*3.3. Decarburization Modeling*

The current decarburization modeling focuses on the refining process after the solid scrap is completely melted down into a flat bath. Therefore, the scrap melting phenomenon is not included in the present study for the sake of simplicity. The model proposed here considers a liquid steel-oxygen two-phase reacting flow system inside the flat bath and the simulation domain, as mentioned above, only includes the bottom section of EAF with estimated jet penetration cavities, through which the oxygen enters the domain to react with carbon and other impurities. The injected oxygen also results in two main effects on the system including the stirring of the liquid steel bath and the bath temperature rise due to the heat released by the oxidation reactions. In the present study, the oxidations of the carbon, iron and manganese as a mixture of liquid steel by the injected oxygen are listed in Table 1:

**Table 1.** Oxidation reactions (A) to (C) considered in present study.


Oxidation reactions take place in cells of the simulation domain that contain the oxygen. The oxidation rates of carbon, iron and manganese at high carbon content are mainly limited by the amount of oxygen contained in the same cell. If the oxygen is sufficient, the rate equations can be written as suggested by Wei and Zhu [24]:

$$-\frac{\mathcal{W}\_{\rm s}}{100 \, M\_{\rm C}} \frac{d[\% \mathcal{C}]}{dt} = \frac{2\eta\_{\rm C} Q\_{O\_2}}{22 \, 400} \mathbf{x}\_{\rm C} \tag{20}$$

$$-\frac{\mathcal{W}\_s}{100\,\mathrm{M}\_{\mathrm{Mu}}}\frac{d[\%\mathrm{M}n]}{dt} = \frac{2\eta\_{\mathrm{Mu}}Q\_{\mathrm{O}\_2}}{22\,\mathrm{A}00}\mathbf{x}\_{\mathrm{Mu}}\tag{21}$$

where *Ws* and *QO*<sup>2</sup> is the mass of liquid steel and the volume of oxygen in the corresponding cell, respectively; *Mi* is the mole mass of each substance; η*<sup>i</sup>* is the efficiency factor of each substance, which is a function of total mixing of the system and can be estimated based on the work done by Shukla et al. [25]; *xi* is the oxygen distribution ratios of each substance and is assumed to be proportional to the Gibbs free energies of corresponding oxidation reactions:

$$\mathbf{x}\_{\mathbb{C}} = \frac{\Delta \mathbf{G}\_{\mathbb{C}}}{\Delta \mathbf{G}\_{\mathbb{C}} + \Delta \mathbf{G}\_{\mathbb{F}t} + \Delta \mathbf{G}\_{Mn}} \tag{22}$$

$$
\Delta \mathbf{x}\_{Mn} = \frac{\Delta \mathbf{G}\_{Mn}}{\Delta \mathbf{G}\_{\mathbf{C}} + \Delta \mathbf{G}\_{\mathbf{F}t} + \Delta \mathbf{G}\_{Mn}} \,' \tag{23}
$$

where the Gibbs free energies Δ*Gi* of respective substance can be defined as:

$$
\Delta G\_{\text{C}} = \Delta G\_{\text{C}}^{0} + RT \ln \left[ \frac{P\_{\text{CO}}}{a\_{\text{C}} \cdot a\_{O\_2}^{0.5}} \right] \tag{24}
$$

$$
\Delta G\_{\rm Fe} = \Delta G\_{\rm Fe}^{0} + RT \ln \left[ \frac{a\_{\rm FeO}}{a\_{\rm Fe} \cdot a\_{O\_2}^{0.5}} \right] \tag{25}
$$

$$
\Delta G\_{Mn} = \Delta G\_{Mn}^{0} + RT \ln \left[ \frac{a\_{MnO}}{a\_{MnO} a\_{O\_2}^{0.5}} \right] \tag{26}
$$

where Δ*G*<sup>0</sup> *<sup>i</sup>* and *ai* is the standard Gibbs free energy and the activity of each substance in the bath respectively; *R* is gas constant; *PCO* is the partial pressure of carbon monoxide.

At low carbon content, the oxidation rate of carbon is no longer controlled by the oxygen contained in the cell. Instead, the mass carbon transfer rate to liquid steel will directly impact the decarburization rate, which can be expressed as:

$$-\mathcal{W}\_s \frac{d[\% \mathcal{C}]}{dt} = -\rho\_5 k\_{\mathcal{C}} A\_{inter} \left( [\% \mathcal{C}] - [\% \mathcal{C}]\_{\mathfrak{e}} \right), \tag{27}$$

where *Ainter* is the bubble inter-surface area; [%*C*]*<sup>e</sup>* is carbon equilibrium concentration in the molten bath; *kC* is the carbon mass transfer coefficient through the oxygen bubble surface which can be calculated by [26]:

$$k\_c = 0.59 \cdot \left[ D\_{\mathbb{C}^\circ} (\mu\_{rel} / d\_{\mathbb{B}}) \right]^{0.5},\tag{28}$$

where *DC* is the diffusion coefficient of carbon; *urel* is relative velocity of liquid steel; *dB* is the bubble diameter.

The oxides formed through Reaction (A) to Reaction (C) may gradually float upwards to the top surface, which is the lower surface of the slag layer that is not included in the simulation domain. Practically, the oxides will accumulate at the slag layer and have further reactions there. The absorption of the oxides by slag layer can be achieved computationally by removing the corresponding oxides that are in contact with the domain top surface. The process described above is illustrated in Figure 5.

**Figure 5.** Oxygen injection through jet penetration cavities and absorption of oxides.

During the refining stage, the temperature of the liquid steel bath increases due to the energy released by the oxidation reactions. The amount of energy released to the bath can be estimated by the oxidation rates and the oxidation enthalpies Δ*H*<sup>i</sup> of each reaction, where Δ*H*<sup>i</sup> is a function of bath temperature and taken from reference [27]. Thus the rate of energy-generating in a cell due to the oxidations can be expressed as:

$$\frac{dE\_{\text{reac}}}{dt} = \sum \Delta H\_i W\_s \frac{d[\%i]}{dt} \,\tag{29}$$

where *i* represents the carbon, iron and manganese considered in the liquid steel bath.

The current liquid steel-oxygen two-phase reacting flow system was solved in the numerical simulation domain given in Figure 4, which has 2.5 million computational cells totally. By adopting the Eulerian model with the appropriate source terms compiled through user-defined function (UDF) code, the model is able to achieve the above-described simulation of in-bath oxidation reactions and heat release. The total computational time needed to simulate 1000 s decarburization process is around 50 h if using 0.05 s time step size and 80 cores in the HPC cluster.
