen and sulfur in molten steel reduce the agglomeration force between the alumina inclusions, although the interfacial active effect of oxygen is much greater than that of sulfur.³⁾ These series of basic studies suggest that the alumina inclusions, formed in molten steel by aluminum deoxidation, agglomerate and coalesce into alumina clusters on the basis of the agglomeration force due to the cavity bridge force. This agglomeration property is assumed to be influenced by the interfacial properties through the agglomeration force.

To understand the agglomeration mechanism of alumina inclusions from the standpoint of interfacial chemical interaction, Al deoxidation experiments were performed in this study with the oxygen and sulfur concentrations in the molten steel controlled using a mildly stirred steel bath, while minimizing the effect of fluid dynamic action. By these experiments, the effect of the interfacial active elements on the agglomeration and removal of alumina inclusions in molten steel were investigated. The agglomeration and removal mechanisms of alumina inclusions in molten steel were studied by analyzing the experimental results based on the kinetics of alumina inclusion removal and the interfacial chemical interaction between alumina inclusions in molten steel.

2. Experimental Methods

2.1. Al Deoxidation Experiments of Molten Steel

In the experiment, an indirect resistance heating furnace comprising a high-frequency induction-heated (30 kW, 20 kHz) graphite cylinder was used to minimize the effect of molten steel flow on the agglomeration and removal of inclusions.^1,2,3) 500 g of electrolytic iron (C concentration = 0.001 mass%, S concentration = 0.0001 mass%, O concentration = 0.005 mass%, Mn concentration = 0.0001 mass%) was placed in an alumina crucible with 40 mm inner diameter and 150 mm height, and melted in an Ar gas atmosphere. The molten steel temperature was kept constant at 1600°C. Here, the stirring state close to a still bath, which is obtained under this melting condition, is referred to as a mildly stirred state.

The amount of Al added to deoxidize the 0.018 mass% O is 0.02 mass% from the stoichiometric ratio of the Al deoxidation reaction shown in Eq. (1).   

$$2\underset{\_}{\mathrm{A}\mathrm{l}}+3\underset{\_}{\mathrm{O}}=\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$$(1)

Strong and weak Al deoxidation experiments were performed based on this stoichiometric ratio to evaluate the effect of the O concentration on the agglomeration and removal behaviors of Al₂O₃ inclusions in molten steel. In the strong Al deoxidation experiment, 0.04–0.06 mass% excess Al was added to the molten steel, whose O concentration was adjusted to the target of 0.018 mass%. In the weak Al deoxidation experiment, 0.02 mass% Al was added to the molten steel, whose O concentration was adjusted to be between higher 0.022–0.044 mass%. The amounts of Al₂O₃ inclusions formed immediately after Al deoxidation were adjusted to the same level and the O concentration [O] during the experiment was changed in the range of 0.0006–0.0261 mass% in this way. In the strong Al deoxidation experiment, the S concentration [S] was changed in the range of 0.018–0.073 mass% to examine the effect of the S concentration on the agglomeration and removal behaviors of the Al₂O₃ inclusions in molten steel. In all the Al deoxidation experiments, molten steel was sampled with a transparent quartz tube of 6 mm inner diameter at 1 min after the Al addition, and this point in time was set as the start of the Al deoxidation experiment. The experiment duration was set to 10 min. Molten steel samples were taken at appropriate intervals during this time and analyzed for the total oxygen, Al, and S concentrations. To accurately grasp the Al₂O₃ inclusion content immediately after Al deoxidation, molten steel samples were also taken immediately before Al deoxidation and analyzed for the total oxygen concentration.

2.2. Microscopic Observation of Inclusions in Molten Steel

A 10 mm long cylindrical sample for inclusion observation was cut from the remainder of each sample analyzed for composition, was embedded in resin so that its circular cut section would become the observation surface, and was mirror polished. The inclusion observation position of the sample was basically near the center in the longitudinal direction. An optical microscope was used to observe the inclusions present in the cross section of 5 mm diameter, excluding the outermost circumference, at 100x magnification and the inclusions present in an area of 1–4 mm² at 1000x magnification. It is possible to detect cluster inclusions with a diameter of 10 μm or more for 100x observation and single inclusions with a diameter of 0.5 μm or more for 1000x observation. The particle size distribution of both inclusions was examined in this way. The particle size distribution of inclusions was measured on one cross section per molten steel sample in principle. When the detected number of cluster inclusions was extremely small, one more cross section was added as the observation surface using the remnant of the molten steel sample. The detected number of single inclusions was 52 or more per molten steel sample, and the detected number of cluster inclusions was 5 or more per molten steel sample. The average particle diameter and volume number density of both inclusions were calculated by DeHoff’s equation⁴⁾ from the obtained particle size distribution of single and cluster inclusions. The inclusion filling rate of cluster inclusions was evaluated from optical micrographs by an image analyzer. The composition analysis of inclusions in some of the molten steel samples was performed with an electron probe micro-analyzer (EPMA).

3. Experimental Results

3.1. Observation Results of Inclusions in Molten Steel

Figure 1 shows an optical micrograph of inclusions in the Al deoxidation experiment. According to 100x and 1000x observations with an optical microscope, the inclusions in the Al deoxidized molten steel were classified into single inclusions, independently present with a diameter of a few micrometers, and agglomerations of single inclusions or coarse cluster inclusions with a diameter of 10 μm or more. When these inclusions were analyzed with the EPMA, FeO·Al₂O₃ (hercynite) was observed at the outer periphery of some of the Al₂O₃ particles comprising the clusters only in the weak Al deoxidation experiment with a 0.0261 mass% O concentration. However, all other inclusions consisted of Al₂O₃. According to McLean and Ward,⁵⁾ the O concentration at which thermodynamically stable FeO·Al₂O₃ forms at 1600°C is 0.058 mass% or more. Thus, in this study, the inclusions present in the Al deoxidized molten steel (O ≤ 0.0261 mass%) are mainly Al₂O₃. The Al₂O₃ filling rate of the cluster inclusions was at an average of 24%, regardless of the O and S concentrations. Observations with the EPMA at higher magnification than the optical microscope showed that small spherical FeO particles with a diameter of less than 0.5 μm were present in the weak Al deoxidation experiment samples, and that fine FeS particles with a diameter of less than 0.5 μm were present in the strong Al deoxidation experiment samples containing S. The solubility of O and S in solid iron is very low; thus, these FeO and FeS particles are considered to be secondarily-formed inclusions precipitated during the solidification or cooling process after solidification. Therefore, it is confirmed that all Al₂O₃ inclusions with a diameter of 0.5 μm or more, which are detected by microscopic observation, do not contain secondarily-formed inclusions.

Fig. 1.

Optical micrograph of inclusions in the Al deoxidation experiment.

Figure 2 shows the changes in the average diameter d_CI and the volume number density N_V,C of Al₂O₃ cluster inclusions in molten steel with time. Figure 3 shows the changes in the average diameter d_SI and the volume number density N_V,S of Al₂O₃ single inclusions in molten steel with time. As shown in Fig. 2, d_CI is between 20 and 50 μm. It gradually increases after the start of the experiment, reaches the maximum value in about 4 min and then decreases. N_V,C is distributed in the range of 8–100 mm⁻³. It rapidly decreases up to about 4 min after the start of the experiment and then slowly decreases. However, Fig. 3 shows that d_SI hardly increases between 1.8 and 3.0 μm. N_V,S is in the range of 5000–100000 mm⁻³ and tends to decrease rapidly in the first half of the experiment and to slowly decrease thereafter like N_V,C.

Fig. 2.

Changes in the average diameter d_CI and the volume number density N_V,C of Al₂O₃ cluster inclusions in molten steel with time.

Fig. 3.

Changes in the average diameter d_SI and the volume number density N_V,S of Al₂O₃ single inclusions in molten steel with time.

To evaluate the agglomeration property of Al₂O₃ inclusions, the relation between the O and S concentrations in molten steel and the maximum average diameter d_CI,MAX of Al₂O₃ cluster inclusions in every experiment is organized and shown in Fig. 4. d_CI,MAX decreases with the increase of O and S concentrations in molten steel. The rate of decrease in d_CI,MAX is greater for the O concentration; thus, O reduces the agglomeration property of Al₂O₃ inclusions more significantly than S.

Fig. 4.

Relation between the O and S concentrations in molten steel and the maximum average diameter d_CI,Max of Al₂O₃ cluster inclusions.

3.2. Decrease Rate in Concentrations of Total Al₂O₃ Inclusions, Al₂O₃ Cluster Inclusions and Al₂O₃ Single Inclusions in Molten Steel

Figure 5 shows the typical changes in the Al concentration [Al] in molten steel with time in the strong Al deoxidation experiment. The Al concentration is almost constant during the experiment; thus, the reoxidation reaction of molten steel is considered to have not occurred either in the weak Al deoxidation experiment with the same atmosphere and crucible conditions as those of the strong Al deoxidation experiment.

Fig. 5.

Typical changes in the Al concentration [Al] in molten steel with time in the strong Al deoxidation experiment.

The O concentration during the experiment was calculated by using the thermodynamically reevaluated value of Al deoxidation equilibrium by Itoh et al.⁶⁾ from the analyzed Al concentration in the strong Al deoxidation experiments. It was also calculated by using both of the thermodynamically reevaluated value of Al deoxidation equilibrium, and of the mass balance on the basis of the O concentration before the Al addition and the amount of Al added in the weak Al deoxidation experiments. The oxygen concentration [I.O]_T of total Al₂O₃ inclusions is equivalent to the concentration of oxygen contained in all Al₂O₃ inclusions in molten steel and obtained by subtracting the O concentration during the experiment from the analytical value of the total oxygen concentration at each point of time. The oxygen concentration [I.O]_C of Al₂O₃ cluster inclusions and the oxygen concentration [I.O]_S of Al₂O₃ single inclusions in molten steel are respectively calculated from Eqs. (2) and (3) by using d_CI, d_SI, N_V,C and N_V,S obtained from microscopic observation of inclusions.   

$$\begin{array}{l}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}=100\cdot (4\pi /3)\cdot {({\mathrm{d}}_{\mathrm{C}\mathrm{I}}/2)}^3\mathrm{\hspace{0.17em} }\cdot \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{N}}_{\mathrm{V},\mathrm{C}}\cdot \epsilon \cdot 3{\mathrm{M}}_{\mathrm{O}}\cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/({\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\rho }_{\mathrm{F}\mathrm{e}})\end{array}$$(2)

  

$$\begin{array}{l}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}=100\cdot (4\pi /3)\cdot {({\mathrm{d}}_{\mathrm{S}\mathrm{I}}/2)}^3\cdot \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{N}}_{\mathrm{V},\mathrm{S}}\cdot 3{\mathrm{M}}_{\mathrm{O}}\cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/({\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\rho }_{\mathrm{F}\mathrm{e}})\end{array}$$(3)

where M_O is the atomic mass of O, M_Al2O3 is the molecular mass of Al₂O₃, ρ_Al2O3is the density of Al₂O₃ inclusions and 3970 kg·m⁻³, ρ_Fe is the density of steel and 7880 kg·m⁻³ at room temperature, and ε is the Al₂O₃ filling rate of Al₂O₃ cluster inclusions.

Figure 6 shows the relation between [I.O]_T and [I.O]_C + [I.O]_S. In the region with low oxygen concentration of total Al₂O₃ inclusions, [I.O]_T tends to slightly exceed [I.O]_C + [I.O]_S because the detection of Al₂O₃ cluster inclusions with an especially large diameter become difficult in the last half of the experiment where the number density of Al₂O₃ cluster inclusions in molten steel greatly decrease. However, the relation of [I.O]_T = [I.O]_C + [I.O]_S generally holds in all Al deoxidation experiments.

Fig. 6.

Relation between the oxygen concentration [I.O]_T of total Al₂O₃ inclusions and the summations [I.O]_C + [I.O]_S of the oxygen concentrations of Al₂O₃ cluster and single inclusions.

Figure 7 shows the changes in the logarithms of [I.O]_T, [I.O]_T–[I.O]_S, and [I.O]_S with time. Here, [I.O]_T–[I.O]_S (= [I.O]_C) with smaller variability than [I.O]_C is adopted to indicate the change in the oxygen concentration of Al₂O₃ cluster inclusions. The logarithms of [I.O]_T, [I.O]_T–[I.O]_S, and [I.O]_S decrease linearly with time. Then, each decrease rate constant is obtained by applying the first-order kinetics or Eqs. (4), (5), (6) to the changes in [I.O]_T, [I.O]_T–[I.O]_S, and [I.O]_S with time.   

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{T}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}$$(4)

  

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{C}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}$$(5)

  

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{S}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}$$(6)

where k_T is the decrease rate constant (s⁻¹) for all the Al₂O₃ inclusions, k_C is the decrease rate constant (s⁻¹) for the Al₂O₃ cluster inclusions, and k_S is the decrease rate constant (s⁻¹) for the Al₂O₃ single inclusions.

Fig. 7.

Changes in the logarithms of (a) [I.O]_T of total Al₂O₃ inclusions, (b) [I.O]_T–[I.O]_S of Al₂O₃ cluster inclusions, and (c) [I.O]_S of Al₂O₃ single inclusions with time.

Figure 8 shows the effects of O and S concentrations in molten steel on k_T, k_C, and k_S. As shown in Fig. 8, k_T, k_C, and k_S almost agree with each other. Their decrease rate constants decrease as the O and S concentrations increase; however, the decrease is significant with respect to the O concentration.

Fig. 8.

Effect of O and S concentrations in molten steel on the decrease rate constant k_T for total Al₂O₃ inclusions, the decrease rate constant k_C for Al₂O₃ cluster inclusions, and the decrease rate constant k_S for Al₂O₃ single inclusions.

4. Discussion

4.1. Validity of Present Experiments as Model Experiments Concerning Agglomeration and Removal of Al₂O₃ Inclusions in Molten Steel

When [I.O]_T, immediately after deoxidation, is obtained by subtracting the O concentration during the experiment from the total oxygen concentration before deoxidation in all of the present experiments, it is 0.0161–0.0177 mass% and its average value is 0.017 mass%. Okumura et al.^7,8) conducted experiments on the SiO₂ inclusion removal from molten Cu to slag under mechanical stirring and found that the agglomeration and coalescence of the inclusions were promoted and the rate of decrease in the inclusions increased with increasing initial oxygen concentration of all the inclusions. However, since [I.O]_T immediately after deoxidation is adjusted to about 0.017 mass% in the present experiments, there is no need to consider the effect of initial [I.O]_T on the agglomeration and removal of Al₂O₃ inclusions.

The Al deoxidation experiments were performed under mild stirring conditions using an indirect resistance heating furnace. This means that there is low possibility that the Al₂O₃ inclusions floated and removed to the molten steel surface would be engulfed again in the molten steel. As shown in Fig. 5, the Al concentration does not decrease during the experiment. Thus, it is evident that the formation reaction of new Al₂O₃ inclusions because of the reoxidation of molten steel does not proceed. As shown in Fig. 6, [I.O]_T obtained from the analytical value of the total oxygen concentration agrees with [I.O]_C + [I.O]_S obtained from the particle size distribution of inclusions in all of the Al deoxidation experiments. Thus, it is considered that the particle size distribution of primary deoxidation inclusions by morphology is evaluated with reasonable accuracy and that [I.O]_C, [I.O]_S, and [I.O]_T correspond to the Al₂O₃ cluster inclusion content, the Al₂O₃ single inclusion content and the total Al₂O₃ inclusion content, respectively.

According to the above considerations, the present experiments with properly controlled Al deoxidation conditions enable to focus only on the Al₂O₃ inclusions derived from primary deoxidation, and to grasp the agglomeration and removal behaviors of the Al₂O₃ inclusions, which are associated with the concentrations of the interfacial active elements O and S, from the changes in the total inclusion content and the contents of the inclusions by morphology with time.

4.2. Removal Mechanism of Al₂O₃ Inclusions in Molten Steel

In the Al deoxidation experiments, the Al₂O₃ inclusions agglomerate and float in a mildly stirred steel bath close to still molten steel. When the flotation distance within an experimental time of 10 min is calculated by Stokes law⁹⁾ (Eq. (14) given later) applicable to the flotation of inclusions in still molten steel by assuming a minimum particle diameter of 20 μm for Al₂O₃ cluster inclusions and a maximum particle diameter of 3 μm for Al₂O₃ single inclusions, it is 19.3 and 1.8 mm for the Al₂O₃ cluster inclusions and the Al₂O₃ single inclusions, respectively, with respect to the molten steel depth of 56.8 mm. This means that the Al₂O₃ single inclusions can hardly float in the present Al deoxidation experiments. Considering this, the experimental results of Figs. 2 and 3 can be explained as follows. In the initial stage of the experiment, Al₂O₃ cluster inclusions formed in molten steel agglomerate and coalesce with fine suspended Al₂O₃ single inclusions in the middle of floating, and thus d_CI increases and N_V,S decreases. Simultaneously, the Al₂O₃ cluster inclusions float and separate, and N_V,C also decreases. Thereafter, relatively large Al₂O₃ cluster inclusions quickly float and are preferentially removed. Thus, d_CI decreases after reaching the maximum particle diameter. N_V,C and N_V,S continue to slowly decrease. On the other hand, since the Al₂O₃ single inclusions are finely dispersed in molten steel, d_SI hardly increases within the experimental time.

From the above considerations, it is assumed as an Al₂O₃ inclusion removal model a mechanism whereby large Al₂O₃ cluster inclusions formed by Al deoxidation of molten steel float and separate while repeating agglomeration and coalescence with mainly suspended fine Al₂O₃ single inclusions. The rate at which new Al₂O₃ cluster inclusions are formed as fine Al₂O₃ single inclusions in molten steel agglomerate and coalesce with Al₂O₃ cluster inclusions varies in proportion to the concentration of single Al₂O₃ inclusions and is expressed by k_SC·[I.O]_S. Here, k_SC is the rate constant at which Al₂O₃ single inclusions agglomerate with Al₂O₃ cluster inclusions as well as the formation rate constant of Al₂O₃ cluster inclusions from Al₂O₃ single inclusions. The flotation and separation rate of Al₂O₃ cluster inclusions in molten steel varies in proportion to their concentration and is expressed as k_CF·[I.O]_C by using the flotation and separation rate constant k_CF. According to the above Al₂O₃ inclusion removal model, the decrease rate of Al₂O₃ cluster inclusions in molten steel is the difference between ① the flotation and separation rate of Al₂O₃ cluster inclusions and ② the formation rate of Al₂O₃ cluster inclusions from Al₂O₃ single inclusions and is given by Eq. (7).   

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{C}\mathrm{F}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}-{\mathrm{k}}_{\mathrm{S}\mathrm{C}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}$$(7)

This study quantitatively discusses the removal mechanism of Al₂O₃ inclusions in the molten steel by analyzing the experimental results based on the Al₂O₃ inclusion removal model expressed by Eq. (7).

4.2.1. Organization of Experimental Results Based on Al₂O₃ Inclusion Removal Model

As shown in Fig. 6, the total Al₂O₃ inclusion content in the molten steel is equal to the sum of the Al₂O₃ cluster and single inclusion content. Thus, Eq. (8) holds.   

$${[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}={[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}-{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}$$(8)

Concerning the decrease rate of Al₂O₃ cluster inclusions in molten steel, Eq. (9) is obtained when both sides of Eq. (8) are time differentiated, and d[I.O]_T/dt and d[I.O]_S/dt in the right side of Eq. (8) are respectively rewritten using Eqs. (4) and (6). The term α is the proportion of the oxygen concentration of Al₂O₃ cluster inclusions to the oxygen concentration of total Al₂O₃ inclusions and is given by Eq. (10).   

$$\begin{array}{l}-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{T}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}-{\mathrm{k}}_{\mathrm{S}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}={\mathrm{k}}_{\mathrm{T}}/\alpha \cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}-{\mathrm{k}}_{\mathrm{S}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}\end{array}$$(9)

  

$$\alpha ={[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}=1-{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}/{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}$$(10)

If the term α is considered to be independent of time and almost constant during the Al deoxidation experiment, k_CF and k_SC respectively become Eqs. (11) and (12) from the comparison of Eqs. (7) and (9).   

$${\mathrm{k}}_{\mathrm{C}\mathrm{F}}={\mathrm{k}}_{\mathrm{T}}/\alpha$$(11)

  

$${\mathrm{k}}_{\mathrm{S}\mathrm{C}}={\mathrm{k}}_{\mathrm{S}}$$(12)

Although the Al₂O₃ single inclusions in molten steel decrease according to Eq. (6), the decrease content k_S·[I.O]_S is equal to the newly formed Al₂O₃ cluster inclusion content k_SC·[I.O]_S from Eq. (12). When Eqs. (5) and (6) are rearranged into first-order kinetics concerning [I.O]_T by using the constant α and are each compared with Eq. (4), the relation of Eq. (13) is obtained.   

$${\mathrm{k}}_{\mathrm{T}}={\mathrm{k}}_{\mathrm{C}}={\mathrm{k}}_{\mathrm{S}}$$(13)

The relation of Eq. (13) holds in Fig. 8; thus, it is possible to regard α as constant during the Al deoxidation experiments. When the average value of α was examined at each point of time in all the experimental data by Eq. (10), it was 0.53 despite the passage of time.

Therefore, k_CF and k_SC in the Al₂O₃ inclusion removal model can be obtained by Eq. (11), in which α is 0.53, and Eq. (12) according to the experimental results of Fig. 8.

4.2.2. Flotation and Separation Rate Constant of Al₂O₃ Cluster Inclusions in Molten Steel

The flotation and separation rate constant of Al₂O₃ cluster inclusions in molten steel is estimated based on a simple inclusion flotation mechanism without agglomeration and coalescence as described below. It is assumed that Al₂O₃ cluster inclusions with the representative average diameter d_CI,R alone exist in molten steel and molten steel is completely mixed, but the stirring is mild and the Al₂O₃ cluster inclusions do not agglomerate and coalesce with each other. The flotation and separation rate of Al₂O₃ cluster inclusions is given by Eq. (15) if the flotation rate v_C (m·s⁻¹) of Al₂O₃ cluster inclusions in molten steel follows Stokes law of Eq. (14) because the value of d_CI,R is kept constant during the experiment.   

$${\mathrm{v}}_{\mathrm{C}}=2({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})\cdot {({\mathrm{d}}_{\mathrm{C}\mathrm{I},\mathrm{R}}/2)}^2\cdot \mathrm{g}/(9\mu )$$(14)

  

$$-\mathrm{d}{\mathrm{N}}_{\mathrm{V},\mathrm{C}}/\mathrm{d}\mathrm{t}=(\mathrm{A}/\mathrm{V})\cdot {\mathrm{v}}_{\mathrm{C}}\cdot {\mathrm{N}}_{\mathrm{V},\mathrm{C}}$$(15)

where ρ_Fe is 7000 kg·m⁻³ for molten steel, g is the acceleration of gravity (m·s⁻²), μ is the coefficient of viscosity of molten steel and 0.005 Pa·s, A is the surface area of molten steel (m²), V is the volume of molten steel (m³), and ρ_C is the density of Al₂O₃ cluster inclusions (kg·m⁻³) and expressed by Eq. (16) using the Al₂O₃ filling rate of Al₂O₃ cluster inclusions.   

$${\rho }_{\mathrm{C}}=\mathrm{\epsilon }\cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}+(1-\mathrm{\epsilon })\cdot {\rho }_{\mathrm{F}\mathrm{e}}$$(16)

When Eq. (15) is rearranged by deleting N_V,C using the relation of Eq. (2), Eq. (17) is obtained.   

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/\mathrm{d}\mathrm{t}=(\mathrm{A}/\mathrm{V})\cdot {\mathrm{v}}_{\mathrm{C}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}$$(17)

The real flotation and separation rate of Al₂O₃ cluster inclusions in molten steel is given by Eq. (18) or Eq. (7), from which the Al₂O₃ cluster inclusion formation term k_SC·[I.O]_S is deleted.   

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}/\mathrm{d}\mathrm{t}={\mathrm{k}}_{\mathrm{C}\mathrm{F}}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}}$$(18)

When Eq. (14) is applied to k_CF obtained by comparing Eqs. (18) and (17), Eq. (19) is obtained.   

$${\mathrm{k}}_{\mathrm{C}\mathrm{F}}=(\mathrm{A}/\mathrm{V})\cdot {\mathrm{v}}_{\mathrm{C}}=(\mathrm{A}/\mathrm{V})\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})\cdot \mathrm{g}\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I},\mathrm{R}}{}^2/(18\mu )$$(19)

Therefore, when d_CI,R for the Al₂O₃ cluster inclusions that really float and separate from molten steel is estimated and substituted into Eq. (19) with the experimental conditions, approximate k_CF can be evaluated. Additionally, since the decrease rate of Al₂O₃ cluster inclusions in molten steel is expressed by Eq. (5), the same equation as Eq. (19) can be also obtained for k_C from the comparison with Eq. (17). k_C is the apparent flotation rate constant in the case where both of the flotation and the formation of Al₂O₃ cluster inclusions occur simultaneously in molten steel. Therefore, when d_CI,R corresponding to the average particle diameter of Al₂O₃ cluster inclusions reflecting both effects is adopted, Eq. (19) is considered to give a reasonable value of k_C.

4.2.3. Agglomeration Rate Constant of Al₂O₃ Single Inclusions in Molten Steel

The agglomeration rate at which the Al₂O₃ single inclusions that are suspended in molten steel adhere to the floating Al₂O₃ cluster inclusions is formulated as change in the concentration of Al₂O₃ single inclusions in molten steel. It is assumed that Al₂O₃ cluster inclusions with the average diameter d_CI are uniformly dispersed in molten steel, which have the Al₂O₃ single inclusion content [I.O]_S, with the volume number density N_V,C at a certain time. In the case, the molten steel can be divided into small polyhedral elements where one large Al₂O₃ cluster inclusion is positioned at the center and fine Al₂O₃ single inclusions are suspended around it. When the small polyhedral elements are approximated by spherical elements of equal volume, the radius R_E (m) is given by Eq. (20).   

$${\mathrm{R}}_{\mathrm{E}}={\{3/(4\pi \cdot {\mathrm{N}}_{\mathrm{V},\mathrm{C}})\}}^{1/3}$$(20)

Assuming the same mechanism as that of mass transfer of O in molten steel, the agglomeration rate of Al₂O₃ single inclusions with an Al₂O₃ cluster inclusion is expressed by Eq. (21) using the oxygen concentration of Al₂O₃ single inclusions.¹⁰⁾   

$$\mathrm{d}{\mathrm{n}}_{\mathrm{O}}/\mathrm{d}\mathrm{t}={\mathrm{A}}_{\mathrm{C}}\cdot {\mathrm{k}}_{\mathrm{m}}\cdot \{{\rho }_{\mathrm{F}\mathrm{e}}/(100{\mathrm{M}}_{\mathrm{O}})\}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}$$(21)

where n_O is the number of moles (mol) of oxygen atoms contained in Al₂O₃ single inclusions adhering to one Al₂O₃ cluster inclusion, k_m is the molten steel-side mass transfer coefficient (m·s⁻¹) of Al₂O₃ single inclusions to the Al₂O₃ cluster inclusion floating through molten steel, and A_C is the surface area (m²) of the Al₂O₃ cluster inclusion and 4π·(d_CI/2)². The agglomeration rate of the Al₂O₃ single inclusions with an Al₂O₃ cluster inclusion is equal to the decrease rate of Al₂O₃ single inclusions in the spherical element of molten steel and given by Eq. (22).   

$$\mathrm{d}{\mathrm{n}}_{\mathrm{O}}/\mathrm{d}\mathrm{t}=-{\mathrm{V}}_{\mathrm{E}}\cdot \{{\rho }_{\mathrm{F}\mathrm{e}}/(100{\mathrm{M}}_{\mathrm{O}})\}\cdot (\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}/\mathrm{d}\mathrm{t})$$(22)

where V_E is the volume (m³) of the spherical element and 4/3π·R_E³. The right-hand sides of Eqs. (21) and (22) are equal; thus, the decrease rate of Al₂O₃ single inclusions in molten steel is given by Eq. (23).   

$$-\mathrm{d}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}/\mathrm{d}\mathrm{t}=({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\cdot {\mathrm{k}}_{\mathrm{m}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{S}}$$(23)

As evident from its derivation process, Eq. (23) can predict only the change in [I.O]_S in such a short time that both A_C and V_E are regarded as almost constant. The specific surface area A_C·V_E⁻¹ is given by Eq. (24). As the coarsening and the floating separation of the Al₂O₃ cluster inclusions proceed at the same time, Eq. (23) can be applied to the present experiments if the change in A_C·V_E⁻¹ during the experimental time is sufficiently small to be regarded as a constant.   

$${\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}}=\pi \cdot {\mathrm{N}}_{\mathrm{V},\mathrm{C}}\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I}}{}^2$$(24)

In that case, when Eqs. (6) and (23) are compared by considering Eq. (12), the agglomeration rate constant of Al₂O₃ single inclusions is given by Eq. (25).   

$${\mathrm{k}}_{\mathrm{S}\mathrm{C}}=({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\cdot {\mathrm{k}}_{\mathrm{m}}$$(25)

When the penetration theory is applied to the mass transfer of Al₂O₃ single inclusions in molten steel, k_m is given by Eq. (26).   

$${\mathrm{k}}_{\mathrm{m}}=2{\pi }^{-1/2}\cdot {({\mathrm{D}}_{\mathrm{E}\mathrm{q}}\cdot {\mathrm{v}}_{\mathrm{C}}/{\mathrm{d}}_{\mathrm{C}\mathrm{I},\mathrm{M}\mathrm{a}\mathrm{x}})}^{1/2}$$(26)

where D_Eq is the equivalent diffusion coefficient (m²·s⁻¹) of Al₂O₃ single inclusions, and d_CI,Max is used as the representative average diameter to calculate k_m because it best represents the agglomeration property of Al₂O₃ inclusions in each experiment. When Eq. (25) is rearranged with the application of Eqs. (26) and (14), k_SC is ultimately given by Eq. (27).   

$${\mathrm{k}}_{\mathrm{S}\mathrm{C}}=2({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\cdot {\{{\mathrm{D}}_{\mathrm{E}\mathrm{q}}\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})\cdot \mathrm{g}\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I},\mathrm{M}\mathrm{a}\mathrm{x}}/(18\pi \cdot \mu )\}}^{1/2}$$(27)

The above discussions indicate that k_SC can be approximately evaluated by applying appropriate values of A_C·V_E⁻¹ and D_Eq to Eq. (27).

4.2.4. Study on Removal Mechanism of Al₂O₃ Inclusions in Molten Steel

Figure 9 shows the effect of d_CI,R on k_CF and k_C obtained experimentally. The average particle diameters d_CI,Max15, d_CI,Max20, and d_CI,Max25 were recalculated by DeHoff’s equation⁴⁾ for cluster inclusions with diameters of 15 μm or more, 20 μm or more, and 25 μm or more in the particle size distribution of d_CI,Max, respectively. The triangles (△), open circles (○) and squares (□) indicate the experimental values of k_CF when d_CI,Max15, d_CI,Max20, and d_CI,Max25 are plotted as d_CI,R, respectively. The diamonds (◇) indicate the experimental values of k_C when d_CI,Max are plotted as d_CI,R, and the solid line indicates the values calculated by Eq. (19). The experimental values and calculated values of k_C agree well with each other. Thus, it is confirmed that the flotation and separation rate constant of the target Al₂O₃ cluster inclusions in molten steel can be estimated from Eq. (19) by applying the appropriate d_CI,R. The calculated values of k_CF by Eq. (19) are evaluated smaller than the experimental values when d_CI,Max15 is taken as d_CI,R and evaluated greater than the experimental values when d_CI,Max25 is taken as d_CI,R. However, the calculated values best duplicate the experimental values when the intermediate value d_CI,Max20 is taken as d_CI,R. These results indicate that the particle diameter of Al₂O₃ cluster inclusions, which practically floated and separated from molten steel, is 20 μm or more and that the experimentally obtained value of k_CF is a reasonable value which is theoretically predicted from Eq. (19) based on the average diameter d_CI,Max20 for Al₂O₃ cluster inclusions with a diameter of 20 μm or more.

Fig. 9.

Effect of the representative average diameter d_CI,R of Al₂O₃ cluster inclusions on the flotation and separation rate constant k_CF and the decrease rate constant k_C for Al₂O₃ cluster inclusions.

The relation between A_C·V_E⁻¹, which is calculated by Eq. (24) from d_CI and N_V,C obtained by optical microscopic observation, and the detected number of Al₂O₃ cluster inclusions is distinguished based on the elapsed time from the start of the experiment and is shown in Fig. 10. A_C·V_E⁻¹ until 4 min from the start of the experiment does not depend on the detected number of Al₂O₃ cluster inclusions and is distributed within a certain range with an average value of 195 m⁻¹. A_C·V_E⁻¹ after 6 min from the start of the experiment decreases as the detected number of Al₂O₃ cluster inclusions decreases when the detected number of Al₂O₃ cluster inclusions is less than 25. The above results suggest that since the number density of Al₂O₃ cluster inclusions in molten steel after 6 min or more greatly decreases, the detection of Al₂O₃ cluster inclusions with an especially large diameter becomes difficult and A_C·V_E⁻¹ is underestimated as the detected number decreases. Therefore, in this study, the decrease in A_C·V_E⁻¹ is considered to be relatively small even in the latter half of the experiment and the average value of 195 m⁻¹ until 4 min from the start of the experiment is adopted as A_C·V_E⁻¹ during the experiment. This means that A_C·V_E⁻¹ during the experimental time can be regarded as almost constant because the coarsening and the floating separation of the Al₂O₃ cluster inclusions proceed at the same time, as is clear from Eq. (24).

Fig. 10.

Relation between the specific surface area A_C·V_E⁻¹, obtained by optical microscopic observation of Al₂O₃ inclusions, and the detected number of Al₂O₃ cluster inclusions.

Figure 11 shows the experimentally obtained relation between k_SC and d_CI,Max. The equivalent diffusion coefficient of fine Al₂O₃ particles suspended in molten steel is unknown. For this reason, the diffusion coefficient 2.3 × 10⁻⁹ m²·s^{−1 11)} of O in molten steel and the Brown diffusion coefficient 2.3 × 10⁻¹³ m²·s⁻¹ of fine particles, which is expressed by the Stokes-Einstein Eq. (28), are assumed as two extreme cases. The values of k_SC in the two extreme cases calculated by Eq. (27) are respectively represented by a dotted line and a dashed-dotted line in Fig. 11.   

$${\mathrm{D}}_{\mathrm{E}\mathrm{q}}={\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T}/(3\pi \cdot \mu \cdot {\mathrm{d}}_{\mathrm{S}\mathrm{I}})$$(28)

where k_B is the Boltzmann constant and T is the absolute temperature. Because the average diameter of Al₂O₃ single inclusions ranged from 1.8 to 3.0 μm, the intermediate value of 2.4 μm was used as d_SI in Eq. (28) to estimate the Brown diffusion coefficient. The experimental values of k_SC lie between both of the agglomeration rate constants calculated by supposing the diffusion of the solute O and the Brown diffusion of fine particles, and agree with the values (indicated by the solid line) obtained from Eq. (27) by using the intermediate value 3.3 × 10⁻¹¹ m²·s⁻¹ of both diffusion coefficients as D_Eq. Thus, the experimentally obtained values of k_SC are reasonable results as predicted by Eq. (27) derived by assuming a mechanism similar to the mass transfer of solute elements for the agglomeration of Al₂O₃ single inclusions with Al₂O₃ cluster inclusions.

Fig. 11.

Relation between the agglomeration rate constant k_SC of Al₂O₃ single inclusions and the maximum average diameter d_CI,Max of Al₂O₃ cluster inclusions.

As discussed earlier, the decrease rate of Al₂O₃ cluster inclusions in molten steel can be appropriately represented by Eq. (7). Thus, in this study, it is evident that the removal of Al₂O₃ inclusions from the mildly stirred molten steel can be reasonably explained by the mechanism whereby large Al₂O₃ cluster inclusions formed by Al deoxidation float and separate while repeatedly agglomerating and coalescing with fine Al₂O₃ single inclusions suspended in molten steel.

4.3. Agglomeration Mechanism of Al₂O₃ Inclusions in Molten Steel

4.3.1. Agglomeration Force Due to Cavity Bridge Force between Al₂O₃ Inclusions in Molten Steel

In order to organize the relation between the agglomeration property and the agglomeration force of Al₂O₃ inclusions, the agglomeration force due to the cavity bridge force acting between Al₂O₃ single inclusions in molten steel^1,2,3) is evaluated below. The agglomeration force F_A,S (N) acting between two isospherical Al₂O₃ inclusions in contact with each other while forming a cavity bridge in molten steel is given by Eq. (29) as the sum of the pressure difference ΔP_Fe (Pa) between the cavity bridge and molten steel, and a force due to the surface tension σ_Fe (N·m^-1) of molten steel.   

$${\mathrm{F}}_{\mathrm{A},\mathrm{S}}=\pi \cdot {\mathrm{R}}_4{}^2\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}+2\pi \cdot {\mathrm{R}}_4\cdot {\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}$$(29)

where R₄ is the radius (m) of the cavity bridge neck and can be calculated from Eq. (30).   

$$\begin{array}{l}{\mathrm{R}}_4=\{-3{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}+{(9{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}{}^2-8{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{r}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}})}^{0.5}\}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}/(2\mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}})\end{array}$$(30)

where ΔP_Fe is 3.86 × 10³ Pa, and r is the radius (m) of Al₂O₃ inclusions and set at 1 μm assuming a single inclusion. θ_Al2O3-Fe is the contact angle between molten steel and Al₂O₃, and given by Eq. (31).   

$${\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}((4{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}{}^2-{\mathrm{F}}_{\mathrm{A}}{}^2)/(8{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{\Delta }{\mathrm{P}}_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{r}}_{\mathrm{C}\mathrm{Y}}))$$(31)

where F_A is the measured value (N·m⁻¹) of the agglomeration force acting between the Al₂O₃ cylinders, and r_CY is the radius (m) of the Al₂O₃ cylinders in the measurement of the agglomeration force and 4 mm.³⁾ The effects of the O and S concentrations on the surface tension of molten steel are formulated by Eq. (32).³⁾   

$${\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}=1.94-0.291\cdot \mathrm{l}\mathrm{n}(1+237{\mathrm{a}}_{\mathrm{O}})-0.235\cdot \mathrm{l}\mathrm{n}(1+185{\mathrm{a}}_{\mathrm{S}})$$(32)

where a_O is the activity of O and a_S is the activity of S.

The agglomeration force acting between Al₂O₃ inclusions with a particle radius of 1 μm in molten steel thus can be obtained from the measured value of F_A by calculating R₄ from Eq. (30) using σ_Fe in Eq. (32) and θ_Al2O3-Fe in Eq. (31), and by substituting the obtained value of R₄ in Eq. (29).

4.3.2. Free Energy Change with Agglomeration of Al₂O₃ Inclusions

The free energy change ΔG_Ag (J·m⁻²) with the agglomeration of Al₂O₃ inclusions themselves in molten steel can be used as an indicator of the agglomeration tendency when the agglomeration mechanism of Al₂O₃ inclusions is discussed on the basis of thermodynamics. ΔG_Ag for two lamellar Al₂O₃ inclusions is given by Eq. (33).¹²⁾   

$$\mathrm{\Delta }{\mathrm{G}}_{\mathrm{A}\mathrm{g}}={\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}-2{\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$$(33)

where σ_Al2O3-Fe is the interfacial tension (N·m⁻¹) between the molten steel and Al₂O₃ inclusions, and σ_Al2O3-Al2O3 (N·m⁻¹) is the interfacial tension between Al₂O₃ inclusions, which is given by Eq. (34) using the surface tension σ_Al2O3 (N·m⁻¹) and the dihedral angle ϕ between Al₂O₃ inclusions.   

$${\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}=2{\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot \mathrm{c}\mathrm{o}\mathrm{s}(\varphi /2)$$(34)

When Eq. (33) is rewritten by using Eq. (34) and Young’s equation or Eq. (35), Eq. (36) is obtained.   

$${\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}={\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}-{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$$(35)

  

$$\mathrm{\Delta }{\mathrm{G}}_{\mathrm{A}\mathrm{g}}=2{\mathrm{\sigma }}_{\mathrm{F}\mathrm{e}}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}-2{\mathrm{\sigma }}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}(1-\mathrm{c}\mathrm{o}\mathrm{s}(\varphi /2))$$(36)

where σ_Fe and θ_Al2O3-Fe are given from Eqs. (32) and (31), respectively. σ_Al2O3 is 0.75 N·m^{−1 13)} and ϕ is 150°¹⁴⁾ between oxides of the same sort. ΔG_Ag can be calculated according to the O and S concentrations by substituting these values into Eq. (36).

4.3.3. Study of Agglomeration Mechanism of Al₂O₃ Inclusions in Molten Steel

In a system where fine Al₂O₃ single inclusions are dispersed in molten steel, the interfacial area between the Al₂O₃ inclusions and molten steel is large and the interfacial tension between Al₂O₃ inclusions and molten steel is also large. Therefore, the free energy of the entire system lies in a very high state. The Al₂O₃ single inclusions in molten steel are considered to have a strong agglomeration tendency because the entire system intends to change toward a lower free energy state according to thermodynamics. Therefore, the effect of ΔG_Ag on both d_CI,Max and k_SC indicating the agglomeration properties is organized and is shown in Fig. 12. Irrespective of whether the interfacial active element is O or S, d_CI,Max and k_SC increase linearly as ΔG_Ag decreases. It is evident that the more the free energy of the system decreases with agglomeration, the stronger the Al₂O₃ inclusions in molten steel exhibit the agglomeration tendency. Therefore, the agglomeration property of Al₂O₃ inclusions in molten steel can be well explained from a macroscopic perspective based on thermodynamics.

Fig. 12.

Effect of the free energy change ΔG_Ag with agglomeration of Al₂O₃ inclusion on the maximum average diameter d_CI,Max of Al₂O₃ cluster inclusions and the agglomeration rate constant k_SC of Al₂O₃ single inclusions.

However, since thermodynamics explains only the tendency of the system to change toward a lower free energy state, that is, the agglomeration tendency, it is difficult to elucidate a mechanism whereby the agglomeration of Al₂O₃ inclusions actually occurs from thermodynamics alone. For this reason, it is also important to study from a microscopic perspective based on the interaction between particles. The author clarified that the agglomeration force due to the cavity bridge force, which is much stronger than the van der Waals force in molten steel and the capillary force on the molten steel surface, acts between the Al₂O₃ inclusions in molten steel.²⁾ The author also clarified that once the Al₂O₃ inclusions come into contact with each other, they maintain their agglomeration state without being easily separated even under a molten steel flow because the agglomeration force is larger than the buoyancy and drag acting on the Al₂O₃ inclusions.¹⁾ Focusing on the cavity bridge force as interaction between particles in the agglomeration of Al₂O₃ inclusions, the effects of F_A,S on d_CI,Max and k_SC are organized and are shown in Fig. 13. d_CI,Max and k_SC increase in proportion to F_A,S, irrespective of the type of interfacial active elements of O and S, and are organized by a linear relation. This is explained as follows. As the agglomeration force acting between an Al₂O₃ single inclusion suspended in molten steel and an Al₂O₃ single inclusion comprising the cluster inclusions floating through molten steel increases, the agglomeration rate increases. Consequently, the particle diameter of the Al₂O₃ cluster inclusions increases.

Fig. 13.

Effect of the agglomeration force F_A,S between two isospherical Al₂O₃ inclusions of 1 μm radius on the maximum average diameter d_CI,Max of Al₂O₃ cluster inclusions and the agglomeration rate constant k_SC of Al₂O₃ single inclusions.

As the Al₂O₃ inclusions with thermodynamically agglomerating tendency approach each other, the agglomeration force due to the cavity bridge force acts between the inclusions. Therefore, the Al₂O₃ inclusions attract and contact each other strongly. This mechanism explains the agglomeration of the Al₂O₃ inclusions in the mildly stirred steel bath of this study in a unified manner, including the effect of the interfacial active elements. In addition, it is difficult for a pair of Al₂O₃ inclusions approaching each other in molten steel to contact each other only by fluid dynamic interaction while overcoming the viscosity resistance of the molten steel.¹⁵⁾ Considering that the agglomeration force due to the cavity bridge force is a very strong attractive force, the agglomeration force due to the cavity bridge force is considered to play an important role concerning the agglomeration of Al₂O₃ inclusions, even in a strongly stirred molten steel.

5. Conclusions

The author performed Al deoxidation experiments with a mildly stirred steel bath by controlling O and S concentrations in molten steel, studied the agglomeration and removal mechanisms of Al₂O₃ inclusions in molten steel based on the experimental results, and obtained the following conclusions:

(1) In the Al deoxidation experiments of mildly stirred molten steel, the initial inclusion content was controlled constant, the reoxidation of molten steel was prevented, and the secondarily formed inclusions were excluded. Thus, according to the experimental results in this study, the agglomeration and removal of Al₂O₃ inclusions originating from primary deoxidation can be kinetically treated in consideration of the effects of the interfacial active elements O and S.

(2) The decrease rate constants of total Al₂O₃ inclusions, Al₂O₃ cluster inclusions and Al₂O₃ single inclusions in molten steel as well as the maximum average diameter of the Al₂O₃ cluster inclusions decrease as the O and S concentrations in molten steel increase. However, the effect of O is greater than that of S.

(3) The removal of Al₂O₃ inclusions in molten steel can be reasonably explained by a mechanism whereby large Al₂O₃ cluster inclusions, formed by Al deoxidation, float and separate while repeatedly agglomerating and coalescing with fine Al₂O₃ single inclusions suspended in the molten steel.

(4) When Al₂O₃ inclusions with thermodynamically agglomerating tendency approach each other in molten steel, the agglomeration force due to the cavity bridge force acts between the Al₂O₃ inclusions. Therefore, the Al₂O₃ inclusions strongly attract and contact each other. This mechanism explains the agglomeration of the Al₂O₃ inclusions in molten steel in a unified manner, including the effects of the interfacial active elements of O and S.

(5) It is difficult for a pair of Al₂O₃ inclusions to approach and contact each other in molten steel with fluid dynamic interaction alone while overcoming the viscosity resistance of the molten steel. Considering that the agglomeration force due to the cavity bridge force is a very strong attractive force, the agglomeration force due to the cavity bridge force plays an important role in the agglomeration and coalescence of the Al₂O₃ inclusions, even in strongly stirred molten steel.

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