# An Improved Dynamic Model of the Vanadium Extraction Process in Steelmaking Converters
## 2 Improved Mathematical Model of the Vanadium Extraction Process
### The Slag-Metal Interface  
When the molten pool is strongly stirred, the slag is broken and incorporated into the molten steel to form the slag-metal interface. Within the system composed of molten metal and slag, due to different physical properties or flow velocities, there exist boundary layers [22] representing the mass transfer resistances on both sides of the phase interface. As shown in Figure 2, due to the concentration difference, each component \([\mathrm{M}]\) (with a concentration of \(c_{[\mathrm{M}]}\)) and \((\mathrm{FeO})\) (with a concentration of \(c_{(\mathrm{FeO})}\)) diffuse to the phase interface, where their concentrations are denoted by \(c_{[\mathrm{M}]}^{*}\) and \(c_{(\mathrm{FeO})}^{*}\), respectively. Similarly, each component \((\mathrm{S})\) (with a concentration of \(c_{(\mathrm{S})}^{*}\)) and \([\mathrm{Fe}]\) (with a concentration of \(c_{[\mathrm{Fe}]}^{*}\)) in the phase interface diffuse to the slag and molten steel, where their concentrations are denoted by \(c_{(\mathrm{S})}\) and \(c_{[\mathrm{Fe}]}\), respectively. The whole chemical reaction process consists of reactant diffusion, interface reaction, and product diffusion. As an oxidant, the \((\mathrm{FeO})\) generated by top-blown oxygen oxidizes \([\mathrm{V}]\), \([\mathrm{Ti}]\), \([\mathrm{Si}]\), and \([\mathrm{Mn}]\) at the slag-metal interface. The corresponding chemical reaction equations are  
\[[\mathrm{V}]+\frac{3}{2}\left(\mathrm{FeO}\right) =\frac{1}{2}\left(\mathrm{V}_{2}\mathrm{O}_{3}\right)+\frac{3}{2 }\left[\mathrm{Fe}\right], \tag{3}\] \[[\mathrm{Ti}]+2\left(\mathrm{FeO}\right) =\left(\mathrm{TiO}_{2}\right)+2\left[\mathrm{Fe}\right],\] (4) \[[\mathrm{Si}]+2\left(\mathrm{FeO}\right) =\left(\mathrm{SiO}_{2}\right)+2\left[\mathrm{Fe}\right],\] (5) \[[\mathrm{Mn}]+\left(\mathrm{FeO}\right) =\left(\mathrm{MnO}\right)+\left[\mathrm{Fe}\right]. \tag{6}\]  
In order to establish the rate equation of the chemical reactions (3)-(6), the rate limiting elements of the chemical reactions are analyzed from the two following aspects:  
1. Interface Chemical Reaction: The interface chemical reaction is carried out rapidly at a high temperature, and the chemical reaction process does not become the rate limiting step. 2. Mass Transfer of Reactant and Resultant: Generally, the concentrations of the reactant \((\mathrm{FeO})\) and the resultant \([\mathrm{Fe}]\) are high, and their mass transfer processes will not be the rate limiting step. Therefore, the mass transfer processes of the reactants \([\mathrm{M}]\) and the products \((\mathrm{S})\) are the rate limiting steps. The rates of the interface reactions are regulated by each other. It is generally considered that the diffusion processes on both sides of the slag-metal interface satisfy the steady-state principle [22]. According to the definition of a rate limiting step, the diffusion rates of \([\mathrm{M}]\) can reveal the rates of the interface reactions (3)-(6). By [PERSON]'s law [23], we have  
\[J_{[\mathrm{V}]}^{\mathrm{sm}} =-\frac{1}{A^{\mathrm{sm}}}\frac{\mathrm{d}n_{[\mathrm{V}]}^{ \mathrm{sm}}}{\mathrm{d}t} =\beta_{[\mathrm{V}]}(c_{[\mathrm{V}]}-c_{[\mathrm{V}]}^{*\mathrm{sm} }), \tag{7}\] \[J_{[\mathrm{Ti}]}^{\mathrm{sm}} =-\frac{1}{A^{\mathrm{sm}}}\frac{\mathrm{d}n_{[\mathrm{Ti}]}^{ \mathrm{sm}}}{\mathrm{d}t} =\beta_{[\mathrm{Ti}]}(c_{[\mathrm{Ti}]}-c_{[\mathrm{Ti}]}^{* \mathrm{sm}}), \tag{8}\]  
Figure 2: Slag–metal phase interface diagram.