*2.2. The Impact Zone*

The impact zone in the BOF is the region where the oxygen jet impinges on the molten metal bath (Figure 1) causing a deformation on the liquid surface that is assumed to have the shape of a paraboloid of height *hc* and maximum diameter *dc*. The number of impact zones is equivalent to the number of nozzles *nn* in the lance as long as there is no coalescence of the oxygen jet. The area of one impact zone is equal to the surface area of the paraboloid:

$$A\_{iz} = \frac{d\_c \pi}{12h\_c^2} \left[ \left( \frac{d\_c^2}{4} + 4h\_c^2 \right)^{1.5} - \frac{d\_c^3}{8} \right] \tag{16}$$

where *dc* and *hc* are calculated using the correlations proposed in [15]. For the current work, the decarburization rates previously used by Dogan et al. [6] are modified by a parameter *α<sup>p</sup>* in order to account for the difference between the conditions at which the equation rates were derived and the BOF operating conditions. These parameters can be tuned using process data. Therefore, the decarburization rate via O2 (Equation (1)) is -2*rO*2,*iz*, where:

$$1 - r\_{O\_2, iz} = a\_{O\_2, t}^p A\_{iz} n\_{\eta} k\_{O\_2} \ln(1 + P\_{O\_2}) \tag{17}$$

with:

$$a\_{O\_2,t}^p = \frac{a\_{O\_2}^p}{1 + a\_{Si,C}^p[\%Si]} \tag{18}$$

The partial pressure in Equation (17) is in atm, and the parameter *α<sup>p</sup> Si*,*<sup>C</sup>* in Equation (18) accounts for the inhibiting effect that silicon has on the rate of carbon oxidation [14]. Similarly, the decarburization rate via CO2 (Equation (2)) is given by:

$$1 - r\_{\text{CO}\_2,iz} = a\_{\text{CO}\_2}^p A\_{iz} n\_n k\_a P\_{\text{CO}\_2} \tag{19}$$

In the above expressions, *ka* and *kO*<sup>2</sup> are the rate coefficients calculated using the correlations found in Dogan et al. [6] and *P* is the partial pressure calculated as:

$$P\_i = \frac{F\_i}{\sum\_{\vec{i}} F\_{\vec{i}}} P\_d \quad \text{for } i \in \{O\_2, CO, CO\_2, N\_2/Ar\} \tag{20}$$

where *Pa* is the ambient pressure. In Equation (20), *FO*<sup>2</sup> and *FN*2/*Ar* are the inlet molar flow rate of oxygen and bottom stirring gas, and *FCO* and *FCO*<sup>2</sup> are the molar flux of carbon monoxide and carbon dioxide formed from the decarburization reaction in Equations (1) and (2) and the post-combustion reaction in Equation (6).

When the carbon content of the metal bath [%*C*] becomes lower than the critical carbon content *CC*, the decarburization rate is given by [6]:

$$-r\_{\mathbb{C}\_c, i\mathbb{Z}} = a\_{\mathbb{C}\_c}^p k\_{\mathbb{H}} \frac{A\_{iz} n\_{\mathbb{H}}}{V\_b} [\% \mathbb{C}] \tag{21}$$

where *Vb* is the volume of liquid metal and *km* is the rate constant calculated using the correlation developed by Kitamura et al. [16].

The desiliconization rate is calculated using the expression found in Rout et al. [17] and modified by a parameter *α<sup>p</sup> Si* to give:

$$-r\_{Si,iz} = \alpha\_{Si}^p k\_{\mathcal{W}} \rho\_b(\left[\%Si\right] - \left[\%Si\_{cq}\right]) A\_{iz} n\_{\mathcal{W}} \tag{22}$$

where *ρ<sup>b</sup>* is the density and [%*Si*] is the silicon content of the liquid metal. Rout et al. [9] found that the equilibrium silicon content of the metal bath ([%*Sieq*]) is approximately zero, therefore it is neglected in the calculations. For the current study, it is assumed that the rate of iron oxidation is proportional to the partial pressure of oxygen and is given by:

$$1 - r\_{\text{Fe},iz} = a\_{\text{Fe}}^p A\_{iz} n\_n P\_{O\_2} \tag{23}$$

All the oxygen injected in the system via the lance that is not used for decarburization, desiliconization or iron oxidation, is assumed to be used in the post-combustion of CO.

#### *2.3. Scrap Melting*

In the BOF, the heat generated by the oxidation reactions (Equations (1) and (4)–(6)) is much higher than that required to reach the targeted end-point temperature. Therefore, scrap metal is usually added in order to absorb part of the surplus heat via melting. In this section the model used to compute the scrap melting rate *W*˙ *sc* as well as the heat absorbed *Qsc* is presented.

Consider a scrap metal plate of half-thickness *L*, initial temperature *Tsc*<sup>0</sup> and melting temperature *Tm*, as shown in Figure 3a.

**Figure 3.** Schematic representation of the melting of scrap. (**a**) Schematic representation of a plate. (**b**) Schematic representation of temperature gradient between hot metal and a cold metal plate.

This plate is then submerged in a metal bath at temperature *Tb* > *Tm* and carbon content [%*C*]. Assuming that heat transfer occurs only in the axial direction and constant physical properties for the plate, a heat balance at the interface between the solid plate and the metal bath gives [11,18–20]:

$$-k\_{\text{sc}}A\_{\text{sc}}\frac{\partial T\_{\text{sc}}(0,\text{t})}{\partial \text{x}} - h\_{\text{sc}}A\_{\text{sc}}(T\_{\text{b}} - T\_{\text{m}}') = \frac{dL}{dt}\rho\_{\text{sc}}(\Delta H\_{\text{sc}} + \mathbb{C}\_{\text{P,sc}}(T\_{\text{b}} - T\_{\text{m}}'))A\_{\text{sc}} \quad T\_{\text{b}} \ge T\_{\text{m}} \tag{24}$$

where *ksc* is the thermal conductivity of the scrap plate, *hsc* is the heat transfer coefficient, *ρsc* is the scrap density, Δ*Hsc* is the latent heat of melting of the scrap, *CP*,*sc* is the specific heat of the scrap, *Asc* is the interfacial area, *x* is the distance of a point within the scrap from the interface, *t* is time and *Tsc*(*x*, *t*) is the scrap temperature at a position *x*. A schematic representation can be found in Figure 3b.

Equation (24) can be solved using the Quasi-Static approach to yield Equations (25)–(27) [21]:

$$\frac{dL}{dt} = \frac{\sum\_{n=1}^{\infty} \frac{-A\_n}{n\pi} (1 - \cos(n\pi)) \lambda^2 k\_{\text{sc}} L \ e^{-\lambda^2 a\_{\text{sc}}t} - h(T\_b - T\_m')}{\rho\_{\text{sc}} (\Delta H\_{\text{sc}} + \mathbb{C}\_{P, \text{sc}} (T\_b - T\_m')) - \sum\_{n=1}^{\infty} \frac{A\_n}{n\pi} (1 - \cos(n\pi)) (\rho\_{\text{sc}} \mathbb{C}\_{P, \text{sc}} + 2\lambda^2 k\_{\text{sc}} t) e^{-\lambda^2 a\_{\text{sc}}t}} \tag{25}$$

$$
\lambda = \left(\frac{n\pi}{2L(t)}\right) \tag{26}
$$

$$A\_{\pi} = 2(T\_{\text{sc}\_0} - T\_m') \frac{1 - \cos(n\pi)}{n\pi} \tag{27}$$

where *α* = *ksc*/*ρscCPsc* is the thermal diffusivity. The scrap melting rate *W*˙ *sc* is then given by:

$$
\dot{W}\_{\text{sc}} = -n\_{\text{sc}} 2\rho\_{\text{sc}} A\_{\text{sc}} \frac{dL}{dt} \tag{28}
$$

where *nsc* is the total number of scrap plates.

At the beginning of the BOF operation, the temperature of the molten metal may not be high enough to melt the scrap, in which case the interface temperature *T <sup>m</sup>* is equal to the bath temperature *Tb*. The interface temperature is otherwise assumed to correspond to the melting temperature. This is shown in Equation (29) :

$$T\_m' = \begin{cases} T\_b & T\_b < T\_m \\ T\_m & T\_b \ge T\_m \end{cases} \tag{29}$$

The melting temperature *Tm* is dependent on the carbon content *Cm* at the melting interface and can be calculated using Equation (30) [22]:

$$T\_{\rm II} = \begin{cases} 1810 - 90\% \text{C}\_{\rm m} & 0 \le \% \text{C}\_{\rm m} \le 4.27\%\\ 1425 & \% \text{C}\_{\rm m} > 4.27\% \end{cases} \tag{30}$$

Dogan [22] assumed *Cm* to be equal to the carbon content in the bulk metal [%*C*]. This assumption reflects, to some extent, what happens in the BOF: Carbon in the metal bath migrates to the scrap surface lowering its melting temperature [21,23]. One consequence of such an assumption is that scrap types of similar physical properties will all melt at the same time, independent of their carbon content. On the other hand, if *Cm* is assumed to be equal the carbon content of the scrap *Csc*, low-carbon content scrap types will only melt towards the very end of the blow, which is not observed in practical BOF operations. For the current work, *Cm* is defined as:

$$\mathcal{C}\_m = 0.8[\% \mathcal{C}] + 0.2 \mathcal{C}\_{\text{sc}} \tag{31}$$

The heat transfer coefficient *hsc* in Equation (25) is calculated according to the correlation proposed by Gaye et al. [24] and given by:

$$h\_{\kappa} = 5000\dot{\epsilon}^{0.2} \tag{32}$$

where ˙ is the mixing power due to bottom stirring and top blowing in Wm−3, and *hsc* is in units of Wm−2K−1. The rate *Qsc* at which heat is absorbed by the scrap is given by the heat conduction term in Equation (24) before melting starts, and by the convective term once melting starts as shown in Equation (33).

$$Q\_{\mathfrak{sl}} = 2n\_{\mathfrak{sl}}A\_{\mathfrak{sl}} \begin{cases} -\frac{\partial T\_{\mathfrak{sl}}(0,t)}{\partial \mathbf{x}} & T\_b < T\_m \\ h(T\_b - T\_m') & T\_b \ge T\_m \end{cases} \tag{33}$$
