*2.6. Decarburization in the Emulsion Zone*

In Figure 5, the initial carbon composition and diameter of a metal droplet are represented by *Ci*,0 and *D*0, respectively. The metal droplet is formed from the interaction of the oxygen jet and the metal bath at the impact zone at time *t*0, and ejected to the slag–metal–gas emulsion. The droplet stays in the emulsion zone for *Rt* seconds, then it returns to the metal bath at time *t*<sup>0</sup> + *Rt* having a composition *Ci*, *<sup>f</sup>* and diameter *Df* . While in the emulsion zone, carbon in the metal droplet can reduce FeO in slag according to reaction 3.

**Figure 5.** Droplet decarburization.

The droplet generation rate *RB*, given by Equation (46), is calculated using the empirical expression proposed by Subagyo et al. [31] modified by a parameter *pDG* to account for possible differences between the conditions at which the correlation was derived and a given BOF operation:

$$R\_B = p\_{DC} \frac{\dot{V}\_{O\_2} N\_B^{3.2}}{(2.6 \times 10^6 + 2 \times 10^{-4} N\_B^{12})^{0.2}} \tag{46}$$

where *V*˙ *<sup>O</sup>*<sup>2</sup> is the volumetric flow rate of oxygen and *NB* is the blowing number [31].

Since millions of droplets are generated at every point, it can become computationally expensive to track the composition of the individual metal droplets. Therefore, an algebraic equation to calculate the final carbon content of the metal droplets was developed and the following assumptions were made in order to make the problem tractable:


The first-principles model developed by Kadrolkar and Dogan [10] for the droplet decarburization was used to generate data for the final carbon content (*CC*, *<sup>f</sup>*) of individual droplets with respect to its initial carbon content *CC*,0, the slag temperature *Ts* and composition. Data were generated for the range of initial carbon content between 0.3% and 5% and slag temperature of 1623–2153 K, and for the slag composition provided by Cicutti et al. [13]. Equations (47) and (48) give a good description of the generated data with an R<sup>2</sup> of 0.96 and 0.92, respectively.

$$
\Delta \mathcal{C}\_{\mathcal{C}} = \mathcal{C}\_{\mathcal{C},0} - \mathcal{C}\_{\mathcal{C},f} = 0.9514 \left( \frac{0.001 T\_s}{\mathcal{C}\_{\mathcal{C},0}} \right)^{-1.4345} \quad \text{(\%FeO)} > 10 \tag{47}
$$

$$\mathcal{L}\_{\mathbb{C},f} = 0.67492 \mathcal{C}\_{\mathbb{C},0}^{1.2261 \mathcal{C}\_{\mathbb{C},0}/(\%FeO)} \quad (\%FeO) \le 10 \tag{48}$$

In Figure 6, the average final carbon content of the droplets reported by Cicutti et al. [13], the values predicted by Kadrolkar and Dogan [10]'s model and what was obtained using Equation (47) are shown. Similar to the first-principles model proposed by Kadrolkar and Dogan [10], the final carbon content of the droplets predicted by Equation (47) is largely within the range of values reported by Cicutti [13] for a real BOF operation. These results indicate that Equation (47) approximates the kinetics of droplet decarburization reasonably well and can therefore be used within the BOF model to describe the kinetics of decarburization in the emulsion zone.

**Figure 6.** Comparison of final carbon content of metal droplets in the emulsion reported by Cicutti et al. [13], and the values predicted using Kadrolkar and Dogan [10]'s first-principles model and Equation (47) as a function of blow time.

Using Equations (47) and (48) and the previously stated assumptions, and introducing a term *αp <sup>μ</sup>*,*<sup>D</sup>* to account for changes in the slag viscosity, it is possible to obtain Equation (49) for the rate of carbon removal:

$$\dot{W}\_{\text{C,s}} = R\_B \left( 1 - \frac{1 - 0.01 \text{C}\_{\text{C,0}}}{1 - 0.01 \text{C}\_{\text{C,f}}} \right) a\_{\mu,D}^p \tag{49}$$

with *α<sup>p</sup> <sup>μ</sup>*,*<sup>D</sup>* given by:

$$a\_{\mu,D}^p = \exp(-a\_{\mu}^p \mu\_s) \tag{50}$$

where *<sup>W</sup>*˙ *<sup>C</sup>*,*<sup>s</sup>* is the rate of carbon removal, *<sup>α</sup><sup>p</sup> <sup>μ</sup>* is a parameter and *μ<sup>s</sup>* is the slag viscosity. The initial carbon content of the droplet *CC*,0 is equal to the carbon content of the metal bath *Cb* at the time of ejection. The term *α<sup>p</sup> <sup>μ</sup>*,*<sup>D</sup>* is necessary because the effect of slag viscosity is not taken into account in Equations (47) and (48). However, it is likely that the high slag viscosity at the beginning of the blow significantly decreases the decarburization rate. At high viscosities the slag becomes less fluid, negatively impacting the rate of FeO mass transfer to the droplet surface and potentially leading to FeO depletion in the neighborhood of the droplet, decreasing the decarburization rate. The value of *α<sup>p</sup> μ* can be determined using dynamic data.

The mass of carbon *WC*,*<sup>s</sup>* in the emulsion is then given by:

$$\frac{d\mathcal{W}\_{\mathbb{C},s}}{dt} = (0.01R\_B\mathbb{C}\_{\mathbb{C},0} - \dot{W}\_{\mathbb{C},s}) - 0.01\overline{\mathcal{C}}\_{\mathbb{C},s}\frac{\mathcal{W}\_{D,s}}{R\_t} \tag{51}$$

where the first term on the right hand side is the amount of carbon entering the emulsion zone minus the carbon removed via decarburization, and the second term is the flow rate of carbon returning to the metal bath. *CC*,*<sup>s</sup>* is the average carbon content of the droplets in the emulsion calculated by:

$$
\overline{\mathbf{C}}\_{\mathbf{C},s} = 100 \frac{\mathcal{W}\_{\mathbf{C},s}}{\mathcal{W}\_{\mathbf{D},s}} \tag{52}
$$

and *WD*,*<sup>s</sup>* is the total mass of droplets in the emulsion determined as:

$$\frac{d\mathcal{W}\_{D,s}}{dt} = \left(R\_B - \dot{\mathcal{W}}\_{C,s}\right) - \frac{\mathcal{W}\_{D,s}}{R\_t} \tag{53}$$

The droplet residence time *Rt* is calculated using the following empirical expression:

$$R\_t = 5 + 20\varepsilon^{-0.3\frac{(\%FeO)}{C\_{C,0}}} \tag{54}$$

As long as the residence time *Rt* of the droplets in the emulsion is small, *CC*,*<sup>s</sup>* is approximately equal to *CC*, *<sup>f</sup>* and the simplifying assumption of uniform carbon content for the droplets in the emulsion does not significantly impact the final result, while avoiding the need to track the behavior of individual droplets.
