A model for the reduction stage (Article III)

i=1

A²_iλiexp(−16λiFoM)

∑∞ i=1

A²_iexp(−16λiFoM) , (261)

Sh=0.65 µS

µS+µL

¹/2

Re^1/2Sc^1/2, (262)

where FoMis the Fourier number for mass transfer. The values for parametersAi

andλiwere obtained from the literature [329]. Here, the average residence time of the metal droplets (tmd) was employed as the characteristic time in Fourier number for mass transfer (FoM) and in the corresponding Fourier number for heat transfer (FoH). By making use of the analogue of heat and mass transfer, the heat transfer correlations were obtained by substituting Sh, Sc, and FoMwith Nu, Pr, and FoH, respectively [279].

6.3 A model for the reduction stage (Article III)

The aim of the reduction stage is decrease the losses of valuable metallics to the top slag in order to improve the overall economy of the process. For this purpose, reductants and fluxes are added to the vessel [128]. Typical ferrous reductants have a density lower than that of iron (see Table 20) and are thus subject to buoyancy induced by the density difference. Reductants with a density higher than that of the top slag should, in principle, eventually settle between the metal and slag phases, while lighter reductants should rise on top of the slag phase. Nevertheless, the trajectories of the reductant particles are likely to be affected considerably by the fluid flows in the metal bath. In a physical modelling study of the reduction stage, Guthrieet al. [128] found that the wooden reductant particles were generally well-mixed in a "metal–slag" foam,viz.a mixture of the zinc chloride and silicone oil used to represent the metal and slag phases, respectively. The particles showed a tendency to return back to the plume along the sidewalls, only to be lifted by the plume back to the foam [128].

The reduction stage is characterised by efficient mixing induced by vigorous argon stirring [62, 99, 102, 103]. Physical [102, 128] and numerical [62, 101] modelling studies on the reduction stage of the AOD process suggest a relatively strong emulsification of the top slag. In this work, it was assumed that the reactions during the reduction stage of the AOD process take place solely between emulsified slag droplets and metal bath.

Table 20. Properties of ferrous reductants, liquid steel and slag.

Material State Density [kg/m³] Melting point [K] Reference

50FeSi Solid 6100 1683 [532]

75FeSi Solid 2800 1589 [533]

SiMn Solid 6120 1488 [533]

Steel (0.1 wt-% C) Liquid (at 1873 K) 7050 1803 [533]

CaO-SiO₂-Cr₂O₃slag Liquid 3200–3500 1758 [533]

6.3.1 Conservation of species, mass, and energy

The parallel reversible reactions considered by the model are the same as in the top-blowing model,i.e.Eqs. 213–219 (p. 152), and are not repeated here. Because the reactions are formulated as reversible, the simple oxidation reactions form a complex reaction system, in which species with a higher oxygen affinity are able to reduce oxides of species with a lower oxygen affinity, thereby establishing a competitive equilibrium at the reaction interface. The rate expressions were formulated according to the modified law of mass action, while the forward reaction rate coefficients were treated according to the reaction quotient method. The mass transport terms were formulated according to thestationary medium approach[279]. Consequently, the conservation of speciesiin phaseψ at the slag droplet interface (sd) was defined as

βi,ψ,sdρψ,sd yi,B−y^∗_i,sd

| {z }

mass transport

+

∑

R⁰⁰_k,sdνi,k

| {z }

chemical reactions

=0, (263)

whereβ is the mass transfer coefficient,ρis the density,yis the mass fraction,y^∗is the interfacial mass fraction,νis the mass-based stoichiometric coefficient, andR⁰⁰is the reaction rate flux. Employing theimplicit Euler methodfor time integration, the conservation of speciesiof phaseψin bulk phaseBwas defined according to

−βi,ψ,sdρψ,sdAsd yi,B−y^∗_i,sd

| {z }

mass transport

−mByi,B−m^t_B^−∆ty^t_i,B^−∆t

| {z∆t }

accumulation of mass

=0, (264)

whereAis the reaction area and∆tis the time step. The conservation of total masses in the bulk phases was defined by summation of the mass transport terms of individual species. The conservation of heat at the slag droplet interface was defined as

αL,sd(Tbath−T_sd^∗) +αG,sd Tplume−T_sd^∗

+αS,sd Tslag−T_sd^∗

| {z }

heat transport

−

∑

R⁰⁰_k,sd∆hk

| {z }

chemical reactions

=0,

(265) where∆hkis the specific enthalpy change of reactionk. In addition, the conservation of energy was defined in each of the bulk phases (metal bath, plume gas, and top slag).

Their mathematical treatment, however, is not repeated here. The effect of refractory wear on the slag composition is calculated by assuming a fixed dissolution rate of refractory material into the slag. The dissolution rate is calculated from the loss of refractory lining thickness per heat.

6.3.2 Emulsification of slag

At the metal–slag interface, the turning flow of the molten metal can detach slag droplets, which brings about a considerable increase in the metal–slag interfacial area [1, 234–236, 534]. A schematic illustration of the emulsification mechanism is shown in Fig 25. Because the density of the slag droplets is much lower than that of molten steel, the buoyancy force eventually lifts the slag droplets back to the top slag [1, 234–236, 534]. There is little information on the size distributions and residence times of slag droplets in converter and ladle processes. Some useful experimental information has been provided by Lachmundet al.[535], who analysed slag droplet size distributions in a ladle treatment using a metal plate, which was lowered into the ladle. In contrast to the lack of data from actual processes, emulsification has been studied extensively with physical models [365, 534, 536–542]. Although the results are not directly applicable to industrial processes, these studies have provided valuable information on the physical properties and mechanisms affecting the emulsification phenomena. In recent years, CFD-based methods have also been applied for detailed studies of emulsification [234–236, 542, 543].

The description of emulsification employed in this work is based on the fundamental principles of emulsification proposed by Wei and Oeters [1, 534] and no consideration is given to periodic changes due to plume oscillation or possible disintegration of the slag droplets. The condition for droplet formation can be described according to the following force balance [1, 534]: