**2. Mathematical Model**

### *2.1. Mass and Energy Balances*

The phenomena taking place in the BOF are quite complex, therefore several assumptions are made for the derivation of the mass and energy balances. Flux, iron ore, metal droplets and gas bubbles are considered to be uniformly dispersed in the slag phase. The metal droplets, gas bubbles and slag form an emulsion for which the continuous phase is assumed to be the slag. The oxidation reactions take place mainly at the impact zone (IZ), the interface between the oxygen jet and the metal bath, and there is no resistance to the diffusion of oxides from the metal bath to the slag. A schematic representation of the flow of material in the BOF is shown in Figure 2.

**Figure 2.** Material flow in the BOF assumed for the current study.

The reactions modelled by the current work are as follows [14], where heats of reaction, Δ*Hrxn*, were obtained from FactSage at 1900 K:

Decarburization:

$$\text{H}\_{\text{(C)}} + \frac{1}{2}\text{O}\_{2(\text{g})} \longrightarrow \text{CO}\_{\text{(g)}} \qquad \Delta H^{rxn} = -118.613 \text{ kJ/mol} \tag{1}$$

$$\text{H}\_{\text{(C)}} + \text{CO}\_{2(g)} \longrightarrow 2\text{CO}\_{(g)} \qquad \Delta H^{rxn} = 160.623 \text{ kJ/mol} \tag{2}$$

$$\text{(FeO)} + \text{[C]} \longrightarrow \text{[Fe]} + \text{CO}\_{(6)} \qquad \Delta H^{rxn} = 124.644 \text{ kJ/mol} \tag{3}$$

Desiliconization:

$$\text{[Si]} + \text{O}\_{2(g)} \longrightarrow \text{(SiO}\_2\text{)} \qquad \Delta H^{rxn} = -945.172 \text{ kJ/mol} \tag{4}$$

Iron Oxidation:

$$\text{(Fe)} + \frac{1}{2}\text{O}\_{2(g)} \longrightarrow \text{(FeO)} \qquad \Delta H^{rxn} = -242.783 \text{ kJ/mol} \tag{5}$$

Post-combustion:

$$\text{CO} + \frac{1}{2}\text{O}\_2 \longrightarrow \text{CO}\_{2(g)} \qquad \Delta H^{rxn} = -277.927 \text{ kJ/mol} \tag{6}$$

Argon or nitrogen can be injected at the bottom of the BOF to improve mixing. Even though the injected nitrogen can undergo reaction at BOF operating conditions, the amount of nitrogen compounds formed is small and can be neglected. Therefore, it is assumed that the stirring gas leaves the furnace unreacted. At the impact zone, iron, carbon and silicon react with the injected oxygen forming iron oxide, carbon monoxide and silica as shown in Equations (1)–(5). Some of the CO may further react with O2 forming CO2 (Equation (6)).

The mass of dissolved flux and oxides is promptly incorporated by the slag. Similarly, the mass of melted scrap is assimilated by the metal bath. All Fe in the iron ore is assumed to be in the form of magnetite (Fe3O4) that is reduced to FeO by carbon in the metal bath (Equation (7)).

$$\text{Fe}\_2\text{O}\_4 + [\text{C}] \longrightarrow (3\,\text{FeO}) + \text{CO}\_{(\text{g})} \qquad \Delta H^{rxn} = 184.660 \text{ kJ/mol} \tag{7}$$

The impact of the oxygen jet on the liquid metal causes millions of metal droplets to be ejected in the emulsion zone. Carbon in the metal droplets can reduce FeO in the slag according to the reaction shown in Equation (3). The refined droplets then fall back to the metal bath.

The gases CO, CO2, N2/Ar and unreacted oxygen O2 form the off-gas stream. Owing to the high temperatures, the residence time for the gases is assumed to be negligible in this work. A mass balance for the metal bath gives:

$$\frac{d\mathcal{W}\_b}{dt} = \dot{\mathcal{W}}\_{\rm sc} - \dot{\mathcal{W}}\_{\rm C,IO} + \dot{\mathcal{W}}\_{D,\varepsilon-b} - \dot{\mathcal{W}}\_{D,b-\varepsilon} - \dot{\mathcal{W}}\_{\rm C,b-\varepsilon} - \dot{\mathcal{W}}\_{\rm Fe,b-\varepsilon} - \dot{\mathcal{W}}\_{\rm Si,b-\varepsilon} \tag{8}$$

where *Wb* represents the mass of the metal bath. *W*˙ *sc* is the scrap melting rate, *W*˙ *<sup>C</sup>*,*IO* is the rate at which carbon in the metal bath is consumed to reduce magnetite in the iron ore to FeO, *<sup>W</sup>*˙ *<sup>D</sup>*,*b*−*<sup>e</sup>* and *<sup>W</sup>*˙ *<sup>D</sup>*,*e*−*<sup>b</sup>* are the mass flow rate of droplets from the metal bath to the emulsion and from the emulsion to the metal bath, respectively. *<sup>W</sup>*˙ *<sup>C</sup>*,*b*−*e*, *<sup>W</sup>*˙ *Fe*,*b*−*e*, *<sup>W</sup>*˙ *Si*,*b*−*<sup>e</sup>* are the mass flow rate of carbon, iron and silicon out of the metal bath and equals the oxidation rate of the respective element at the impact zone. A similar mass balance on the slag–metal–gas emulsion gives:

$$\frac{d\mathcal{W}\_{\text{Sure}}}{dt} = \dot{\mathcal{W}}\_{\text{Diss}\text{Flux}} + \dot{\mathcal{W}}\_{\text{FeO,IO}} + \dot{\mathcal{W}}\_{\text{FeO,b-e}} + \dot{\mathcal{W}}\_{\text{SiO}\_2,b-e} + \dot{\mathcal{W}}\_{\text{D},b-e} - \dot{\mathcal{W}}\_{\text{D},c-b} - \dot{\mathcal{W}}\_{\text{CO,EZ}} \tag{9}$$

where *Wsme* is the mass of the slag and droplets in the emulsion. *W*˙ *DissFlux* is the rate of lime and dolomite dissolution, *<sup>W</sup>*˙ *FeO*,*IO* denotes the mass flow rate of iron oxide coming from iron ore, *<sup>W</sup>*˙ *FeO*,*b*−*e*, *<sup>W</sup>*˙ *SiO*2,*b*−*<sup>e</sup>* are the mass flow rate of iron oxide and silicon dioxide into the slag and are equal to the rate of formation of the respective component at the impact zone. *W*˙ *CO*,*EZ* is the rate carbon monoxide is formed at the emulsion zone due to droplet decarburization.

Heat is generated in the BOF by the oxidation and post-combustion reactions (Equations (1) and (4)–(6)) and is consumed by scrap melting, to heat the fluxes and iron ore, by decarburization in the emulsion (Equation (3)) and carbon dioxide reduction (Equation (2)). The total heat *Qrxn gen* generated by the oxidation reactions is given by

$$Q^{rxn}\_{\mathcal{S}^{rxn}} = Q^{rxn}\_{\text{FeO}} + Q^{rxn}\_{\text{SiO}\_2} + Q^{rxn}\_{\text{CO}} + Q^{rxn}\_{\text{CO}\_2} \tag{10}$$

where *Qrxn FeO* is the heat from iron oxidation (Equation (5)), *<sup>Q</sup>rxn SiO*<sup>2</sup> is the heat from silicon oxidation (Equation (4)), *Qrxn CO* is the net heat generation from decarburization at the impact zone (Equations (1) and (2)), *Qrxn CO*<sup>2</sup> is the heat of CO post-combustion. *<sup>Q</sup>rxn <sup>i</sup>* is obtained by multiplying the rate of formation of compound *i* at the impact zone by the respective heat of reaction.

Individual energy balances were derived for the emulsion and metal bath. To simplify the problem, the following assumptions are made:


An energy balance for the slag–metal–gas emulsion yields:

$$\begin{aligned} \left(\mathcal{W}\_{\boldsymbol{\theta}}\mathbb{C}\_{P,\boldsymbol{\varepsilon}} + \mathcal{W}\_{D}\mathbb{C}\_{P,b}\right) \frac{d\boldsymbol{T}\_{\boldsymbol{s}}}{dt} &= \left(1 - a\_{1}^{p}\right)Q\_{\mathcal{S}\boldsymbol{\varepsilon}\boldsymbol{m}}^{\mathrm{r}\boldsymbol{m}} - Q\_{\mathrm{Fe},\boldsymbol{E}\boldsymbol{Z}}^{\mathrm{r}\boldsymbol{m}} - Q\_{D,b-\boldsymbol{\varepsilon}} \\ &- Q\_{\mathrm{Fe},b-\boldsymbol{\varepsilon}} - Q\_{\mathrm{SiO}\_{2},b-\boldsymbol{\varepsilon}} - Q\_{\mathrm{CO},b-\boldsymbol{\varepsilon}} - Q\_{\mathrm{CO}\boldsymbol{2},b-\boldsymbol{\varepsilon}} \\ &- Q\_{\mathrm{IO}} - Q\_{\mathrm{c}-b} - Q\_{\mathrm{flux}} - Q\_{\mathrm{Ni}/A}\boldsymbol{r}\_{\boldsymbol{s}}\boldsymbol{b} - \boldsymbol{\varepsilon} - Q\_{\mathrm{O}\_{2},b-\boldsymbol{\varepsilon}} \end{aligned} \tag{11}$$

with

$$Q\_{i,b-\epsilon} = \dot{W}\_{i,b-\epsilon} \mathbb{C}\_{P,i}(T\_s - T\_b) \quad \text{for } i \in \{\text{FeO, SiO}\_2, \text{CO, CO}\_2, \text{N}\_2/\text{Ar}, \text{O}\_2\} \tag{12}$$

In the above, *Ws* is the mass of slag, *WD* is the mass of the droplets in emulsion, *CP*,*s*, *CP*,*b*, *CP*,*<sup>i</sup>* are the heat capacity of the slag, molten metal and component *i*, respectively. *QFe*,*EZ* is the heat consumed by the decarburization reaction in the emulsion, *QD*,*b*−*<sup>e</sup>* is the heat required to raise the temperature of the incoming droplets to the temperature of the slag–metal–gas emulsion; *QFeO*,*b*−*e*, *QSiO*2,*b*−*e*, *QCO*,*b*−*e*, *QCO*,*b*−*<sup>e</sup>* and *QCO*2,*b*−*<sup>e</sup>* are the heat required to raise the temperature of the iron, silicon, carbon oxide and carbon dioxide formed at the impact zone to the emulsion temperature, respectively; *Qe*−*<sup>b</sup>* is the heat lost from the emulsion to the metal bath, *Qflux* and *QIO* are the heat consumed by the flux and iron ore additions, *QN*2/*Ar*,*b*−*<sup>e</sup>* is the heat required to raise the temperature of the stirring gas from the bath to the emulsion temperature, *QO*2,*b*−*<sup>e</sup>* is the heat required to raise the temperature of the non-reacted oxygen from the bath to the emulsion temperature. The rate of heat transfer between the slag and metal bath is given by:

$$Q\_{\mathfrak{s}-b} = h\_{\mathfrak{s}-b} \pi R^2 (T\_{\mathfrak{s}} - T\_b) \tag{13}$$

where *R* is the furnace diameter and *hs*−*<sup>b</sup>* is the heat transfer coefficient, which can be estimated using dynamic data. Applying an energy balance for the metal bath gives:

$$\mathcal{W}\_b \mathcal{C}\_{P,b} \frac{dT\_b}{dt} = Q\_{c-b} + Q\_{D,c-b} - Q\_{O\_2,in} - Q\_{N\_2/Ar,in} - Q\_{\mathbb{S}\mathcal{C}} \tag{14}$$

where *QD*,*e*−*<sup>b</sup>* is the heat transferred by the droplets coming from the emulsion to the metal bath, and *Qsc* is heat used to melt the scrap, *QO*2,*in* and *QN*2/*Ar*,*in* are the heat consumed to raise the temperature of oxygen and stirring gas to the bath temperature:

$$Q\_{i,in} = F\_i \mathbb{C}\_{P,i}(T\_b - T\_{i,0}) \quad \text{for } i \in \{N\_2/Ar, O\_2\} \tag{15}$$

where *Fi*, *Ti*,0 are the inlet mass flow rate and temperature of gas *i*.
