# An Improved Dynamic Model of the Vanadium Extraction Process in Steelmaking Converters
## 3 Numerical Simulation and Discussion  
The values of the mass transfer coefficients of each element in the model are given in Table 1[20]. The values of the standard Gibbs free energy of each element in the model are given in Table 2[22]. The equilibrium constant of the chemical reactions (13) and (31) are \[K^{\text{sm}}_{\text{(FeO)}} =\exp\left(-\frac{\Delta G^{\text{sm}}_{\text{(FeO)}}}{RT^{\text{sm} }}\right),\] \[K^{\text{es}}_{\text{(FeO)}} =\exp\left(-\frac{\Delta G^{\text{es}}_{\text{(FeO)}}}{RT^{\text{ res}}}\right),\]  
where \(\Delta G^{\text{sm}}_{\text{(FeO)}}\) and \(\Delta G^{\text{es}}_{\text{(FeO)}}\) are the Gibbs free energies of the chemical reactions (13) and (31) in Table 2, \(R=8.314)/(\text{mol}\cdot\text{K})\) is the ideal gas constant, and \(T^{\text{sm}}\) and \(T^{\text{es}}\) are the average temperatures of the slag-metal interface and the emulsion system, respectively. According to [1], \(T^{\text{sm}}\) and \(T^{\text{es}}\) can be taken as  
\[T^{\text{sm}} =1573\text{ K},\] \[T^{\text{es}} =1673\text{ K}.\]  
The values of the initial conditions of each element in the model are given in Table 3[20; 22]. The proportionality coefficients in the model are given in Table 4[1; 22]. In addition, the oxygen flow \(Q_{\text{O}_{2}}\) can be taken as [20]  
\[Q_{\text{O}_{2}}=\frac{8}{3}\text{ m}^{3}/\text{s}.\]  
Furthermore, the densities of molten steel (\(\rho_{\text{m}}\)) and of slag (\(\rho_{\text{s}}\)) can be taken as [22]  
\[\rho_{\text{m}} =7\times 10^{3}\text{ kg}/\text{m}^{3},\] \[\rho_{\text{s}} =3\times 10^{3}\text{ kg}/\text{m}^{3}.\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(n^{0}_{\text{[C]}}\) & \(m^{0}_{\text{[Y]}}\) & \(m^{0}_{\text{[T]}}\) & \(m^{0}_{\text{[S]}}\) & \(m^{0}_{\text{[Mn]}}\) & \(m^{0}_{\text{[Fe]}}\) & \(m^{0}_{\text{[O]}}\) & \(W^{0}_{m}\) & \(W^{0}_{s}\) \\ \hline
**Value** & 3014.2 & 210.7 & 113.4 & 205.8 & 210.7 & 66,210 & 14 & 70,000 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Proportionality coefficient [1; 22]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \(\beta_{\text{[Fe]}}\) & \(\beta_{\text{[C]}}\) & \(\beta_{\text{[Y]}}\) & \(\beta_{\text{[T]}}\) & \(\beta_{\text{[Si]}}\) & \(\beta_{\text{[Mn]}}\) \\
**Value** & \(1\times 10^{-3}\) & \(3\times 10^{-6}\) & \(3.5\times 10^{-5}\) & \(2.4\times 10^{-5}\) & \(1.9\times 10^{-5}\) & \(2.65\times 10^{-5}\) \\ \hline
**Parameters** & \(\beta_{\text{(FeO)}}\) & \(\beta_{\text{[O]}}\) & \(\beta_{\text{(FeO)}}\) & \(\beta_{\text{(FeO)}}\) & \(\beta_{\text{(SO)}}\) & \(\beta_{\text{(MnO)}}\) \\
**Value** & \(3.4\times 10^{-6}\) & \(5\times 10^{-6}\) & \(3.2\times 10^{-8}\) & \(7\times 10^{-8}\) & \(5\times 10^{-8}\) & \(3.4\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mass transfer coefficients [20] (\(\text{m}\cdot\text{s}^{-1}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Chemical Reaction** & \(A\)/\(J\cdot mol^{-1}\) & \(B\)/\(J\cdot mol\cdot K\)\({}^{-1}\) \\ \hline \((\text{FeO})=[\text{Fe}]+[\text{O}]\) & 121,009 & \(-52.35\) \\ \([\text{V}]+\frac{3}{2}\) (FeO) & \(\frac{1}{2}\) (V\({}_{2}\text{O}_{3})+\frac{3}{2}\) [Fe] & \(-452,710\) & 137.55 \\ \([\text{Ti}]+2\) (FeO) & \((\text{TiO}_{2})+2\) [Fe] & \(-835,421\) & 125.87 \\ \([\text{Si}]+2\) (FeO) & \((\text{SiO}_{2})+2\) [Fe] & \(-386,769\) & 202.3 \\ \([\text{Mn}]+(\text{FeO})=(\text{MnO})+[\text{Mn}]\) & \(-123,307\) & 56.48 \\ \([\text{V}]+\frac{3}{2}\) [O] & \(=\frac{1}{2}\) (V\({}_{2}\text{O}_{3})\) & \(-428,826\) & 168.11 \\ \([\text{Ti}]+2\) [O] & \(=(\text{TiO}_{2})\) & \(-932,145\) & 167.84 \\ \([\text{Si}]+2\) [O] & \(=(\text{SiO}_{2})\) & \(-594,285\) & 229.76 \\ \([\text{Mn}]+[\text{O}]=(\text{MnO})\) & \(-244,316\) & 106.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standard Gibbs Free Energies [22] (\(\Delta G=A+B\cdot T\ /\ J\cdot mol^{-1}\)). According to the parameters and initial conditions above, the MATLAB [26] numerical simulation results and discussion are as follows.