A Novel Approach for Numerical Modeling of the CAS-OB Process: Process Model for the Heat-Up Stage

Abstract

The objective of this work was to develop a process model for the CAS-OB (Composition Adjustment by Sealed argon bubbling-Oxygen Blowing). The CAS-OB is designed to homogenize and control the steel composition and temperature before the casting. In the heating mode (OB) studied here, a refractory bell is lowered and submerged 30 cm below the liquid steel surface of the ladle and under this well-defined sealed volume, oxygen gas is injected to oxidize solid aluminum particles that are fed and molten at the surface. Under consideration were the melting of the solid aluminum particles, the oxidation of pure molten aluminum, and the oxidation of dissolved species, in this case Al, Mn, C and Si, and the solvent Fe. We also considered the formation and oxidation of steel droplets formed in the blowing when they pass through and react with the surface slag and also the reaction of pure aluminum on the top of the slag layer. Based on our simulations, only 30–40% of the chemical energy can be used to heat up the steel. A fraction of 0.8–0.85 of the O₂ can be utilized in the process; these values correspond to those obtained in previous work. The main part of the heating energy comes from the oxidation of the fed Al. FeO is primarily an intermediate product of the reactions. The model was tested against industrial trials for steel temperature and compositions of slag and steel, and it succeeded in capturing correct trends and absolute accuracy within the analyzing accuracy.

1. Introduction

The CAS-OB (Composition Adjustment by Sealed Argon Bubbling with Oxygen Blowing) process was developed in the 1980s by Nippon Steel Corporation to improve on the existing argon rinsing stations used for deoxidizing and alloying molten steel. The CAS-OB process enables consistently high alloy recoveries and the reheating of steel using the exothermic reaction between oxygen and aluminum. With this capability of good chemical composition control, steel homogeneity, and reheating, the CAS-OB becomes an ideal buffer station in the secondary metallurgy of steelmaking. The objective of the CAS-OB process is to homogenize and control the steel composition and temperature. It has been reported that the CAS-OB process enables a better scheduling, improved temperature control, and higher inclusion purity.¹⁾ Figure 1 presents a typical process treatment schedule; the oxygen blowing stage is highlighted.

Fig. 1.

Typical schedule of the CAS-OB process.²⁾

In the heating mode (OB) studied here, the refractory bell is lowered and submerged 30 cm below the liquid steel surface of the ladle and under this well-defined volume, oxygen gas is typically injected at the rate of 2300 m³/h (29 mol/s) at the Ruukki Metals Oy, Raahe Works CAS-OB stations. At the same time, solid aluminum particles are typically fed at the rate of 40 kg/min (25 mol/s) (see Fig. 2). The molar ratio of O₂ to Al is then 1.16. The stoichiometric O₂/Al ratio is 0.75, and therefore this means that there is some O₂ left over to oxidize other species as well. Of course, some residual O₂ will escape the system together with the dust and Ar via the de-dusting duct. Typically, heating rates of 10 K/min can be obtained. This value is limited by the thermal stresses developed in the wall structures in the ladle and bell. Therefore, accurate control of the heating is essential.

Fig. 2.

Parts of the CAS-OB station. 1. Supersonic O₂ lance, 2. De-dusting duct, 3. Al-feeding pipe, 4. Surface slag, 5. Ladle, 6. Porous plug for Argon bottom stirring, 7. Steel, 8. Bell, 9. Lance lifting system and 10. Bell lifting system.

As far as the authors are aware, there are no process models available in the open literature for CAS-OB. There are some papers published on flow simulations in the process (e.g.^3,4)), but no reaction source terms are included in these. The objective here was to develop a model that describes the relevant unit processes in the CAS-OB process. At this stage, the most important ones are the reactions taking place at the lance gas jet impact area and the reactions between metal droplets and slag and the related heat and mass transfer. There is a significant amount of droplets formed during the super-sonic lance blowing, and when these land on and pass through the surface slag layer, a significant conversion takes place, and droplets also act as an efficient media for heat transfer. Melting of the solid aluminum, oxidation of pure liquid aluminum, and oxidation/reduction of dissolved species (in this case, Al, Mn, C and Si, and the dissolvent Fe) were to be considered. Energy equations for the gas volume, liquid melt and different sections of the gas-liquid interface will be solved to get their temperatures. Because a significant fraction of the generated heat can be transferred and accumulated into the ladle and bell structures, we also developed simplified heat transfer models for these.

2. Description of the Model

Our simulation model is based on the numerical solution of the time-dependent differential equations of species and energy in the CAS-OB process. These are formulated for all species and temperatures in all the important volumes and reaction surfaces, including volumes of gas in the bell, liquid steel bath, top slag layer, and metal droplets dispersed in the slag, for a total of 48 unknown variables and an equal number of equations to be solved. Chemical reactions may occur at the jet hit depression area, the molten aluminum-gas surface, the molten aluminum-slag surface, the aluminum particle oxygen surface, and the metal droplet-slag surface. We also consider heat transfer in the bell and ladle structures in order to have accurate heat-up and cooling characteristics for the process. Figure 3 presents the principles of the model.

Fig. 3.

Principles of the process model. G=gas, L=liquid, o= slag/liquid Al surface, f = open eye, B1-4, bell calculation nodes, L1-4 = ladle calculation nodes, S1-2 = top slag nodes.

2.1. Modeling the Operation of the Top Lance

The driving force for the process is the injection of oxygen via a supersonic oxygen top lance. The supersonic gas jet produces a depression on the surface of the steel bath. The depth and diameter of the depression are defined by the momentum flux in the jet, lance distance H, and material properties on both sides of the surface. We assume here that the surface in the depression is kept clean by the jet, referred to as the free surface “f”, and any slag formed is pushed to the surrounding ring surface, the occupied surface “o” (see Fig. 3). As the liquid aluminum is lighter than the slag, we assume that it stays on the top of the slag layer.

The correlations proposed by Koria and Lange⁵⁾ were used here to calculate the dimensions of the depression. The gas jet was assumed to be perpendicular to the surface. The flow rate of the blown gas is measured and the jet velocity is available based on our CFD simulations.⁶⁾ These were calculated for an AOD lance, but a similar approach was used for the CAS-OB lance. The momentum flux leaving the nozzle can be calculated as follows.   

$$\dot{M}={\dot{m}}_J{u}_J=\frac{\pi d_n^2}{4}{\rho }_G{u}_J^2={\dot{m}}_G{u}_J={\dot{V}}_{G.NTP}{\rho }_{NTP}{u}_J$$(1)

In our case, d_n = 30.5 mm and the velocity at the nozzle exit was obtained with CFD simulations, u_J = 475 m/s. From these, the non-dimensional momentum flow rate can be calculated as follows.⁵⁾   

$$\dot{\underset{\_}{M}}=\frac{\dot{M}}{{\rho }_{L}g{H}^3}$$(2)

The depth and diameter of the depression are then calculated as follows.⁵⁾   

$$\frac{h_f}{H}=4.469{\underset{\_}{\dot{M}}}^{0.66}$$(3)

  

$$\frac{d_f}{H}=2.813{\underset{\_}{\dot{M}}}^{0.282}$$(4)

Based on the findings of Sharma et al.,⁷⁾ it was assumed that the geometry of the depression is unaffected by reactions and the interference of top slag. A typical lance height is 1.5 m, and as the bell diameter under the bell is 1.28 m, we then have the typical geometry of the depression: h_f = 104 mm and d_f = 712 mm. We consider the increase in surface area of the open eye due to the convex surface assuming the spherical cap geometry here, and therefore $S_f=\pi \left(a_f^2+h_f^2\right)$ where a_f is the radius of the depression perimeter a_f = 0.5d_f.

The surface under the bell is divided into two parts: free surface S_f (open eye) and occupied area S_o by liquid; the surface slag and aluminum are on the top of it. The free liquid steel surface has no mass. The area covered by liquid slag is assumed to have a constant thickness, h_S. Based on the slag layer area and mass, the thickness of the liquid layer can be calculated as follows.   

$$h_s=\frac{m_S}{{\rho }_S{S}_o}$$(5)

Metal droplets are formed at the gas jet impact area while blowing, and these eject from the bath and land on the surrounding slag surface. The degree and mode of spattering is defined by the blowing number N_B⁸⁾   

$$N_B=\frac{{\rho }_G{u}_G^2}{2\sqrt{{\mathrm{\sigma }}_{L}g{\rho }_L}}$$(6)

The initial gas velocity u_G for this specific CAS-OB geometry was taken from CFD simulations.⁶⁾ Based on these, the centerline gas velocity attenuates from the nozzle level to the surface vicinity before the flow turns, in this case by a factor of 0.432. In addition, only a fraction of 0.447 can be utilized to spatter the liquid.⁸⁾ Therefore, the critical gas velocity is equal to u_G = 0.432 × 0.447 × u_J = 0.193 × u_J. The rate of droplet formation R_B (kg/s) is defined as follows:⁸⁾   

$$\frac{R_B}{\dot{V}}=\frac{N_B^{3.2}}{{\left(2.6\times {10}^6+2\times {10}^{-4}{N}_B^{12}\right)}^{0.2}}$$(7)

wherein $\dot{V}$ is the lance blow rate in Nm³/s. At this point, we only consider the constant blowing rate and the mass of droplets in the slag-metal emulsion at time t is then calculated as   

$$m_D=min\left({\tau }_D,t\right)R_B$$(8)

wherein ${\tau }_D=\frac{h_S}{u_{t.D}}$ is the residence time of droplets in the slag and h_s is the slag layer thickness. The terminal velocity of the droplets u_t.D was solved from the droplet force balance analytically as follows. All numeral values are combined for simplicity here.⁹⁾   

$$u_{t.D}=0.153\times {\left(\frac{d_{D}g\left({\rho }_M-{\rho }_S\right)}{{\rho }_S}{\left(\frac{d_D{\rho }_S}{{\mu }_S}\right)}^{0.6}\right)}^{1/1.4}$$(9)

This equation is applicable in the range Re_D = 1–1000. From the mass, we can solve the overall surface area of the emulsified steel droplets as follows.   

$$S_D=\frac{6m_D}{d_D{\rho }_L}$$(10)

2.2. Chemical Reactions

We consider the melting of pure solid aluminum, primary oxidation reactions of pure liquid and dissolved aluminum (Al and Al, respectively), and oxidation of Si, Mn, C, and Fe. In addition to these, the secondary reactions of the metal droplets in the slag are considered. Our main objective is to develop an effective model to consider parallel surface reactions with mass transfer limitations. (s), (l), and (S) refer to solid, liquid, and slag, respectively. The following reactions are considered in the gas jet impact area.   

$$\mathrm{A}\mathrm{l}\left(\mathrm{s}\right)\leftrightarrow \mathrm{A}\mathrm{l}\left(\mathrm{L}\right)$$R0

  

$$2\mathrm{A}\mathrm{l}\left(\mathrm{L}\right)+3/2{\mathrm{O}}_2\leftrightarrow \mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R1

  

$$2\underset{\_}{\mathrm{A}\mathrm{l}}\left(\mathrm{L}\right)+3/2{\mathrm{O}}_2\leftrightarrow \mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R2

  

$$\underset{\_}{\mathrm{S}\mathrm{i}}\left(\mathrm{L}\right)+{\mathrm{O}}_2\leftrightarrow \mathrm{S}\mathrm{i}{\mathrm{O}}_2\left(\mathrm{S}\right)$$R3

  

$$\underset{\_}{\mathrm{M}\mathrm{n}}\left(\mathrm{L}\right)+1/2{\mathrm{O}}_2\leftrightarrow \mathrm{M}\mathrm{n}\mathrm{O}\left(\mathrm{S}\right)$$R4

  

$$\mathrm{F}\mathrm{e}\left(\mathrm{L}\right)+1/2{\mathrm{O}}_2\leftrightarrow \mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)$$R5

  

$$\underset{\_}{\mathrm{C}}+1/2\mathrm{\hspace{0.17em} }{\mathrm{O}}_2\leftrightarrow \mathrm{C}\mathrm{O}\left(\mathrm{g}\right)$$R6

The surface slag is formed in the above reactions R1–5. In addition to steel droplets forming in the blowing, Al is assumed to form droplets as soon as it is molten. We assume that the rate of Al droplet formation is equal to the difference between the rate of melting and dissolution. As the density of the Al is lower than that of the top slag, liquid Al is assumed to stay on the top of slag layer surface. The following reactions take place across the area S_D formed by the steel droplets when they fall through the surface slag layer.   

$$\mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)+2/3\underset{\_}{\mathrm{A}\mathrm{l}}\left(\mathrm{L}\right)\leftrightarrow \mathrm{F}\mathrm{e}+1/3\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R8

  

$$\mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)+1/2\underset{\_}{\mathrm{S}\mathrm{i}}\left(\mathrm{L}\right)\leftrightarrow \mathrm{F}\mathrm{e}+1/2\mathrm{S}\mathrm{i}{\mathrm{O}}_2\left(\mathrm{S}\right)$$R9

  

$$\mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)+\underset{\_}{\mathrm{M}\mathrm{n}}\left(\mathrm{L}\right)\leftrightarrow \mathrm{F}\mathrm{e}+\mathrm{M}\mathrm{n}\mathrm{O}\left(\mathrm{S}\right)$$R10

  

$$\mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)+\underset{\_}{\mathrm{C}}\leftrightarrow \mathrm{F}\mathrm{e}+\mathrm{C}\mathrm{O}\left(\mathrm{g}\right)$$R11

In addition to these, important reactions are reduction of FeO, SiO₂, and MnO when pure liquid Al is “sprayed” on the top of the slag layer. We assumed that these are immediate and are limited by the delivery rate of Al and equilibrium.   

$$\mathrm{F}\mathrm{e}\mathrm{O}\left(\mathrm{S}\right)+2\mathrm{A}\mathrm{l}\left(\mathrm{L}\right)\leftrightarrow 3\mathrm{F}\mathrm{e}+\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R12

  

$$3/2\mathrm{S}\mathrm{i}{\mathrm{O}}_2\left(\mathrm{S}\right)+2\mathrm{A}\mathrm{l}\left(\mathrm{L}\right)\leftrightarrow 3/2\mathrm{S}\mathrm{i}+\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R13

  

$$\mathrm{M}\mathrm{n}\mathrm{O}\left(\mathrm{S}\right)+2\mathrm{A}\mathrm{l}\left(\mathrm{L}\right)\leftrightarrow 3\mathrm{F}\mathrm{e}+\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(\mathrm{S}\right)$$R14

The rates of all chemical reactions were calculated by the method presented in our previous publications.^10,11,12) The rate is derived based on the modified law of mass action using activities. Unknown forward rate coefficients are used as free variables that are set to sufficiently high values to guarantee mass transfer control and chemical equilibrium at the reaction surface. As an example, the rate of reaction 1 is calculated as follows and a similar approach is used for all the others. Please see our previous work^10,11,12) for more details on this method.   

$$R_1=k_{f.1}\left(a_{Al}^2{a}_{O2}^{1.5}-\frac{a_{A{l}_2{O}_3}}{K_1}\right),\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{\hspace{0.17em} }{k}_{\mathrm{f}.1}\to \infty .$$(11)

It is known that the rate of CO post-oxidation, R7 (not considered here), is highly dependent on the H₂/H₂O content of the gas.¹³⁾ In typical hydro-carbon combustion applications, there is always H₂/H₂O present. But here, pure O₂ is used, and in principle these catalyzing gases are not present. Under fully dry conditions, the rate of this reaction is very slow, and therefore it is not reasonable to assume infinite reaction kinetics. Instead, actual rate expression is required. At the moment, as far as the authors are aware of, there is no such equation available, so this reaction is not included here. However, the role of the post-combustion of CO will be studied later when validating the model with experiments. Based on our previous experimental work,¹⁴⁾ only a small amount of C is released, so this reaction has no significant importance for the process operation, and at the moment, we assume that only CO is present in the system.

2.3. Conservation of Gaseous Species

Conservation equations of the gaseous oxygen in the gas volume defined by the snorkel and liquid steel surface are given below. Positive flow direction is towards the reaction surface. There is also a flow of Ar out from the surface, but as it is much smaller than the O₂ flow, its effect on the gas side bulk flow rate is neglected. However, the net flow of O₂ (i.e., the Stefan flow) is considered. We have used the first-order upwind method to model the convective flux here.¹⁵⁾ Total mass flow rates are simply calculated by summing up the contributions from all reactions.   

$$\begin{array}{l}{\dot{m}}_{O_2,in}-{\dot{m}}_{G,out}{y}_{O_2}\\ -h_{O_2}{S}_f{\rho }_G\left(y_{O_2}-y_{O_2,f}\right)-max\left({\dot{m}}_{G.f},0\right)S_f{y}_{O_2}\\ +max\left(-{\dot{m}}_{G.f},0\right)S_f{y}_{O_2,f}-h_{O_2}{S}_o{\rho }_G\left(y_{O_2}-y_{O_2,o}\right)\\ -max\left({\dot{m}}_{G.o},0\right)S_o{y}_{O_2}+max\left(-{\dot{m}}_{G.o},0\right)S_o{y}_{O_2,o}=V_G{\rho }_G\frac{d{y}_{O_2}}{dt}\end{array}$$(12)

The total gas flow rate at the exit can be calculated from the inlet mass flow rates and reaction rates in the process. The continuously stirred tank reactor model was used here as a first assumption to solve this.¹⁶⁾ In this model, it is assumed that the gas volume in the snorkel is fully mixed and homogenous. Argon, CO, and the residual O₂ flow out from this volume by convection. The effect of diffusion is assumed to be negligible at the exit as composition is fully mixed. Conservation equations of gaseous oxygen at the liquid steel surface and at the pure liquid Al surfaces are given by the Eqs. (13) and (14), respectively.   

$$\begin{array}{l}{h}_{O_2}{\rho }_G\left(y_{O_2}-y_{O_2,f}\right)+max\left({\dot{m}}_{G.f},0\right)y_{O_2}-max\left(-{\dot{m}}_{G.f},0\right)y_{O_2.f}\\ +v_{O_2,2}{R}_2+v_{O_2,3}{R}_3+v_{O_2,4}{R}_4+v_{O_2,5}{R}_5+v_{O_2,6}{R}_6=0\end{array}$$(13)

  

$$\begin{array}{l}{h}_{O_2}{\rho }_G\left(y_{O_2}-y_{O_2,o}\right)+max\left({\dot{m}}_{G.o},0\right)y_{O_2}\\ -max\left(-{\dot{m}}_{G.o},0\right)y_{O_2,f}+v_{O_2,1}{R}_1=0\end{array}$$(14)

Here, ν_j,k is the stoichiometric coefficient of species j in reaction k. In addition to O₂ and Ar, gas phase contains also some CO as carbon is removed by oxidation from the liquid surface and also when carbon reduces FeO(S) in the slag, CO is formed. To consider the oxidation of C and the formation of CO, conservation equations for these at the surface and at the bulk gas are needed. The conservation equation of the dissolved C is presented later. The mass balance of CO in the bulk gas and at the surface is as follows.   

$$\begin{array}{l}-{\dot{m}}_{G,out}{y}_{CO}-h_{CO}{S}_f{\rho }_G\left(y_{CO}-y_{CO,f}\right)-max\left({\dot{m}}_{G.f},0\right)S_f{y}_{CO}\\ +max\left(-{\dot{m}}_{G.f},0\right)S_f{y}_{CO.f}-h_{CO.D}{S}_D{\rho }_G\left(y_{CO}-y_{CO,D}\right)\\ -max\left({\dot{m}}_{G.D},0\right)S_o{y}_{CO}+max\left(-{\dot{m}}_{G.D},0\right)S_o{y}_{CO.D}=V_G{\rho }_G\frac{d{y}_{CO}}{dt}\end{array}$$(15)

  

$$\begin{array}{l}{h}_{CO}{\rho }_G\left(y_{CO}-y_{CO,f}\right)+max\left({\dot{m}}_{G.f},0\right)y_{CO}\\ -max\left(-{\dot{m}}_{G.f},0\right)y_{CO.f}+v_{CO,6}{R}_6=0\end{array}$$(16)

  

$$\begin{array}{l}{h}_{CO.D}{\rho }_G\left(y_{CO}-y_{CO,D}\right)+max\left({\dot{m}}_{G.D},0\right)y_{CO}\\ -max\left(-{\dot{m}}_{G.D},0\right)y_{CO.D}+v_{CO,11}{R}_{11}=0\end{array}$$(17)

For Argon, one more equation is required for each surface, and here the sum of mass fraction is set to 1, and this condition is implemented as additional residual for each location to close the system at all positions.

2.4. Conservation of Metal and Slag Species

In the heating mode, solid Al particles are fed into the process. These will be first molten and then oxidized for heat generation. The rate of melting of a single spherical Al particle can be estimated from a simple heat transfer controlled model by assuming a spherical one-dimensional system. One additional issue that complicates this system is the buoyancy effect. As the density of Al is smaller than steel, a significant part of the particle is above the steel surface. Based on the force balance of a spherical Al particle floating in steel, only a fraction of 0.42 of the particle’s surface is below the liquid steel surface. Therefore, the surface area of a sphere should be multiplied by a factor of 0.42 to correct this effect. The fraction of the particle above the surface is exposed to thermal radiation and melts, but as the gas blow rate is high, the molten Al is rapidly blown away and we assume that this fraction does to contribute to Al dissolution. The analytical solution for the melting time of the Al sphere is equal to   

$${\tau }_m=\frac{d_{Al}^2}{8}\frac{{\rho }_{Al}{l}_m}{{\lambda }_e\left(T-T_m\right)}$$(18)

λ_e is effective steel conductivity, assumed 60 W/mK here,¹⁷⁾ T_m is the melting temperature of Al 660°C, l_m is the melting heat of Al, and T is the local temperature, here T_L. There is also a solid steel crust formed in some cases during the early stages of melting, as the melting temperature of Al is lower than of steel.¹⁷⁾ As the mixing here is very effective and all molten material is rapidly blown away, the effect of crust formation is neglected. There are particles at different stages of melting in the process at the same time. To be precise, the population balances of different sizes of particles should be solved. However, the melting time of a single Al particle is relatively small when compared with the whole process time (5–15 min). We could therefore assume that the average melting rate of solid Al per unit mass, r_m, is equal to r_m = 1/τ_m. The rate of solid Al accumulation is defined as the difference between the input flow and the rate of melting   

$${\dot{m}}_{Al\left(s\right)}-S_{Al,melt}={\dot{m}}_{Al\left(s\right)}-r_m{m}_{Al\left(s\right)}=\frac{d{m}_{Al\left(s\right)}}{dt}$$(19)

Molten Al is dissolved and blown away to the top of the slag layer where it is then oxidized by gaseous O₂, FeO(S), SiO₂(S) or MnO(S). The amount of molten Al on the top of the slag layer is as follows; we assumed here that liquid Al does not accumulate in the open eye area.   

$$\begin{array}{l}{r}_m{m}_{Al\left(s\right)}-h_{Al}{S}_{Al}{\rho }_L\left(1-y_{Al}\right)+S_o{R}_1{v}_{1,Al}+\\ S_o{R}_{12}{v}_{12,Al}+S_o{R}_{13}{v}_{13,Al}+S_o{R}_{14}{v}_{14,Al}=\frac{d{m}_{Al\left(l\right)}}{dt}\end{array}$$(20)

where S_Al is the total surface area of the Al particles that float on the surface. It is calculated from the mass of solid Al as   

$$S_{Al}=\frac{6m_{Al\left(8\right)}}{d_{Al}{\rho }_{Al}}$$(21)

Accumulation rate of dissolved Al is defined as the difference between the rate of pure Al dissolution from the surface of Al particles and the rate of mass transfer to free surface where it is oxidized.   

$$\begin{array}{l}-h_{Al}{S}_{Al}{\rho }_L\left(y_{Al}-1\right)-h_{Al}{S}_f{\rho }_L\left(y_{Al}-y_{Al,f}\right)\\ -max\left({\dot{m}}_{L,f},0\right)S_f{y}_{Al}+max\left(-{\dot{m}}_{L,f},0\right)S_f{y}_{Al,f}\\ -h_{Al,D}{S}_D{\rho }_L\left(y_{Al}-y_{Al,D}\right)-max\left({\dot{m}}_{L,D},0\right)S_D{y}_{Al}\\ +max\left(-{\dot{m}}_{L,D},0\right)S_D{y}_{Al,D}=\frac{d\left(m_L{y}_{Al}\right)}{dt}\end{array}$$(22)

  

$$\begin{array}{l}{h}_{Al}{\rho }_L\left(y_{Al}-y_{Al,f}\right)+max\left({\dot{m}}_{L,f},0\right)y_{Al}\\ -max\left(-{\dot{m}}_{L,f},0\right)y_{Al,f}+v_{Al,2}{R}_2=0\end{array}$$(23)

  

$$\begin{array}{l}{h}_{Al,D}{\rho }_L\left(y_{Al}-y_{Al,D}\right)-max\left({\dot{m}}_{L,D},0\right)y_{Al}\\ +max\left(-{\dot{m}}_{L,D},0\right)y_{Al,D}+v_{Al,8}{R}_8=0\end{array}$$(24)

The mass conservation equations of Si, Mn, C, and Fe at the steel bath and at the free metal surface all have the same format. Only Si is presented here.   

$$\begin{array}{l}-h_{Si}{S}_f{\rho }_L\left(y_{Si}-y_{Si,f}\right)-max\left({\dot{m}}_{L,f},0\right)S_f{y}_{Si}\\ +max\left(-{\dot{m}}_{L,f},0\right)S_f{y}_{Si,f}-h_{Si,D}{S}_D{\rho }_L\left(y_{Si}-y_{Si,D}\right)\\ -max\left({\dot{m}}_{L,D},0\right)S_D{y}_{Si}+max\left(-{\dot{m}}_{L,D},0\right)S_D{y}_{Si,D}=\frac{d\left(m_L{y}_{Si}\right)}{dt}\end{array}$$(25)

  

$$\begin{array}{l}{h}_{Si}{\rho }_L\left(y_{Si}-y_{Si,f}\right)+max\left({\dot{m}}_{L,f},0\right)y_{Si}\\ -max\left(-{\dot{m}}_{L,f},0\right)y_{Si,f}+v_{Si,3}{R}_3=0\end{array}$$(26)

  

$$\begin{array}{l}{h}_{Si,D}{\rho }_L\left(y_{Si}-y_{Si,D}\right)-max\left({\dot{m}}_{L,D},0\right)y_{Si}\\ +max\left(-{\dot{m}}_{L,D},0\right)y_{Si.D}+v_{Si,9}{R}_9=0\end{array}$$(27)

The overall mass balance of the liquid steel can be solved simultaneously by summing all individual mass balances of components. It is assumed that there is only one volume of each slag component in the system, which means that Al₂O₃ that is formed (either at the free surface or at the free liquid Al surface) ends up in the same place and only a single value of activity is required. Conservation equations of the slag species are given as follows. Equations for MnO and FeO are similar to SiO₂, and therefore are not presented here. The total mass of the slag can be calculated by summing all slag components.   

$$\begin{array}{l}{v}_{A{l}_2{O}_3,1}{R}_1{S}_o+v_{A{l}_2{O}_3,2}{R}_2{S}_f+v_{A{l}_2{O}_{3}8}{R}_8{S}_D\\ +v_{A{l}_2{O}_3,12}{R}_{12}{S}_o=\frac{d{m}_{A{l}_2{O}_3}}{dt}\end{array}$$(28)

  

$$v_{Si{O}_2,3}{R}_3{S}_f+v_{Si{O}_2,9}{R}_9{S}_D=\frac{d{m}_{Si{O}_2}}{dt}$$(29)

2.5. Energy Balances

One of the main objectives of the CAS-OB process is to chemically heat up the steel melt effectively. Therefore, the accurate prediction of the reactor temperature is an essential requirement for the model. For this, energy equations for the system are derived and implemented in the following. Reaction enthalpies were taken from.^18,19) Thermal radiation between surfaces in the bell is considered; the effect of volumetric radiation of the gas and dust is here excluded, but could be considered later. The energy conservation equation for gas in the bell is as follows.   

$$\begin{array}{l}{\dot{m}}_{O_2,in}{c}_{p,O_2}\left(T_{in,O_2}-T_G\right)+{\dot{m}}_{Ar,in}{c}_{p,Ar}\left(T_f-T_G\right)\\ -{\alpha }_{G.o}{S}_o\left(T_G-T_o\right)-{\alpha }_{G.f}{S}_f\left(T_G-T_f\right)-{\alpha }_w{S}_w\left(T_G-T_{B1}\right)\\ +R_6{v}_{CO,6}{S}_f{c}_{p,CO}\left(T_f-T_G\right)=m_G{c}_{p,G}\frac{d{T}_G}{dt}\end{array}$$(30)

The first two terms describe the heat-up of oxygen and argon to bulk gas temperature, respectively. All argon is assumed to pass through the open eye. The 3, 4, and 5 terms present the convective heat transfer to the slag surface, the open eye, and the bell walls, respectively, and the 6 term is the heat-up of the CO formed to bulk temperature. The right-hand side is the accumulation of heat in the gas. The energy equation for the liquid bulk melt was formulated as follows.   

$$\begin{array}{l}{\dot{m}}_{Ar,in}{c}_{p,Ar}\left(T_{in,Ar}-T_L\right)\\ -\frac{m_{Al\left(s\right)}}{{\tau }_m}\left(c_{p,Al\left(s\right)}\left(T_m-T_{in}\right)+l_m\left(T_m\right)+c_{p,Al\left(l\right)}\left(T_L-T_m\right)\right)\\ -{\alpha }_{L.o}{S}_o\left(T_L-T_o\right)-{\alpha }_{L.f}{S}_f\left(T_L-T_f\right)\\ -{\alpha }_{wL}{S}_{wL}\left(T_L-T_{L1}\right)-{\alpha }_{L.o}{S}_e\left(T_L-T_{S1}\right)\\ -{\alpha }_{B.sub}{S}_{sub}\left(T_L-T_{B2}\right)\\ -max\left(-R_2,0\right)v_{2,Al}{S}_f{c}_{p,Al}\left(T_f-T_L\right)\\ -max\left(-R_3,0\right)v_{3,Al}{S}_f{c}_{p,Si}\left(T_f-T_L\right)\\ -max\left(-R_4,0\right)v_{4,Min}{S}_f{c}_{p,Mn}\left(T_f-T_L\right)\\ -max\left(-R_5,0\right)v_{5,Fe}{S}_f{c}_{p,Fe}\left(T_f-T_L\right)\\ -max\left(-R_6,0\right)v_{6,C}{S}_f{c}_{p,C}\left(T_f-T_L\right)\\ +R_{12}{v}_{12,Fe}{S}_o{c}_{p,Fe}\left(T_o-T_L\right)+R_{13}{v}_{13,Si}{S}_o{c}_{p,Si}\left(T_o-T_L\right)\\ +R_{14}{v}_{14,Min}{S}_o{c}_{p,Mn}\left(T_o-T_L\right)\\ +\sum _{i=0}^{n_L}\left(max\left(R_D{y}_i+\sum _{k=8}^{11}\left(R_k{v}_{k,i}\right)S_D,0\right)\right)\hspace{0.17em}{c}_{p,i}\left(T_o-T_L\right){\eta }_D\\ =m_L{c}_{p.L}\frac{d{T}_L}{dt}\end{array}$$(31)

where ${\eta }_D=1-exp\left(-\frac{6{\alpha }_D{\tau }_D}{d_D{c}_{p,L}{\rho }_L}\right)$ is the droplet heat transfer efficiency, i.e., how much of the available temperature difference T_o-T_L can be utilized when the droplet passes the slag. The larger the heat transfer coefficient and the smaller the droplet size and residence time, the better the efficiency. The 1 term is the heat-up of argon to bulk liquid temperature. The 2 term describes the heat consumption for fed Al particles to heat up and melt, and we assume here that the required energy is taken from the liquid melt. The next three terms are the convective heat flux to the surrounding surfaces. The 6 term is very important; it describes the heat loss from the free steel/slag surface exposed to the surroundings, S_e is the surface area outside the bell and when the bell is raised; S_e equals the overall ladle surface area. 7 term is the heat loss to bell when it is submerged. The following five equations present the cooling of dissolved elements in the case of a backwards reaction. The 13, 14, and 15 terms describe the incoming heat from the reduced iron, silicon, and manganese, respectively. The last term on the left is the incoming heat from hot metal droplets exiting the slag. The term on the right side is the accumulation of heat. The energy balance for the free liquid steel surface, i.e., the open eye, is as follows; we assume here that slag species are not returned to the free surface from the top slag layer.   

$$\begin{array}{l}{\alpha }_L\left(T_L-T_f\right)+{\alpha }_G\left(T_G-T_f\right)-\left(J_f-J_{B1}\right)-\sum _{k=2}^6{R}_k\mathrm{\Delta }{h}_k\left(T_f\right)\\ +\sum _{k=2}^6\sum _{i=0}^{n_L+n_G}max\left(-R_k{v}_{i,k},0\right)\hspace{0.17em}{c}_{p.i}\left(T_{G/L}-T_f\right)=0\end{array}$$(32)

J_f and J_B1 are the radiosities of the free liquid surface and the internal wall area of the bell, respectively. These are solved from a separate thermal radiation model, presented later. Energy equation for the surface slag element is a bit more complex. We assume that both surface slag and the liquid Al layer on the top of it are at the same temperature T_o.   

$$\begin{array}{l}\frac{d{m}_{Al(l)}}{dt}{c}_{p,Al(l)}\left(T_L-T_o\right)\\ +{\alpha }_{L.o}{S}_o\left(T_L-T_o\right)+{\alpha }_{G.o}{S}_o\left(T_G-T_o\right)-S_o\left(J_o-J_{B1}\right)\\ +\sum _{i=0}^{n_L}{R}_D{y}_i{c}_{p,\mathrm{\hspace{0.17em} }i}\left(T_L-T_o\right){\eta }_D-\sum _{k=8}^{11}{R}_k{S}_D\mathrm{\Delta }{h}_k\left(T_o\right)\\ -R_{12}{S}_o\mathrm{\Delta }{h}_{12}\left(T_o\right)-R_{13}{S}_o\mathrm{\Delta }{h}_{13}\left(T_o\right)-R_{14}{S}_o\mathrm{\Delta }{h}_{14}\left(T_o\right)\\ +\sum _{i=0}^{n_S}\sum _{k=1}^7{R}_k{v}_{k,\mathrm{\hspace{0.17em} }i}{S}_f{c}_{p,\mathrm{\hspace{0.17em} }i}\left(T_f-T_o\right)=\left(m_{Al(l)}{c}_{p,Al(l)}+m_s{c}_{p,s}\right)\frac{d{T}_o}{dt}\end{array}$$(33)

The first term describes the heat-up of incoming Al to T_o, and the following three terms are the convective and radiative heat transfer rates. The 5 and 6 terms describe the heat-up and reactions of the incoming metal droplets, respectively. The 7, 8, and 9 terms are the reaction heat from Al oxidation by the slag species, and the 10 term is the heat-up of the incoming slag from the open eye reactions. The right side is the accumulation rate of heat on the surface layer.

2.6. Heat Transfer Models for Ladle and Bell

This section describes the heat transfer mechanisms within the bell and ladle structures, including thermal radiation in the bell, convection heat transfer between gas and walls, conduction of heat in the solid structures, and finally, accumulation of the heat in the structures. We assumed that both the ladle and the bell are formed of a steel shell covered by a ceramic insulation on the inner side (see Figs. 3 and 4). We also assume that only the gradients in the normal direction are important. Correct masses and surface areas are used for the layers, leaving shell thickness to be solved. Heat transfer within the wall material is described by a one-dimensional energy equation:   

$$-\frac{\partial }{\partial x}\left(k\frac{\partial T}{\partial x}\right)=c_p\rho \frac{\partial T}{\partial t}$$(34)

Energy equation is discretized by the control volume method. Only one average temperature in the tangential direction is used for each layer. More nodal points could be solved to get better accuracy, but then the thermal radiation model becomes much more complex. For the ladle, the boundary condition for the internal surface is liquid convection heat transfer. At the outer surface, convection and radiation cools the surface. For the bell, the external surface boundary conditions are the same, but at the inner surface, we also consider the thermal radiation heat transfer from the free steel and pure aluminum surfaces as follows:   

$$\begin{array}{l}-k{S}_{B1}\frac{\partial T}{\partial x}={\alpha }_{Bell.in}{S}_{Bell.in}\left(T_G-T_{B1}\right)+\\ S_o\left(J_o{F}_{o-B1}-J_{B1}{F}_{o-B1}\right)+S_f\left(J_f{F}_{f-B1}-J_{B1}{F}_{f-B1}\right)\\ ={\alpha }_{Bell.in}{S}_{Bell.in}\left(T_G-T_{B1}\right)+S_o\left(J_o-J_{B1}\right)+S_f\left(J_f-J_{B1}\right)\end{array}$$(35)

B1 refers to the inner surface of the bell. Radiosities J_o, J_f and J_B1 are calculated as follows:¹⁹⁾   

$$J_{B1}=\frac{E_b\left(T_o\right){\epsilon }_o{F}_{B1-o}+E_b\left(T_f\right){\epsilon }_f{F}_{B1-f}+E_b\left(T_{B1}\right)\frac{{\epsilon }_B}{1-{\epsilon }_B}}{{\epsilon }_o{F}_{B1-o}+{\epsilon }_f{F}_{B1-f}+\frac{{\epsilon }_B}{1-{\epsilon }_B}}$$(36)

  

$$J_o=E_b\left(T_o\right){\epsilon }_o+J_{B1}\left(1-{\epsilon }_o\right)$$(37)

  

$$J_f=E_b\left(T_f\right){\epsilon }_f+J_{B1}\left(1-{\epsilon }_f\right)$$(38)

where E_b = σ T⁴ is the black body emissive power at temperature T. View factors F were calculated based on,²⁰⁾ F_B1–o = S_o / S_B1, F_B1–f = S_f / S_B1. Data for emissivity of the different surfaces were taken from,^20,21,22) ε _o = 0.9, ε _B1 = 0.28, and ε_f = 0.4. A significant fraction of the heat is lost from the slag surface outside the bell. This is calculated from the balance.   

$${\alpha }_{L.o}{S}_e\left(T_L-T_{S1}\right)=\frac{k_S{S}_e}{h_S}\left(T_{S1}-T_{S2}\right)={\epsilon }_e\mathrm{\sigma }{S}_e\left(T_{S2}^4-T_{atm}^4\right)$$(39)

Fig. 4.

Close description of the jet hit area during oxygen mode (OB), bell is lowered and sub-merged 30 cm below the steel surface to form a sealed environment. Aluminum particle size d_Al = 38 mm, metal droplet size d_D = 1.5 mm. Aluminum on the slag surface is shown by a dark gray broken line.

Emissivity of the slag surface is calculated from the data,²²⁾ ε _e = 0.9.

2.7. Heat and Mass Transfer Coefficients

Heat and mass transfer coefficients play a very important role in the model. They define the rate of the processes and also partly define the controlling mechanisms. Without proper values, the model cannot be reliably used. Heat and mass transfer in the area where the lance jet hits was studied in our recent paper.⁶⁾ As a result, a new heat and mass transfer correlation model was developed based on an experimentally validated CFD model, Eq. (40).   

$$S{h}_i=\frac{h_{i}D}{D_i}=S{c}_i^{0.42}GF$$(40)

Where G = 0.037–0.0083H and F = 2Re^0.5(1+0.005Re^0.55)^0.5

D_i is the diffusion coefficient of species i. D is the nozzle diameter. H is the vertical nozzle distance from the surface, and Re is the nozzle Reynolds number.

As the Ar bubbles rise, the liquid steel is carried along, and as a result, the melt will be mixed. The liquid rise velocity was obtained from our CFD simulations.^3,6) The residence times τ of the liquid under the different surface regions can be estimated as τ_o = (d_Bell/2 – a_f)/ u_Rise and τ_f = R_f asin(a_f /R_f) / u_Rise, respectively. R_f is the curvature of the depression, R_f = $\left(a_f^2+h_f^2\right)/\left(2h_f\right)$ Based on CFD simulations, the velocity is equal to 0.8 m/s.³⁾ We use the surface penetration model proposed by Higby²³⁾ to calculate the mass transfer during this contact period. Heat transfer coefficients can be obtained by replacing Sh with Nu and D_i with thermal diffusivity. Equation (41) is used for metal side mass transfer for metal-gas (f), metal-slag (o), and metal droplets-slag (D).   

$$h_{i,o/f/D}=2\sqrt{\frac{D_i}{\pi {\tau }_{o/f/D}}}$$(41)

Heat and mass transfer coefficients are similar for falling liquid steel droplets in the slag. We assume that liquid side mass transfer controls the metal droplet-slag reactions R8–11. On the slag side, oxides are present with high mass fractions, while on the liquid side mass fractions are at the ppm level. As the droplet falls, an internal circulation is created in the steel. The surface of the droplet is renewed the rate defined by the slip velocity and droplet size and residence time is calculated as follows.   

$${\tau }_D=\frac{d_D}{u_{t.D}}$$(42)

Slip velocity is calculated by Eq. (9). Heat and mass transfer coefficients can then be calculated from Eq. (41) for the droplets.

Heat and mass transfer around the melting Al particles is different from the metal droplets in slag or gas bubbles. This is a case of liquid passing a solid sphere calotte. The mass transfer coefficient was calculated from the Ranz-Marshall correlation²⁰⁾ as follows   

$$S{h}_{Al}=\frac{h_{Al}{d}_{Al}}{D_{Al}}=2+0.6{Re}^{0.5}S{c}_{Al}^{0.333}$$(43)

The heat transfer coefficient can be obtained by the analogy replacing Nu with Sh and Sc with Pr, respectively. The Reynolds number is calculated from the rise velocity 0.8 m/s.

2.8. Activity of the Species

Our system of species conservation equations was derived on the mass fraction basis. It is therefore necessary to write all activities based on mass fractions. The ideal gas law is assumed to be valid for gases.   

$$a_i=y_i\frac{M_G}{M_i}\frac{p_G}{p_N}$$(44)

Activity coefficients of dissolved species were calculated based on the Unified Interaction Parameter (UIP) Model.²⁴⁾ Wagner’s interaction model was first tested,²⁵⁾ but it was found to be non-suitable, as a numerical solution often jumps to non-dilute regions during the initial steps of iterations where a non-physical trend is predicted. Interaction parameters ${\epsilon }_i^j$ are given in Table 1; the references for these are given in the Table 1.

Table 1. First order molar interaction parameters for the UIP model, Henry’s law constants, Raoult reference state.^24,26)

ε i,j	Fe	Al	Si	Mn	C	O
Fe	0	0	0	0	0	0
Al	0	6	6.9	2.8	6.7	–104
Si	0	6.9	37	0	8.3	–17.3
Mn	0	2.8	0	0	–0.5	0
C	0	6.7	8.3	–0.5	11.4	–22.2
O	0	–104	–17.3	0	–22.2	–13.4
γ °	1	0.029	0.0013	1.3	0.57	1

Activity coefficients were defined as follows.   

$$ln\left({\gamma }_i\right)=ln\left({\gamma }_{Fe}\right)+ln\left({\gamma }_i^0\right)+\sum _{j=1}^N\left({\epsilon }_i^j{x}_j\right)$$(45)

The key difference from Wagner’s model is the first term on the right, i.e., the activity coefficient of the solvent, which in this case is iron. It is defined as follows.   

$$ln\left({\gamma }_{Fe}\right)=-0.5\sum _{j=1}^N\sum _{k=1}^N\left({\epsilon }_j^k{x}_j{x}_k\right)$$(46)

Substituting this into (45) and re-arranging gives us.   

$$ln\left({\gamma }_i\right)=ln\left({\gamma }_i^0\right)+\sum _{j=1}^N\left({\epsilon }_i^j{x}_j-0.5\sum _{k=1}^N{\epsilon }_j^k{x}_j{x}_k\right)$$(47)

Representing mole fractions in the terms of mass fractions   

$$ln\left({\gamma }_i\right)=ln\left({\gamma }_i^0\right)+\sum _{j=1}^N\left(\frac{{\epsilon }_i^{j}M}{M_j}{y}_j-0.5\sum _{k=1}^N\frac{{\epsilon }_j^k{M}^2}{M_j{M}_k}{y}_j{y}_k\right)$$(48)

For the slag phase, the Ban-Ya regular solution model is used.^27,28,29,30) The activity coefficient of species i is defined by   

$$RTln{\left({\gamma }_i\right)}_{RS}=\sum _{j=1}^N\left({\alpha }_{ij}{\overline{\mathrm{\delta }}}_{ij}{X}_j^2+\sum _{k=1}^N\left({\alpha }_{ij}+{\alpha }_{ik}-{\alpha }_{jk}\right)X_j{X}_k{\overline{\mathrm{\delta }}}_{ij}{\overline{\mathrm{\delta }}}_{ik}{\overline{\mathrm{\delta }}}_{jk}\right)$$(49)

  

$$RTln{\left({\gamma }_i\right)}_{s/l}=RTln{\left({\gamma }_i\right)}_{RS}+\mathrm{\Delta }{G}_{RS-l}$$(50)

${\overline{\mathrm{\delta }}}_{ij}$ is equal to inverse Kronecker delta, X is equal to the cation fraction defined as the ratio of anions in species i and anions in all slag species. N_O,i is equal to the number of O atoms in one molecule of species.   

$$X_i=\frac{x_i{N}_{O,\mathrm{\hspace{0.17em} }i}}{\sum _{j=1}^N{x}_i{N}_{O,j}}=\frac{\frac{y_i}{M_i}{N}_{O,\mathrm{\hspace{0.17em} }i}}{\sum _{j=1}^N\frac{y_i}{M_j}{N}_{O,\mathrm{\hspace{0.17em} }j}}$$(51)

and activities are calculated as a_i = γ_ix_i. Our system of slag species are defined below, giving the interaction energies and conversion Gibbs energies. The slag species are FeO, Al₂O₃, MnO, SiO₂, and CaO. Table 2 presents the properties.

Table 2. Slag activity model parameters,^27,28,29,30) conversion energy is between the regular solution state and Raoult state.

	Interaction energies αi,j of cations, J					Conversion energy RS-l, ΔG°, J
	FeO	Al2O3	MnO	SiO2	CaO
FeO	0	–41000	7110	–41840	–31380	–8540 + 7.142 × T
Al2O3	–41000	0	–83680	–127610	–154810	46719*
MnO	7110	–83680	0	–75310	–92050	–86860 + 51.465 × T
SiO2	–41840	–127610	–75310	0	–133890	+17450 + 2.82 × T
CaO	–31380	–154810	–92050	–133890	0	–40880 – 4.703 × T

*  Calculated from the activity data available in slag atlas³⁰⁾

2.9. Physical and Thermo-dynamical Properties

Physical and thermo-dynamical data was obtained from.^18,19,23) We used the effective diffusion model approach,¹⁵⁾ D_L = 4×10^–9 m/s, D_G = 0.21×10^–4 (T/298 K)^1.5(p/101325) m/s. The viscosities of steel and slag were 0.0061 Pa s and 0.0709 Pa s, respectively.²³⁾

2.10. Numerical Solution

The model for the CAS-OB process has a set of non-linear differential equations defined above. The equations are integrated with respect to time using the implicit Euler method. The system is well-defined by 36+12 unknown variables and 36+12 non-linear equations. The full Newton method was used to maximize the numerical efficiency.¹⁴⁾ For the best stability, the energy equations were solved semi-explicitly, which means that when mass transfer and reactions are solved, temperatures are kept constant, but the solution method was implicit itself. In the real industrial CAS-OB process, Al particles, O₂, and Ar are fed with different time periods, mostly overlapping, but also partly at different times and with varying rates. As we wanted to validate the model with these real batches, we needed to consider the feed rates and times accurately. In the model, feed rates are modeled with linear trends, marking the point where changes are made. Examples of these are presented below.

3. Results

The model is compared with process measurements obtained from the CAS-OB process at Ruukki Metals Oy, Raahe Works, Finland. Initial steel temperature is typically ~1600°C, O₂ flow is 2300–2600 m³/h, Al feed is 40–80 kg/min, and Ar flow is 400–800 l/min. Initial Al particle size was 38 mm, based on the measured average mass of used particles.

Although the distribution of steel droplets at their place of birth contains relatively large droplets in excess of 10 mm, these droplets are likely to disintegrate due to contact with the gas jet. The droplet break-up correlation proposed by Koria and Lange³¹⁾ yields that the upper size limit of the non-disintegrating metal droplets is approximately 2–3 mm. Moreover, the residence time of large droplets in the slag is very short, and the mass-transfer rate within the droplet is relatively slow in comparison to small droplets. It may be expected that the effective average size of the droplets residing in the slag is on the order of few millimeters. In this work, a value of 1.5 mm was employed.

Throat diameter of the lance nozzle was 30.5 mm, and gas exit velocity was 475 m/s. Height of the ladle was 2.655 m and liquid surface diameter 2.665 m. Height and diameter of the bell were 1.5 and 1.28 m, respectively. In this work, we used the process data measured during January 2014 at Ruukki Metals Oy Raahe Works in Finland and simulated three Al killed steels. At first, a comparison and temperature and composition predictions are presented, and then we carry out a deeper contribution analysis and energy balance study on process characteristics in more detail for all cases. Table 3 shows the feeding rates and steel temperature before and after the experiment. Table 4 presents the steel and slag composition.

Table 3. Input data and temperature predictions, feed periods are given as time intervals when materials are fed.

	Temperatures		Total input amounts			Feed periods
Batch	time min	t °C	Steel ton	mAl kg	VO2 Nm3	tO2 min	tAl min
A1	0	1620	119.6	94	67	5.33…	5.25…
Exp.	10.0	1628				6.87	6.60
Model	10.0	1628				2633 m3/min	69.6 kg/min
A2	0	1611	118.1	119	67	6.00…	5.85…
Exp.	10.0	1618				7.63	7.33
Model	10.0	1619				2465 m3/min	80.4 kg/min
A3	0	1603	120.3	144	91	7.4…	7.18…
Exp.	12.0	1615				9.6	9.2
Model	12.0	1617				2495 m3/min	71.3 kg/min

Table 4. Melt and slag compositions before and after the heat-up, A = Al-killed steel.

		Steel composition, ppm				Slag composition, kg
Batch	t, min	Al	Si	Mn	C	Al2O3	SiO2	MnO	FeO	CaO
A1	0	500	90	2020	500	386	64	15	41	630
Exp.	10	450 960a	90	1950	510	518 575b	67	29	56	630
Model	10	627	90	2018	500	552	64	16	42	630
A2	0	290	70	1950	390	381	65	23	72	637
Exp.	9	470 976a	90	1830	390	502 564b	72	38	105	637
Model	9	435	70	1948	389	551	64	23	73	637
A3	0	440	110	1710	330	395	60	20	49	646
Exp.	12	490 1267a	110	1580	334	534 650b	61	34	66	646
Model	12	643	109	1707	330	629	60	20	49	646

^a assumed that measured slag masses are correct, ^b assumed that end steel composition is correct

As can be seen from Table 3, temperature predictions are really good and errors are smaller than the accuracy of the temperature measurements. The predictions in steel and slag composition, Table 4, are not as good, but the main trends are clear. Cases a and b present hypothetical situations where we have assumed that a: solves the Al mass fraction, assuming that slag masses are correct; and b: solves Al₂O₃ mass, assuming that the fraction of Al is correct. It can be seen that the simulated values are between the analyzed value and the estimated value, i.e., they are within the errors of the balance limits. The analysis of the steel and slag composition is extremely challenging. For example, based on the analysis of the case A1, there was 94 kg of Al fed into the system, but only 64 kg can be found based on the analyses of steel and slag. For other cases, these values were 119/86 and 144/82. The errors resulting from these are 32%, 27%, and 43% in Al balance. Of course in the model, all Al can be found. Experiments and the model show that FeO is only an intermediate product and is reduced very effectively from the slag. It must be noted that the representative analysis of the slag is extremely difficult.

3.1. Heating Rates

One of the main objectives of the CAS-OB process is to control the steel temperature before the continuous casting. The heat-up rate is typically limited to ~10–15 K/min due to the risk of thermal cracking of the ladle or bell refractories.¹⁴⁾ Figure 5 presents the heat-up rate and fed Al remaining on the surface for the case A1 as an example.

Fig. 5.

Heat-up rate and fed Al at the surface, A1. 0–330 s: Ar-stirring/sampling, 330–450 s: Al feeding and O₂ blowing and 450–600 s: Ar-stirring/sampling.

As can be seen from Fig. 5, the heating rate rapidly reaches a maximum level when O₂ blowing is started, then keeps a constant level of ~10–12 K/min. There is also a peak in the heating rate at the end of the blowing, as the cooling effect of Al melting rapidly disappears. After the Al feed is stopped, the cooling begins and a cooling rate of 1.2–1.5 K/min is reached with no Al melting effects. This corresponds well to the observations from the real process, where typical free cooling rates were 1.5–2 K/min.¹⁴⁾

3.2. Contribution Analysis

In this part, we study the main heat generating reactions and where the energy is consumed. Figure 6 presents the contribution of different reactions and heat-loss terms in the CAS-OB process for the case A1 as an example. It is important to study these in order to make optimization and accurate temperature control possible. Exothermic reaction terms are positive and loss terms (including steel heat–up) are all negative in sign. During the lance blowing, both Al and Fe are oxidized, and then FeO will oxidize Al blown on the top of the side slag layer. Some Si and Mn are also oxidized, but these present only a minor fraction. This means that FeO is primarily an intermediate product in the process. The data given in Table 4 also supports this. Heat generation in a single batch varies between 700 and 1000 kWh. The efficiency of energy use is discussed in more detail below.

Fig. 6.

Generation and consumption of heat, A1.

As can be seen Fig. 6, there is a long waiting period first when the temperature, steel and slag compositions are measured. During this time, the steel is cooling at the rate of 1.3 K/min and energy is lost which is shown in a positive slope on the steel energy curve. When Al feeding is started, both Al and Fe are oxidized. When FeO is formed and Al is spattering on the slag layer, it is immediately oxidized by FeO. Energy is transferred to steel, but also a significant amount of heat is lost to the ladle and bell structures and also a part is lost via the surface slag. Part of the energy is transferred to the slag as it is pushed to the side, and of course part of the reaction heat has already been removed at the free surface by convection and radiation. As Fig. 6 show, as soon as Al feeding and blowing is stopped, the heat accumulated in the slag is recovered to steel.

3.3. Energy and O₂ Efficiency

Finally, we study here the efficiency of energy utilization and O₂ consumption. As an example, in Fig. 7, “Loss” = radiation loss through the top slag surface, “Steel” = heat used to heat up steel, and “Ladle” and “Bell” = heat transfer into the ladle and bell, respectively. “Gas” = energy required to heat up O₂ and Ar, and “Melting” = heat required to melt the fed Al. Thermal efficiency is the fraction of heat used to heat the steel.

Fig. 7.

Thermal efficiency and losses, A1.

As can be seen in Fig. 7, thermal efficiency of the CAS-OB treatment varies between 30–40%, and thus 60–70% of the chemical energy is lost. Most of the energy is lost through the slag surface, the bell structure, and the ladle walls. To minimize these, thicker insulations would be required and the slag surface should be covered with an insulated lid whenever possible and, obviously, to keep the free cooling periods to a minimum. It is known from practice³⁰⁾ that the average free cooling rate is 1.5–2 K/min, and the longer the waiting time, the larger the heat loss. Simulated oxygen efficiencies A1–3 were 0.81, 0.84, and 0.85, respectively.

4. Conclusions

This paper presented a new approach to the model CAS-OB process. More specifically, the oxygen blowing mode (OB) was considered. Melting of the solid aluminum particles, oxidation of pure molten aluminum, and oxidation of dissolved species, here Al, Mn, C, and Si, and the dissolvent Fe were considered. We also implemented energy equations for the liquid melt, the gas in the bell, and different sections of the gas-liquid interface to be solved to get their temperatures. As a significant fraction of the generated heat can be transferred and accumulated into the ladle and bell structures, simplified heat transfer models were developed to take this into account. The new model gives information on the heat-up rate and temperatures of the steel melt, gas, reaction surfaces, and refractory walls. The model yielded accurate predictions for steel composition, heating rate, and O₂ efficiency. For the studied heats, the predicted heating rate was 8–12 K/min versus the ~10 K/min measured average.¹⁴⁾ Based on our simulations, thermal efficiency of the process is greatly affected by the length of the free cooling period; typically 30–40% of the chemical energy can be used to heat the steel. Only a fraction of 0.8–0.85 of the O₂ can be utilized in the process; these correspond to the values observed in previous work.³⁰⁾ Most of the energy comes from the oxidation of the fed Al, and FeO is primarily an intermediate product of the reactions. The model was tested against industrial trials, and it succeeded to capture both main trends and high absolute accuracy. More validation experiments will be carried out in the future. On-line gas analysis measurements are also being planned.

Acknowledgements

This research is part of the Energy Efficiency & Lifecycle Efficient Metal Processes (ELEMET) research program coordinated by the Finnish Metals and Engineering Competence Cluster (FIMECC). Ruukki Metals Oy, Outokumpu Stainless Oy, and the Finnish Funding Agency for Technology TEKES are acknowledged for funding this work. The support provided to this work by the Academy of Finland (projects 258319 and 26495) is also acknowledged. We also want to thank Mr Markus Möttönen and all the Ruukki Metals Oy mill personnel for really nice co-operation during the measurement campaign in Raahe. Pertti Kiiski, Vadim Desyatnyk, Seppo Poimuvirta, Miika Ruutiainen, and Petrus Kiiski are acknowledged for preparing all the required measurement systems at Aalto.

List of symbols

a activity, –

a_f radius of the depression, m

c_p specific heat capacity, J/kg K

d diameter, m

D diffusion coefficient, m²/s

f activity coefficient, –

g gravitational acceleration, 9.81 m/s²

G geometry factor, –

ΔG Gibbs free energy, J/mole

F view factor, flow factor, –

h specific enthalpy, J/kg

h mass transfer coefficient, m/s

h_f depth of the lance depression, m

H lance height, m

J radiosity, W/m²

k reaction rate coefficient, –

K equilibrium constant, –

m mass, kg

$\dot{m}$  mass flow rate, kg/s

$\dot{M}$  momentum flow rate, N

$\underset{\_}{\dot{M}}$  non-dimensional momentum flow rate, –

N_b blowing number, –

Nu Nusselt number, –

p pressure, Pa

Pr Prandt number, –

r_m rate of melting, 1/s

R reaction rate vector, kg/m² s

R_B droplet generation rate, kg/s

Re Reynolds number, –

R_u gas constant, J/mol K

S surface area, m²

Sc Schmidt number, –

Sh Sherwood number, –

T absolute temperature, K

t time, s u velocity, m/s

$\dot{V}$  volume flow rate, m³/s

x mole fraction, –

X cation fraction, –

y mass fraction, –

Greek symbols

α convective heat transfer coefficient, W/m²K

α_i,j interaction energy, J

ε emissivity, –

ε_i,j interaction coefficient, –

δ_i,j Kronecker delta, –

γ activity coefficient, –

λ thermal conductivity, W/mK

η heat transfer efficiency, –

ρ density, kg/m³

σ surface tension, N/m

τ melting time, s

τ_D residence time, s

ν stoichiometric coefficient, –

Subscripts symbols

b black body

B bell

D droplet

f open eye

G gas

J jet

L liquid

NTP normal temperature 298 K and pressure 101325 Pa

s surface, solid

sub submerged

S slag

o slag surface

w wall

∞ bulk condition

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