# An Improved Dynamic Model of the Vanadium Extraction Process in Steelmaking Converters
## 3 Numerical Simulation and Discussion
The composition of the molten steel is shown in Figure 3. From about 0-0.5 min (at the beginning), the slope of each curve is fairly flat, which indicates that the mass fraction remained almost unchanged. The reason for this is that the slag-metal interface and the emulsion system gradually formed at the beginning and, so, the contact area of each reaction was not large enough to visually change the content of each element. As the contact area of the slag-metal interface and the emulsion system are considered to be non-constant, the model in this paper can capture this characteristic. Figure 3a shows that the carbon content changed slightly at the beginning, with the decarburization rate increasing later. The reason for this is that the emulsion system was present at the beginning of generation and the contact area of the decarbonizing reaction limited the decarbonization rate. Figure 3a,b show that the mass fraction of [C] was about 4.02%. Meanwhile, the mass fraction of [V] reached the endpoint control target of the vanadium extraction process [1]:  
\[\omega_{[C]} \geq 3.2\%,\] \[\omega_{[V]} \leq 0.05\%,\]  
where \(\omega_{[M]}\) is the mass fraction of [M]. The mass fractions of [C] and [V] were about 4.02% and 0.05%, respectively, when the endpoint control target was reached; which is consistent with Figure 4 [1]. In Figure 4, the mass fraction of [C] was about 4.1% when the mass fraction of [V] reached 0.05%. Figure 3: Composition changes of the molten pool. The mass fractions of \([C]\) and \([V]\) were about 3.5% and 0 when the curve of \([V]\) became approximately parallel to the horizontal axis. In Figure 4, the mass fraction of \([C]\) was about 3.5% when the mass fraction of \([V]\) reached 0.025%. According to (68) and (69), we have  
\[\frac{1}{r_{C}}\frac{1}{m_{[C]}}\mathrm{d}m_{[C]}=\frac{1}{r_{V}}\frac{1}{m_{[ V]}}\mathrm{d}m_{[V]}.\]  
Integrating both sides of the equation above and substituting the initial conditions, we can obtain the first integrals of \(m_{[C]}\) and \(m_{[V]}\):  
\[\frac{1}{r_{C}}\ln\frac{m_{[C]}}{m_{[C]}^{0}}=\frac{1}{r_{V}}\ln\frac{m_{[V]}} {m_{[V]}^{0}}. \tag{77}\]  
The equation (77) shows that \(m_{[C]}\) and \(m_{[V]}\) changed uniformly. When \(m_{[V]}\) was reduced, the mass of \(m_{[C]}\) was reduced accordingly. Equation (77) implies a contradiction between \"vanadium extraction\" and \"carbon protection\". It is impossible to oxidize all the \([V]\) into the slag and keep enough \([C]\) in the molten steel simultaneously. In fact,  
\[m_{[C]}\to 0\]  
as \(m_{[V]}\to 0\). That is to say, in order to keep enough carbon, there must be a certain amount of \([V]\) left. Similarly to (77), we obtain a series of the first integrals by (69)-(72):  
\[\frac{1}{r_{C}}\ln\frac{m_{[C]}}{m_{[C]}^{0}}=\frac{1}{r_{V}}\ln\frac{m_{[V]} }{m_{[V]}^{0}}=\frac{1}{r_{\Pi}}\ln\frac{m_{[\Pi]}}{m_{[\Pi]}^{0}}=\frac{1}{ r_{\mathrm{Si}}}\ln\frac{m_{[\mathrm{Si}]}}{m_{[\mathrm{Si}]}^{0}}=\frac{1}{r_{ \mathrm{Mn}}}\ln\frac{m_{[\mathrm{Mn}]}}{m_{[\mathrm{Mn}]}^{0}}. \tag{78}\]  
The equation (78) shows that the competitive oxidation between each element is mainly related to their initial value \(m_{[\mathrm{M}]}^{0}\) and parameter \(r_{\mathrm{M}}\). The parameter \(r_{\mathrm{M}}\) is related to the mass transfer coefficient of each element and the Gibbs free energy of each reaction. Thus, the initial value and the mass transfer coefficient of each element and the Gibbs free energy of each reaction work together to control their oxidation rate and oxygen consumption ratio, which affects the competitive oxidation between elements. Equation (78) can be used to estimate the content of each element at the end of vanadium extraction process. For example, it is generally used to measure the carbon content of the iron water after the vanadium extraction process. Then, the other element's contents can be estimated by using (78).