## 3 Results and Discussion  
### Dissolution  
Figure 2a-c show the dissolution curves of equivalent diameters over time in S1, S2, and S3 at 1450, 1500, and 1550 \({}^{\circ}\)C, respectively. The shaded area represents the standard deviation of dissolution time for a particular equivalent diameter in the corresponding experiments. Figure 2d shows the reciprocal of total dissolution time, \(t_{tot}\) (mean of the corresponding experiments), for a 300 \(\upmu\)m particle at three temperatures in S1, S2, and S3. In all cases, time for the complete dissolution of a 300 \(\upmu\)m particle decreases with rising experimental temperature. This is expected because of the increase in diffusivity with temperature, and it agrees with the trend of decreasing slag viscosity with increasing temperature, as shown in Table 1. In the case of magnesia, a dissolution time increase is expected with an increasing C/S ratio. However, at 1450 and 1500 \({}^{\circ}\)C, the sequences of S2 and S3 contradict this expectation. Figure 3a-c show the dissolution rates in terms of the derivative of particle size _dO/dt_ with respect to time in S1, S2, and S3 at 1450, 1500, and 1550 \({}^{\circ}\)C, respectively. In all slags, at a particular temperature, the absolute value of the dissolution rate decreases to a minimum at the beginning owing to the development of a quasi-steady diffusive boundary layer; it then increases because the quasi-steady diffusive boundary layer thickness decreases in accordance with the particle radius. Similar to the total dissolution time, the absolute value of the dissolution rate increases with rising temperature; however, it does not show the expected trend with an increasing C/S ratio. Here, the concentration difference in magnesia between the saturated and bulk slag decreases with an increasing C/S ratio; this parameter acts to decrease the dissolution rate with a rising C/S ratio, but other factors also play important roles. Dissolution is inversely proportional to the \(\delta\), which is influenced by the fluid flow field around the dissolving particle and slag viscosity. According to the process in Ref. [11], average effective diffusive boundary layer thicknesses were determined from  
Figure 2: Dissolution curves for magnesia in (**a**) S1, (**b**) S2, and (**c**) S3 at 1450, 1500, and 1550 \({}^{\circ}\)C; (**d**) reciprocal of total dissolution times; shaded area represents the standard deviation of dissolution time. the experimental mean dissolution curves, presented in Figure 3d. Based on the aforesaid factors, we consider introducing two parameters, \(B/\eta\) and \(B/(\eta\cdot\delta)\), to assess the dissolution trend. The dimensionless concentration difference \(B\) is defined by Equation (1). \[B=\frac{w_{s}-w_{0}}{1-w_{s}}, \tag{1}\]  
where \(w_{0}\) and \(w_{s}\) are the mass fractions of magnesia in bulk and saturated slags, respectively. However, these parameters do not explain the dissolution trend of magnesia in the three slags. For further improvement, the thermodynamic factor of diffusion \(f\), defined by Equation (2), was calculated using the magnesia activity coefficient (\(\gamma\)) calculated from the FactSage(r) results. \[f=1+\frac{dln\gamma}{dlnx} \tag{2}\]  
The extended parameter \(B\cdot f/(\eta\cdot\delta)\) was introduced, and it successfully explains the dissolution trend at 1450 and 1500 \({}^{\circ}\)C. For 1550 \({}^{\circ}\)C, it identifies the most corrosive slag as S1; the ratios \(B/\eta\) and \(B/(\eta\cdot\delta)\) do not successfully quantify this improvement. Notably, \(B\cdot D/\delta\) gives the correct ranking of dissolution times for all slags and temperatures, although the ratio \(f/\eta\) is not sufficiently proportional to diffusivity. This deficiency in the simplified assessment can be observed for 1550 \({}^{\circ}\)C rather than for the other two cases, which follows the form of the lower variance in dissolution times for slag S3. The coefficients of variation for the dissolution times of S1 and S2 are 39.8% and 38.2%, respectively, but only 27.2% for S3.