A Reaction between High Mn–High Al Steel and CaO–SiO₂-Type Molten Mold Flux: Reaction Mechanism Change by High Al Content ([pct Al]₀ = 5.2) in the Steel and Accumulation of Reaction Product at the Reaction Interface

Abstract

A series of laboratory-scale experiments were carried out in order to elucidate the reaction mechanism between high Mn-high Al steel and CaO–SiO₂-type molten mold flux at 1450°C, which represents the reaction taking place during continuous casting of the steel. Compared to the previous study [Kim et al., Metall. Mater. Trans. 44B (2013) 299–308], high Al content in the liquid steel ([pct Al]₀ = 5.2) and high MgO content in the liquid flux ((pct MgO)₀ = 5 to 15) were employed, in order to confirm change of rate-controlling step from mass transport of Al in liquid steel to more complicated steps including mass transport in liquid flux. It was found that Al₂O₃ was rapidly accumulated near the interface of the flux, and SiO₂ and Na₂O were reduced simultaneously, regardless of (pct MgO)₀. At the early stage of the reaction (1 min), MgAl₂O₄ particles were observed in the flux near the interface, then the particles were spreading out into the bulk flux as the reaction time passed. Other solid phases (CaAl₄O₇, Al₂O₃) were also observed due to local depletion of MgO in the flux. The MgAl₂O₄ formation mechanism and its effect on mass transfer in the molten flux were discussed. A series of simple kinetic analyses showed that the mass transport of Al in liquid steel is no more controlling the reaction rate. It was concluded that there were possibilities of mass transport in the flux phase contributing reaction rate controlling step.

1. Introduction

High performance light weight steel has been developed, which consists of high Mn and high Al contents. Continuous casting of this steel grade inevitably involves reaction between the high Mn-high Al steel and CaO–SiO₂ type molten mold flux in the casting mold. The present authors investigated a reaction mechanism between the high Mn-high Al steel and the CaO–SiO₂ type molten mold flux, and it was shown that the following reaction was mostly taking place:^1,2)   

$$4\underset{\_}{\mathrm{A}\mathrm{l}}+3(\mathrm{S}\mathrm{i}{\mathrm{O}}_2)=3\underset{\_}{\mathrm{S}\mathrm{i}}+2(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)$$(1)

Additional oxidation and reduction of Fe_tO and MnO were observed.¹⁾ Employed flux was composed of CaO–SiO₂–Al₂O₃–MgO–Na₂O–F of various (pct CaO)/(pct SiO₂) ratios, (pct Al₂O₃).¹⁾ Reaction rate between the steel and the flux was limited by mass transport of Al in the liquid steel, when the initial content of Al, [pct Al]₀, was not higher than 1.8, in the temperature range of 1440°C to 1550°C. It was found that the rate controlling step was no more valid when [pct Al]₀ was 4.8. Moreover, the flux after the reaction was no more homogeneous, but MgAl₂O₄ solid phases were observed near interface between the steel and the flux. This observation was also reported by Kim and Park for reaction between Fe–Mn–Al alloy and CaO–MgO–SiO₂–Al₂O₃ slag at 1600°C.³⁾ The reaction product (MgAl₂O₄) accumulated at the interface, and it was thought that it may retard the reaction between the steel and the flux. In the previous research, only experimental data of low Al content was utilized in the development of rate equation in order to predict composition evolution in the mold flux during the continuous casting.²⁾

Accumulation of the reaction product MgAl₂O₄ is due to the high Al content in the steel and the MgO in the flux. As new grade of lightweight steel requires high Al content as high as ~10 mass pct.,⁴⁾ and MgO is generally introduced as impurities from raw materials, it is likely that rapid reaction between Al in liquid steel and SiO₂ in the flux takes place, followed by formation of MgAl₂O₄ or other solid phases. This influences composition of remained mold flux, and may change transport property of the liquid flux by changing viscosity. Then it results in shifting rate controlling step from mass transport in liquid steel to mass transport in liquid flux. Previous investigations on SiO₂ reduction by Al in liquid Fe are summarized in Table 1.^{1,2,3,5,6,7,8)} Some including the present authors reported that mass transport of Al in liquid steel was rate controlling step,^2,5,6) while others reported chemical reaction⁷⁾ or mixed control (mass transport in both phases).⁸⁾ It is thought that the rate controlling step during the reaction may not be permanent during the reaction. Depending on the flux chemistry and the reaction product, the rate controlling step would shift to mass transport in the liquid flux.

Table 1. Summary of kinetic studies on SiO₂ reduction in molten slags/fluxes by Al in liquid Fe.

Author	Ooi et al.7)	Sun and Mori5)	Rhamdhani et al.6)	Kim and Park3)	Park et al.8)	Kim et al.1,2)
Steel comp. (mass pct)	Fe-4Al	Fe-(0.25-0.4)Al	Fe-(3.5-5)Al	Fe-(1,3,6)Al-(10-20)Mn	Fe-(0.057-0.45)Al	Fe-13Mn-(0.4-4.8)Al-(0.4-1.7)Si-0.65C
Slag comp. (mass pct.)	40CaO-40SiO2-20Al2O3	CaO-Al2O3-(5,10)SiO2-FeO-MnO (C/A = 1)	40CaO-40SiO2-20Al2O3	30CaO-60SiO2-5Al2O3-5MgO	34CaO-5SiO2-47Al2O3-14MgO	(17-37)CaO-(31-53)SiO2-(0-12)Al2O3-2.5MgO-14Na2O-7.7F
Temperature (°C)	1570	1600	1550–1650	1600	1600	1440–1550
Experimental method	Steel droplet in liquid slag pool	Slag injection on liquid steel pool	Steel droplet in liquid slag pool	Slag injection on liquid steel pool	Slag injection on liquid steel pool	Slag injection on liquid steel pool
Rate controlling step	SiO2 reduction at steel-slag interface	Mass transfer of Al in liquid steel	Mass transfer of Al in liquid steel	–	Mixed transport control	Mass transfer of Al in molten steel ([pct Al]0 ≤ 1.8)
Mass transfer coefficient of Al (m sec−1)	–	2.2–8.1 × 10−4,*	1.3–1.9 × 10−6	3.6 × 10−4–1.5 × 10−2	6 × 10−5	$exp\left(-\frac{14\mathrm{\hspace{0.17em} }200}{T}-1.1107\right)$
Activation energy (kJ mol−1)	–	–	127	–	–	119
Steel & slag comp. range	High Al High SiO2	Low Al Low SiO2	High Al High SiO2	Low-high Al High SiO2	Low Al Low SiO2	Low-high Al High SiO2

*  Calculated by the present authors from $k_{\mathrm{A}\mathrm{l}}^m{\rho }_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}$ with assumption (ρ_steel = 6800 kg m⁻³).

Since reduction of SiO₂ and accumulation of Al₂O₃ in the flux during continuous casting deteriorate cast product quality and process efficiency, prevention of the reaction is required.^9,10) Understanding reaction mechanism between the high Mn-high Al steel and CaO–SiO₂ type mold flux is therefore important, and prediction of the evolution of flux composition would be useful, which may further be utilized to assess functions of the mold flux (crystallization, fluidity, heat transfer, lubrication, etc.). The present article is one of continuing contributions of the present authors in order to understand reactions between the high Mn-high Al steel and CaO–SiO₂ type molten mold flux. Experimental set up and general reaction phenomena for relatively lower [pct Al]₀ (less than 1.8) was presented in Reference 1). Elucidation of reaction mechanism, interfacial morphology, and a simply kinetic model was discussed in Reference 2). The kinetic model was developed in the regime of liquid steel mass transport control due to the relatively lower [pct Al]₀. In the present article, it is shown how the rate controlling step is changed by accumulation of reaction product at the interface. This is induced by higher [pct Al]₀ in the steel and higher (pct MgO)₀ in the flux. All these results can be utilized in order to develop a general reaction model which takes into account reaction kinetics controlled by mass transport in both phases (liquid steel and molten mold flux) and effect of changing the flux composition on mass transfer coefficient.

2. Experimental

2.1. Sample Preparation

Steel sample was prepared by weighing desired amounts of electrolytic iron (99.99 pct, Toho Zinc Co. Ltd., Tokyo, Japan), manganese chip (99.99 pct, LTS Chemical Inc., Orangeburg, NY, USA), aluminum granule (99.99 pct, Kojundo Chemical Co., Saitama, Japan). Fe–C alloy ([pct C] = 5) was prepared separately by melting the electrolytic iron and C powder in a graphite crucible in an induction melting furnace. Pre-fused fluxes with three different (pct MgO)₀ (= 5, 10, 15) were prepared by weighing CaCO₃ (98.0 pct, Kanto chemicals, Tokyo, Japan), SiO₂ (extra pure, Kanto chemicals, Tokyo, Japan), Al₂O₃ (99.0 pct, Kanto chemicals, Tokyo, Japan), and MgO (98.0 pct, Kanto chemicals, Tokyo, Japan) in desired proportions, melting in a graphite crucible in an induction furnace. The liquid flux was then quenched on a stainless steel plate cooled by water, and it was ground into powder. CaF₂ (98.0 pct, Kanto chemicals, Tokyo, Japan) and Na₂CO₃ (99.5 pct, Kanto chemicals, Tokyo, Japan) were then mixed with the pre-fused flux powder. Initial compositions of the steel and the flux samples were given in Table 2.

Table 2. Initial compositions of steel and flux samples used in the present study (mass pct).

Fe	C	Al	Si	Mn
Bal.	1.00	5.17	0.024	20.76

Sample	CaO	SiO2	Al2O3	MgO	Na2O	F
F-5M	32.48	34.51	5.97	5.00	14.17	7.86
F-10M	27.48	34.51	5.97	10.00	14.17	7.86
F-15M	22.48	34.51	5.97	15.00	14.17	7.86

2.2. Sampling Experiment

In order to measure composition change during reaction between molten steel and molten flux, a series of sampling experiment were carried out. Schematic diagram for sampling experiments is shown in Fig. 1(a). 400 g of the prepared steel sample was charged in an Al₂O₃ crucible (OD 60 mm × ID 52 mm × H 100 mm) and melted in an induction furnace equipped with a quartz reaction tube and water-cooled brass end caps. When the steel sample was fully melted, the melt temperature was adjusted to 1450°C by using a B-type thermocouple inserted in the melt. After 30 minutes for homogenization, small amount of steel melt was sampled by using a quartz tube (OD 6 mm × ID 4 mm × H 60 mm) and rapidly quenched into cold water in order to check initial composition of steel. 40 g of flux carried in a boat made from an iron foil was then dropped onto the molten steel in order to initiate the steel/flux reaction. It took about 1 minute for the flux assembly to fully melt. Reaction time was set to zero at this moment. Small amount of steel and flux were taken out periodically up to 20 minutes by using different quartz tubes (OD 6 mm × ID 4 mm × H 60 mm for steel, and OD 10 mm × ID 8 mm × H 60 mm for flux). Those were quenched into cold water. During the experiments, the atmosphere inside the furnace was controlled by supplying Ar gas (99.999 pct purity, further purified by passing through CaSO₄ and Mg chips at 500°C) in order to remove moisture and oxygen in the gas, respectively.

Fig. 1.

Schematic diagrams of experimental apparatus for (a) the sampling experiments and (b) the flux injection experiment.

The steel samples were ground, cut, and analyzed by Inductively Coupled Plasma-Atomic Emission Spectroscopy (ICP-AES) in order to obtain Mn, Si, and Al concentrations in the steel. The flux samples were ground and subjected to X-Ray Flourescence (XRF) spectroscopy in order to obtain CaO, SiO₂, Al₂O₃, MgO, Na₂O, F, MnO, and FeO contents in the flux.

2.3. Flux Injection Experiment

Due to rapid Al₂O₃ accumulation in a CaO–SiO₂-type flux during the reaction, some aluminate could be formed near the steel/flux interface.²⁾ This changes concentration of remained molten flux, then may affect mass transport in the flux phase. Therefore examination of well-preserved steel/flux interface is essential in order to understand the reaction mechanism. Schematic diagram for flux injection experiments is shown in Fig. 1(b). Three Al₂O₃ crucibles (OD 18 mm × ID 15 mm × H 50 mm) containing 16.6 g of steel sample were located in a graphite holder, and put into the induction furnace. After the molten steel was homogenized at 1450°C, the fluxes were dropped onto each molten steel through a carbon tube. After the desired reaction time (1, 2, and 4 minutes), the whole sample assembly was taken out of the furnace and quenched into the water. During water quenching, the steel/flux interface was carefully protected by covering the open ends of the Al₂O₃ crucibles with ceramic caps in order to prevent the sample from damaging by the quenching water.

After the samples were completely dried, those were cold-mounted, cut, and polished in order to examine vertical-sectional area of steel/flux interface. Mircrostructure observation was performed by Scanning Electron Microscope (SEM, JSM-5900, JEOL, Japan). Line scan analysis based on Wavelength-Dispersive Spectroscopy (WDS) was also conducted by using Electron Probe MicroAnalysis (EPMA, JXA-8100, JEOL, Japan).

In both experiments (Sec. 2.2 and Sec. 2.3), the use of Al₂O₃ crucible could result in additional accumulation of Al₂O₃ in the flux by corrosion of the crucible. In a series of preliminary test using different crucibles (MgO and Al₂O₃), it was found that the additional accumulation of Al₂O₃ in the flux was not significant to that caused by the chemical reaction between steel and flux. Therefore, the additional accumulation of Al₂O₃ caused by the use of Al₂O₃ crucible was ignored in the present study.

3. Results

3.1. Composition Evolution in the Steel-flux System

Typical composition evolutions in the molten steel and the flux ((pctMgO)₀ = 5) are shown in Fig. 2. Concentrations of Al in the steel and SiO₂ in the flux decreased, while Si in the steel and Al₂O₃ in the flux increased. This is consistent with the previous investigation,¹⁾ and the Reaction (1) took place. In addition to this reaction, considerable amount of Na₂O was also reduced: it was thought to be reduced mostly by Al. A similar result was observed at 1500°C.¹⁾ It was suggested the following reaction might occur:   

$$2\underset{\_}{\mathrm{A}\mathrm{l}}+3(\mathrm{N}{\mathrm{a}}_2\mathrm{O})=6\mathrm{N}\mathrm{a}+(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)$$(2)

where the reduced Na may be evaporated into gas phase.¹⁾ In the present study, fume was observed in the reaction chamber (quartz tube) during the experiment. Deposits were found on inner surface of the reaction chamber. It was analyzed by EDS and was confirmed to contain Na. Since solubility of Na in liquid Fe is not accurately known but is thought to be very low due to high vapor pressure of Na, it can be concluded that Na in the above reaction was evaporated.

Fig. 2.

Composition evolution in (a) molten steel and (b) molten flux at 1450°C when (pct MgO)₀ was 5. Solid lines are to guide the eye.

After 30 minutes of the reaction, Al₂O₃ concentration increased over 40 pct, while most of Na₂O was reduced. Except for Al₂O₃, SiO₂, and Na₂O, other components in the flux were relatively stable. Similar evolution was observed when (pctMgO)₀ was 10 and 15, respectively. Therefore, it could be concluded that the dominant reactions between high Mn-high Al steel and CaO–SiO₂-type molten mold flux at high Al concentration (5.2 mass pct) and high MgO concentration (5–15 mass pct) are Al₂O₃ accumulation reaction coupled with SiO₂ and Na₂O reduction (Reactions (1) and (2)).

This conclusion is also verified by mass balance between produced Al₂O₃ and reduced SiO₂ and Na₂O. Figure 3 shows relationship between number of moles of produced Al₂O₃ ($\mathrm{\Delta }{\mathrm{n}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$), that of reduced SiO₂ ($\mathrm{\Delta }{\mathrm{n}}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}$), and that of reduced Na₂O ($\mathrm{\Delta }{\mathrm{n}}_{\mathrm{N}{\mathrm{a}}_2\mathrm{O}}$). Produced amount of Al₂O₃ could not be solely accounted for by reduced amount of SiO₂ as shown by closed symbols. Taking into account the reduction of Na₂O (Reaction (2)) improved the mass balance calculation as shown by open symbols. Therefore, the validity of the Reaction (2) was verified. This is practically important result that decrease of Na₂O content in mold flux increases viscosity of the flux and increases melting temperature of the flux. Therefore, gradual decrease of the Na₂O content in mold flux during continuous casting of high Mn-high Al steel is expected to cause serious difficulties in practical operation. This is an additional source of trouble for continuous casting of high Mn-high Al steel, apart from the Reaction (1).

Fig. 3.

Mass balance between Al₂O₃ and weak oxides (SiO₂ and Na₂O) reduced by Al at 1450°C under various (pct MgO)₀ conditions. S: Al₂O₃ formation by reduction of SiO₂ only. N+S: Al₂O₃ formation by reduction of both SiO₂ and Na₂O.

In order to investigate the effect of MgO on Al₂O₃ accumulation rate, composition evolutions of the reactant and the product of the reaction under different (pct MgO)₀ are compared in Fig. 4. It is expected that more (pct MgO)₀ in the mold flux would result in more MgAl₂O₄ to form. Composition evolutions in the molten steel and the flux did not show clear trend. Increasing (pct MgO)₀ from 5 to 10 decreased the reaction rate by examining composition evolution Al, Si, Al₂O₃, and SiO₂. However, further increasing (pct MgO)₀ to 15 increased the reaction rate. It is not clear at the present stage why the effect of MgO on the rate was not simple. Formation of MgAl₂O₄ in the flux, which will be shown in next section, might have played as resistance of mass transport in the flux, or as sink of Al₂O₃ accumulated in the flux.

Fig. 4.

Effect of (pct MgO)₀ on the composition evolution in the molten flux reacted with the molten steel at 1450°C: (a) [pct Al] in molten steel, (b) [pct Si] in molten steel, (c) (pct Al₂O₃) in molten flux, and (d) (pct SiO₂) in molten flux.

Composition evolutions of other weak oxides such as Na₂O, MnO, and FeO are also shown in Fig. 5. Similar to the previous experimental results,¹⁾ concentrations of MnO and FeO showed sharp increase at the beginning of the reaction, followed by rapid decrease, except for (% FeO) change when (%MgO)₀ = 15, which looks certainly in error. From the high Mn in the molten steel and high SiO₂ in the flux, it could be expected that Mn was oxidized, coupled with SiO₂ reduction:   

$$2\underset{\_}{\mathrm{M}\mathrm{n}}+(\mathrm{S}\mathrm{i}{\mathrm{O}}_2)=\underset{\_}{\mathrm{S}\mathrm{i}}+2(\mathrm{M}\mathrm{n}\mathrm{O})$$(3)

Fig. 5.

Effect of (pct MgO)₀ on composition evolution of weak oxides in molten flux reacted with molten steel at 1450°C: (a) (pct Na₂O), (b) (pct MnO), and (c) (pct FeO).

Then, consequent reduction of MnO was followed due to high Al concentration in the molten steel:   

$$2\underset{\_}{\mathrm{A}\mathrm{l}}+3(\mathrm{M}\mathrm{n}\mathrm{O})=3\underset{\_}{\mathrm{M}\mathrm{n}}+(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)$$(4)

At the initial stage of the steel-flux reaction, oxidation of Fe also took place and reduced back in 10 minutes of the reaction time. These observations are similar to what was observed in the previous study for low Al containing steel.¹⁾

3.2. Interfacial Morphology of the Steel-flux Systems at Different (pct MgO)₀

Figure 6 shows SEM images of mold flux near the steel-flux interface at each reaction time up to 4 minutes. Fluxes of different (pct MgO)₀ are shown. It is seen that smaller particles were found in the flux, and it was revealed to be MgAl₂O₄ by WDS analysis. At a given reaction time, number of the MgAl₂O₄ particles was more in the flux of higher (pct MgO)₀. For a given (pct MgO)₀, number of the MgAl₂O₄ particles increased as reaction time passed. And location of MgAl₂O₄ found gradually expanded away from the interface. The formation behavior of MgAl₂O₄ in the fluxes suggests that mass transport of Al₂O₃ from the interface to the bulk flux had occurred, but not as fast as to neglect resistance for the mass transport in the flux. Gradual change of MgAl₂O₄ forming location implies that there were non-ignorable concentration gradient. Development of the concentration gradient will be discussed in Sec. 4.1.

Fig. 6.

Morphologies of the molten flux at different (pct MgO)₀ and reaction times showing MgAl₂O₄ formation.

In addition to the MgAl₂O₄ particles, CaAl₄O₇, Al₂O₃, and CaF₂ phases were also observed near the interface as shown in Fig. 7. MgAl₂O₄ formation caused local depletion of MgO content, while Al₂O₃ was continuously supplied with reduction of SiO₂ or Na₂O. This resulted in alumina or other aluminate formation in the flux. Those reaction products were formed as particles or as layers. Whatever was the shape of the reaction products, they should have affected mass transport in the flux, either by changing viscosity of remained flux, or by increasing apparent viscosity of the local flux-solid mixture. Therefore, it is important to understand formation mechanism of solid reaction products in the molten flux in order to explain how the mass transport in the flux could be changed with the rapid Al₂O₃ accumulation in the molten flux.

Fig. 7.

Morphologies of the molten fluxes near steel-flux interface at 4 minutes of reaction time when (pct MgO)₀ was (a) 5 and (b) 15. CaAl₄O₇, Al₂O₃, and CaF₂ are observed.

4. Discussions

4.1. Formation of MgAl₂O₄ Spinel Particles and Mass Transport in the Molten Flux

In order to examine whether the formation of MgAl₂O₄ has some effect on the mass transport phenomena in the flux, present experimental results of both sampling experiments and flux injection experiments were analyzed along with phase diagram of the flux system. The phase diagram was obtained by FactSage thermodynamic software.^16,17) FTOxid database was used. Phase diagrams of CaO–SiO₂–Al₂O₃–MgO–F systems were calculated, and are projected onto CaO–SiO₂–Al₂O₃ pseudo-ternary phase diagram. This was intended in order to focus on concentration changes of Al₂O₃ and SiO₂ in the flux. For the phase diagram calculations, other components such as MgO, Na₂O, and F (assuming in a form of CaF₂) were fixed to the concentrations obtained by SEM-EDS analysis at each reaction time. The stable aluminates observed in the samples are considered in the thermodynamic calculations. The calculated phase diagram for a flux of (pct MgO)₀ = 15 is shown in Fig. 8. During the reaction between the steel and the flux, Al₂O₃ accumulates in the flux, which then consumes MgO in the flux to form MgAl₂O₄. Therefore, MgO content in the flux gradually decreases marked by different colors (Please see the online version). Full lines represent composition of liquid flux in equilibrium with MgAl₂O₄, and dotted lines represent that with CaAl₄O₇.

Fig. 8.

Composition changes of molten flux measured by sampling experiment and flux injection experiment when (pct MgO)₀ was 15. Calculated CaO–SiO₂–Al₂O₃–MgO–Na₂O–F phase diagram by FactSage^16,17) is also shown for comparison.

Experimental data of (pct MgO)₀ = 15 were selected for comparison with the calculated phase diagram as shown in Fig. 8. Solid symbols are the experimental data obtained from the sampling experiment. This represents overall composition changes of the mixture of molten flux and aluminates. Open symbols are the other experimental data obtained from the flux injection experiment. This represents composition changes of molten flux after formation of aluminates. It is clearly seen that compositions of the molten flux nearby the steel-flux interface (open symbols) shows good agreement with the calculated phase boundary of molten flux in equilibrium with different aluminates. From SEM-EDS analysis, it was determined that MgO content in the molten flux near the steel-flux interface rapidly decreased due to MgAl₂O₄ formation. This local depletion of MgO concentration in the flux results in shift of the liquid composition in equilibrium with MgAl₂O₄ phase as shown in the calculated phase diagram. Consequently, the molten flux can have high Al₂O₃ concentration by the formation of MgAl₂O₄ particles, and mass transport of Al₂O₃ from the interface to the bulk flux can occur successfully without forming subsequent MgAl₂O₄ due to decreased local MgO content. As MgO concentration in the flux decreased, however, calcium aluminate becomes stable, forming the calcium aluminate. This is consistent with the experimental observation. Composition change of molten flux from sampling experiments also shows the similar trend to the flux composition changes from flux injection experiments.

In addition to this, flux composition near the interface was further analyzed by WDS line analysis. The results are shown in Fig. 9 for the sample of (pct MgO)₀ = 15. At the initial stage of steel-flux reaction, Al₂O₃ concentration increased with decrease of SiO₂ nearby the steel-flux interface, and narrow concentration gradient was observed. As the reaction proceeded, the concentration gradient becomes more remarkable, in particular for the Al₂O₃ and SiO₂. Many sharp intensity peaks corresponding mainly to MgAl₂O₄ particles were observed. And overall intensity of SiO₂ and MgO decreased while that of Al₂O₃ increased. From these observations, it is now evident that the reaction between the steel and the flux resulted in accumulation of Al₂O₃ near the interface, forming MgAl₂O₄ as well as developing concentration gradient in the flux.

Fig. 9.

Line scan analysis on the flux sample with 15 of (pct MgO)₀ showing considerable concentration gradient in the flux.

4.2. Reaction Mechanism Investigation

In the previous investigation of the present authors,²⁾ it was concluded that rate controlling step of the Reaction (1) was mass transport of Al in the molten steel, at relatively low Al concentration ([pct Al]₀ ≤1.8) in the temperature range of 1440°C to 1550°C). It was experimentally confirmed that the flux did not show noticeable concentration gradient near the interface. On the other hand, when the Al concentration was high ([pct Al]₀ = 4.8), the reaction was not controlled by the mass transport of Al in the steel. In the present study for high Al content in steel and high MgO content in flux, it was shown that there was significant concentration gradient developed in the flux (Fig. 9), and many reaction product particles existed (Figs. 6 and 7). This observation mean the reaction rate would not be solely controlled by the mass transport of Al in the steel. Assuming rapid chemical reaction at the interface at high temperature, there is a possibility for mass transport in the flux phase to contribute to the rate controlling step.

Figure 10 shows a simple kinetic analysis assuming the rate controlling step would still be the mass transport of Al in the liquid steel:   

$$\frac{d[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}{dt}=-\frac{A{\rho }_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{k}_{\mathrm{A}\mathrm{l}}^m([\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}^i)$$(5)

Fig. 10.

Integrated rate plot of ln (([pct Al]−[pct Al]_eq)/([pct Al]₀−[pct Al]_eq)) with respect to reaction time under various (MgO)₀. Dashed line is calculated by using $k_{\mathrm{A}\mathrm{l}}^m$ derived in the previous study.²⁾

By integraring the above equation, it is obtained:   

$$ln\frac{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_{\mathrm{e}\mathrm{q}}}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_0-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_{\mathrm{e}\mathrm{q}}}=-\frac{A{\rho }_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{k}_{\mathrm{A}\mathrm{l}}^m{\alpha }_{\mathrm{A}\mathrm{l}}t$$(6)

where   

$$\begin{array}{l}{\alpha }_{\mathrm{A}\mathrm{l}}\equiv \\ \frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}(M_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/2){[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_0+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{A}\mathrm{l}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_0}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}(M_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/2)({[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_0-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_{\mathrm{e}\mathrm{q}})+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{A}\mathrm{l}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_0}\end{array}$$(7)

and ⁱ, ₀, and _eq represent interface, initial, and equilibrium, respectively. A, ρ_steel, W_steel, W_flux, $k_{\mathrm{A}\mathrm{l}}^m$, M_Al, $M_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ and t are interfacial area, density of the steel, mass of the steel, mass of the flux, mass transport coefficient of Al in the steel, molecular weight of Al, molecular weight of Al₂O₃, and reaction time, respectively. Derivation steps for the Eqs. (6) and (7) are given in Appendix A.

Experimental data shown in Fig. 4(a) were replotted in the Fig. 10 by symbols. Dashed line in the figure was calculated using the Eq. (6). A was estimated to be a cross-sectional area of the interior of the crucible. To the best knowledge of the present authors, density of the liquid Fe–C–Mn–Al alloys is not available in open literature. Therefore, as a first approximation, density was calculated by the equation proposed by Hoai and Lee¹¹⁾ who measured volume of liquid Fe–C–Mn alloys at various temperatures. Contrary to the previous analysis,²⁾ the [pct Al]_eq is not negligible in the present study due to higher Al content, therefore, it was taken into account in the present analysis. Data used for the calculation is given in Table 3. From the data shown in Fig. 4(a), α_Al was approximated to be 2–3. As can be seen in the figure, decrease of [pct Al] could not be explained by the assumed rate controlling step (mass transport of Al in the liquid steel). Experimental data clearly show that the rate was retarded, probably by additional resistance in the flux phase.

Table 3. Data used for calculation of mass transport coefficient in the present study.

Symbol	Unit	Value	Note
A	m2	2.1 × 10−3	Cross-sectional area of crucible
ρsteel	kg m−3	6.6 × 103	Reference 11)
ρflux	kg m−3	2.6 × 103	Reference 2)
Wsteel	kg	0.4	–
Wflux	kg	0.04	–
[pct Al]0	–	5	–
[pct Al]eq	–	2.8–3.5	for different (pct MgO)0
(pct Al2O3)0	–	6	–
(pct Al2O3)eq	–	40–45	for different (pct MgO)0
[pct Si]0	–	2 × 10−2	–
[pct Si]eq	–	0.8	for different (pct MgO)0
(pct SiO2)0	–	34.51	–
(pct SiO2)eq	–	17	for different (pct MgO)0

As discussed previously, the mass transport in the molten flux could contribute to the rate controlling step, because slow mass transport in the molten flux might resulted in concentration gradient in the flux. In order to examine the possibility of flux phase mass transfer control, the experimental data were analyzed by assuming flux phase mass transport control. If mass transport of Al₂O₃ in the flux might have controlled the reaction rate, then the Al₂O₃ accumulation rate may be formulated as:   

$$\frac{d(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}{dt}=\frac{A{\rho }_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{k}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}^s[{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}^i-(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)]$$(8)

By integrating the above equation, it is obtained:   

$$ln\frac{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_{\mathrm{e}\mathrm{q}}}{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_{\mathrm{e}\mathrm{q}}}=-\frac{A{\rho }_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{\alpha }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}{k}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}^{s}t$$(9)

where   

$$\begin{array}{l}{\alpha }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\equiv \\ \frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}(M_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/2){[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_0+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{A}\mathrm{l}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_0}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}(M_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/2){[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}\mathrm{l}]}_0+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{A}\mathrm{l}}({(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)}_{\mathrm{e}\mathrm{q}})}\end{array}$$(10)

and ρ_flux, $k_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}^s$ are density of the flux and mass transport coefficient of Al₂O₃ in the flux, respectively. Similarly, if mass transport of SiO₂ in the flux might have controlled the reaction rate, then the SiO₂ reduction rate may be formulated as:   

$$\frac{d(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}{dt}=-\frac{A{\rho }_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{k}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}^s[(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}^i]$$(11)

  

$$ln\frac{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_{\mathrm{e}\mathrm{q}}}{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_{\mathrm{e}\mathrm{q}}}=-\frac{A{\rho }_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{\alpha }_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}{k}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}^{s}t$$(12)

where   

$$\begin{array}{l}{\alpha }_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}\equiv \\ \frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}]}_0+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{S}\mathrm{i}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_0}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}]}_0+W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{S}\mathrm{i}}({(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{S}\mathrm{i}{\mathrm{O}}_2)}_{\mathrm{e}\mathrm{q}})}\end{array}$$(13)

Derivation steps for the Eqs. (9) and (12) are given in Appendix A.

The mass transfer coefficient of Al₂O₃ and SiO₂ obtained from the experimental data under 5 to 15 mass pct of (%MgO)₀ in the flux are shown in Fig. 11. $k_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}^s$ in the high MgO flux was estimated to be in the range of 5 × 10⁻⁶ to 10 × 10⁻⁶ m sec⁻¹, while $k_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}^s$ shows similar values in the range of 4 × 10⁻⁶ to 8 × 10⁻⁶ m sec⁻¹. Data used for the calculation is given in Table 3. The $k_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}^s$ and $k_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2}^s$ values estimated in the present study look to be in the similar range of those reported in literature.^12,13,14,15) Nevertheless, it is not thought to be very accurate values, because none of experimental data in Fig. 11 could be explained by a single rate controlling step. Moreover, the flux was not a homogeneous phase. Therefore, the reaction kinetic analyses in this section should not be considered as a quantitative result. Slopes varied as (pct MgO) varied. This might have been affected by formation of reaction products such as MgAl₂O₄ as was discussed in Sec. 4.1.

Fig. 11.

Integrated rate plot for (a) mass transport of Al₂O₃ and (b) mass transport of SiO₂ in molten flux with high (pct MgO)₀ concentration. The apparent rate constant of M_xO_y ($k_{{\mathrm{M}}_x{\mathrm{O}}_y}^s$) calculated from the slope are also shown in the figure.

5. Conclusions

In order to investigate the reaction mechanism between high Mn–high Al steel and CaO–SiO₂-type flux at high Al concentration ([pct Al]₀ = 5.2) and high MgO concentration ((pct MgO)₀ = 5 to 15), a number of steel–flux experiments were carried out with emphasis on the rate controlling step change.

In addition to reduction of SiO₂ by Al, considerable Na₂O reduction by Al was confirmed. This should be considered to deteriorate of mold flux functions by increasing viscosity and affecting crystallization.

Microstructure observation on the steel-flux interface revealed that MgAl₂O₄ particles start to form near the steel-flux interface at the early stage of the reaction and appear in the bulk flux later. At the later stage of the reaction, other aluminates such as CaAl₄O₇ and Al₂O₃ were also observed due to local depletion of MgO in the flux. Regardless of the amount of MgAl₂O₄ particle formation in the molten flux, fast Al₂O₃ accumulation was determined from composition evolution in the molten flux under different (pct MgO)₀ conditions.

MgAl₂O₄ formation mechanism and its effect on mass transport in the molten flux were discussed with consideration of compositions of the molten flux single phase near the MgAl₂O₄ particles, composition changes of the flux-aluminate mixture, and calculated phase diagrams. Development of concentration gradients observed from line scan analysis also suggested that flux composition changes are in close relationship with MgAl₂O₄ formation.

Acknowledgement

This research was financially supported by POSCO.

Appendix A

Rate Equations when Mass Transport of a Species Controls the Reaction Rate

This section provides derivation steps for the rate equations (Eqs. (6), (9), and (12)) used in Sec. 4.2. It is assumed that there are only two phases (steel and flux), no third phase is considered, and chemical equilibrium at the interface is assumed. Reaction rate is controlled either by mass transport of a species X in a boundary layer of the steel, or by mass transport of a species XO_x in the other boundary layer of the flux.

A.1. When Mass Transport of X in Liquid Steel Controls the Reaction Rate

In this case, there is a concentration gradient in the steel phase ([pct X] ≠ [pct X]ⁱ), but no concentration gradient is assumed in the flux phase ((pct XO_x) = (pct XO_x)ⁱ). The equilibrium is assumed:   

$$L=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}^i}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i}=\frac{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i}$$(A1)

where L is the equilibrium partition coefficient. The above relationship should be hold when the two phases approach at the equilibrium. Therefore,   

$$L=\frac{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i}=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}$$(A2)

Reaction rate is formulated as:   

$$\frac{d[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}{dt}=-\frac{A}{V_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}l}}{k}_{\mathrm{X}}^m([\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i)$$(A3)

By mass conservation in the two phase system, it is hold:   

$$[{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0-[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}])\frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{M_{\mathrm{X}}}=((\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0)\frac{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{M_{\mathrm{X}{\mathrm{O}}_x}}$$(A4)

For the above relationship, W_steel and W_flux are assumed not to vary during the reaction. By using the mass conservation,   

$$(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_{\mathrm{x}})={(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_{\mathrm{x}})}_0+\frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{X}{\mathrm{O}}_x}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{X}}}({[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0-[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}])$$(A5)

  

$${(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}={(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0+\frac{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{X}{\mathrm{O}}_x}}{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{X}}}({[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}})$$(A6)

From the Eq. (A2),   

$${[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i=\frac{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)$$(A7)

Substituting the Eqs. (A5) and (A6) into the Eq. (A7), and subsequent substitution into Eq. (A3) results in:   

$$\frac{d[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}{dt}=-\frac{A}{V_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{k}_{\mathrm{X}}^m{\alpha }_{\mathrm{X}}([\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}})$$(A8)

where   

$${\alpha }_{\mathrm{X}}\equiv \frac{M_{\mathrm{X}{\mathrm{O}}_x}{W}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0+M_{\mathrm{X}}{W}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0}{M_{\mathrm{X}{\mathrm{O}}_x}{W}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}({[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}})+M_{\mathrm{X}}{W}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0}$$(A9)

Integration of the above equation yields:   

$$ln\frac{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0-{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}=-\frac{A}{V_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}}{k}_{\mathrm{X}}^m{\alpha }_{\mathrm{X}}t$$(A10)

A.2. When Mass Transport of XO_x in Liquid Flux Controls the Reaction Rate

In this case, [pct X] = [pct X]ⁱ while (pct XO_x) ≠ (pct XO_x)ⁱ. The equilibrium is assumed:   

$$L=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}^i}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}^i}=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}^i}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}$$(A11)

Reaction rate is formulated as:   

$$\frac{d(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}{dt}=-\frac{A}{V_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{k}_{\mathrm{X}{\mathrm{O}}_x}^s((\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}^i)$$(A12)

By using the mass conservation,   

$$[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]={[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0+\frac{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{X}}}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{X}{\mathrm{O}}_x}}({(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0-(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x))$$(A13)

  

$$\begin{array}{l}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}={[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{M}_{\mathrm{X}}}{W_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{M}_{\mathrm{X}{\mathrm{O}}_x}}({(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}})\end{array}$$(A14)

From the Eq. (A11),   

$${(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}^i=\frac{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}{{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_{\mathrm{e}\mathrm{q}}}[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]$$(A15)

Substituting the Eqs. (A13) and (A14) into the Eq. (A15), and subsequent substitution into Eq. (A12) results in:   

$$\frac{d(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}{dt}=-\frac{A}{V_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{k}_{\mathrm{X}{\mathrm{O}}_x}^s{\alpha }_{\mathrm{X}{\mathrm{O}}_x}((\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}})$$(A16)

where   

$$\begin{array}{l}{\alpha }_{\mathrm{X}{\mathrm{O}}_x}\equiv \\ \frac{M_{\mathrm{X}{\mathrm{O}}_x}{W}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0+M_{\mathrm{X}}{W}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0}{M_{\mathrm{X}{\mathrm{O}}_x}{W}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}{[\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}]}_0+M_{\mathrm{X}}{W}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}({(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}})}\end{array}$$(A17)

Integration of the above equation yields:   

$$ln\frac{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}{{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_0-{(\mathrm{p}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }\mathrm{X}{\mathrm{O}}_x)}_{\mathrm{e}\mathrm{q}}}=-\frac{A}{V_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{k}_{\mathrm{X}{\mathrm{O}}_x}^s{\alpha }_{\mathrm{X}{\mathrm{O}}_x}t$$(A18)

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