## 2 Materials and Methods
### Continuous Wear Curve Evaluation and Diffusivity Determination  
Three thousand profiles were averaged in the circumferential direction after preprocessing to obtain a unique, axisymmetric, and representative mean corrosion profile (MCP) [15]. MCP facilitates the dimensional analysis of the sample. The details of profile processing can be found in Ref. [53]. The mass flux density was calculated using dissolution parameters obtained from the MCP and, subsequently, the diffusivity [15]. In conventional post-mortem analysis, the dimensions are measured manually at a small number of locations on the rough corroded samples. In contrast, laser measurements incorporate dimension information from the entire sample surface with a high resolution. Hence, the dissolution parameters from MCPs obtained using laser measurements are anticipated to be precise than those obtained using conventional methods. In the dissolution experiments at all three temperatures, Marangoni grooves were observed at the refractory-slag-atmosphere triple point of the magnesia cylinders. The diffusivity determination method is based on the total mass flux density solely due to forced convection. Therefore, separating the mass fluxes due to Marangoni convection and forced convection is necessary; this separation can be performed easily by identifying the groove length from the MCP. Dissolution from the groove was considered only when the bulk slag composition and the corresponding slag viscosity for different corrosion steps were calculated, as these two parameters are related to the total dissolution. Other dissolution parameters, such as the mean corroded radius, immersion length, and mass flux density were considered for determining the diffusivity; however, the dissolution from the groove was not considered. In this study, the diffusivities of magnesia were determined according to the method reported by [PERSON] et al. [15] using Equation (1). This was derived from the total mass flux density using Sherwood relations for the mantle and bottom of the cylindrical sample [15]. The method reported in Ref. [15] uses the modified Sherwood relation proposed by [PERSON] and [PERSON] [61] (Equation (3)) for the mantle of the cylindrical sample. Instead, herein, the Sherwood relation proposed by [PERSON] et al. [62] (Equation (2)), derived from the simulations for the same experimental setup, was adopted. Moreover, as in Ref. [15], the Sherwood relation reported by [PERSON] [63] was also employed here for the bottom of the sample. \[D=\left[\frac{j\cdot(1+0.566\cdot B)\cdot(2l\cdot R_{1}+R_{t}^{2})}{\rho^{s} \cdot B\cdot\left\{0.2\cdot 1^{0.68}\cdot R_{1}^{0.81}\cdot\omega^{0.65}\cdot v^{( \frac{1}{3}-0.65)}\cdot(R_{2}-R_{1})^{0.49}\cdot BC^{0.32}+0.62\cdot R_{t}^{2} \cdot\omega^{\frac{1}{2}}\cdot v^{\frac{1}{6}}\right\}}\right]^{\frac{3}{2}} \tag{1}\]  
\[Sh_{G}=0.10\cdot Re^{0.65}\cdot Sc^{1/3}\cdot\left\{(R_{2}-R_{1})/R_{1}\right\} ^{0.84}\cdot(BC/l)^{0.32} \tag{2}\]  
\[Sh_{T}=0.21\cdot Re^{1/2}\cdot Sc^{1/3}\cdot\left\{(R_{2}-R_{1})/R_{1}\right\} ^{0.25} \tag{3}\]  
where \(D\) is the effective binary diffusivity; \(j\), the mass flux density; \(B\), the dimensionless concentration difference: \(B=(w_{s}-w_{0})/(1-w_{s})\), where \(w_{0}\) and \(w_{s}\), respectively, denote the mass fraction of dissolving species in the bulk slag and saturated slag; \(R_{1}\), the mean cylinder radius; \(R_{2}\), the crucible radius; \(R_{t}\), the tip radius; \(l\), the immersion length; \(v\), the kinematic viscosity of the slag; \(\omega\), the angular velocity; \(BC\), the bottom clearance i.e., the gap between the sample tip and crucible bottom; and \(\rho^{s}\), the density of the slag. \(Sh\), is the Sherwood number; \(Re\), the Reynolds number; and \(Sc\) the Schmidt number. The characteristic length for \(Sh\) and \(Re\) is the gap width \((R_{2}-R_{1})\) in both Sherwood relations (Equations (2) and (3)). A number of corrosion steps \(i\) were evaluated, and the quantity \(D_{i}\) was calculated for each step according to Equation (1).