*2.5. Flux Dissolution*

Similarly to the decarburization rates at the impact zone, flux dissolution is modelled after Dogan et al. [27]. The flux particles are assumed to have a spherical shape, and the rate of flux dissolution is proportional to the rate of change of the particles' radius.

The rate of lime dissolution is given by Equation (41) [27,28], where *rL* is the particle radius, (%CaO) is the concentration of CaO in slag, (%*CaOsat*) is the saturation concentration of CaO in the slag, *kL* is the mass transfer coefficient, *ρ<sup>s</sup>* and *ρ<sup>L</sup>* are the slag and lime density, respectively. The parameter *α<sup>p</sup> <sup>L</sup>* is introduced here to account for deviations between the experimental conditions at which *kL* was derived, and the BOF operating conditions.

$$\frac{dr\_L}{dt} = a\_L^p k\_L \frac{\rho\_s}{100\rho\_L} ((\% \text{CaO}) - (\% \text{CaO}\_{sat})) \tag{41}$$

For dolomite, the rate of dissolution is given by [27,29]:

$$\frac{dr\_D}{dt} = \begin{cases} a\_D^p k\_D \frac{\rho\_s}{100\rho\_D}}{A} \left(1 + \frac{M\_{\rm MgO}}{M\_{\rm CaO}}\right) \left(\left(\% \rm CaO\right) - \left(\% \rm CaO\_{sat}\right)\right) & \left(\% \rm FeO\right) < 20\%\\ a\_D^p k\_D \frac{\rho\_s}{100\rho\_D}}{A\_D^p k\_D \frac{\rho\_s}{100\rho\_D}} \left(1 + \frac{M\_{\rm FeO}}{M\_{\rm MgO}}\right) \left(\left(\% \rm MgO\right) - \left(\% \rm MgO\_{sat}\right)\right) & \left(\% \rm FeO\right) \ge 20\% \end{cases} \tag{42}$$

where *MMgO* and *MCaO* are the molar mass of MgO and CaO, respectively, (%MgO) is the concentration of MgO in slag, (%*MgOsat*) is the saturation concentration of CaO in the slag and *kD* is the mass transfer coefficient. The rate constants *kL* and *kD* are calculated using the correlations proposed by Dogan et al. [27] with the parameter used to modify the Reynolds number *β* set to the nominal value of 1.

Data for the calcium oxide saturation in slag (%*CaOsat*) for different slag compositions were obtained using the Cell Model [30], a Matlab program that gives the saturation concentration of the individual species in the slag as a function of composition and temperature. Kadrolkar et al. [30] validated the Cell Model results against FactSage*TM* and ThermoCalc*TM*. The function 'fitnlm' in Matlab was then used to fit a curve to the generated data, yielding Equation (43).

$$(\% \text{CaO})\_{\text{sat}} = \frac{3.52 T\_s - 4,823.7 e^{-\frac{2.43}{100}(\% \text{SiO}\_2)} + 12.4(\% \text{FeO}) - 9.71(\% \text{MgO}) + 17.9(\% \text{CaO})}{100} \tag{43}$$

An R2 of 0.9 was obtained for the nonlinear regression. The goodness of Equation (43) to predict the calcium oxide saturation in slag was evaluated using the Cicutti data [13]. The calculated values for *CaOsat* using the Cell Model and Equation (43) are shown in Figure 4, where it can be seen that there is excellent agreement between them.

**Figure 4.** (%*CaOsat*) obtained using the Cell Model and Equation (43) for the Cicutti et al. [13] slag data.

The rate at which heat is absorbed by the flux particles is given by:

$$Q\_{flux} = 4\pi \sum\_{i} r\_i^2 h\_i n\_i (T\_s - T\_i) \quad i \in \{L, D\} \tag{44}$$

where *Ts* is the temperature of the slag, *Ti* is the flux temperature, *hi* is the heat transfer coefficient determined using data and *ni* is the number of flux particles. Assuming a uniform temperature for the flux particle, an energy balance gives Equation (45) for the temperature of an iron ore particle:

$$\mathcal{W}\_i \mathbb{C}\_{P,i} \rho\_i \frac{dT\_i}{dt} = 4\pi r\_i^2 h\_i (T\_5 - T\_i) \quad i \in \{L, D\} \tag{45}$$

where *CP*,*<sup>i</sup>* is the heat capacity and *Wi* = 4/3*ρiπr*<sup>3</sup> is the mass of one single flux particle.
