## 2.2.1 Net chemical reaction rate modeling of main components

Mathematically, let *m*(TiCl4), *Q*m(TiCl4), and *R*(TiCl4) be, respectively, the total amount of gaseous TiCl4, the total amount of fed TiCl4 into the reactor, and the total amount of reduced TiCl4, in the unit of kg·m-2, within the chosen prototype Kroll reduction reactor, at any time τ. And in what follows we shall assume that those variables are continuous functions and differentiable, and the assumption is open to criticism. Thus, the principle of the conservation law of mass can be more precisely stated as follows

$$dm({\text{TiC}}{{\text{l}}_4})=d{Q_m}({\text{TiC}}{{\text{l}}_4}) - dR({\text{TiC}}{{\text{l}}_4})$$

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And the net chemical reaction rate of gaseous TiCl4, *r*(TiCl4), is defined as the total amount of reduced TiCl4 per unit time, in the in the unit of kg·m-2·h-1, and and its mathematical definition is the derivative of *R*(TiCl4) to time τ and given by

$$r({\text{TiC}}{{\text{l}}_4})=\frac{{dR({\text{TiC}}{{\text{l}}_4})}}{{d\tau }}=\frac{{d{Q_m}({\text{TiC}}{{\text{l}}_4})}}{{d\tau }} - \frac{{dm({\text{TiC}}{{\text{l}}_4})}}{{d\tau }}={q_m} - \frac{{dm({\text{TiC}}{{\text{l}}_4})}}{{d\tau }}$$

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where *q*m is the feeding rate of TiCl4, in the unit of kg·m-2·h-1. Similarly, the net chemical reaction rate of gaseous TiCl3, gaseous Mg, and solid TiCl2, in the in the unit of kg·m-2·h-1, could be expressed as

$$r({\text{TiC}}{{\text{l}}_3})=\frac{{dR({\text{TiC}}{{\text{l}}_3})}}{{d\tau }}= - \frac{{dm({\text{TiC}}{{\text{l}}_3})}}{{d\tau }}$$

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$$r({\text{Mg}})=\frac{{dR({\text{Mg}})}}{{d\tau }}= - \frac{{dm({\text{Mg}})}}{{d\tau }}$$

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$$r({\text{TiC}}{{\text{l}}_2})=\frac{{dR({\text{TiC}}{{\text{l}}_2})}}{{d\tau }}= - \frac{{dm({\text{TiC}}{{\text{l}}_2})}}{{d\tau }}$$

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where *m*(TiCl3) and *m*(Mg) are, respectively, the total amount of gaseous TiCl3 and gaseous Mg, and *m*(TiCl2) is the total amount of solid TiCl2 near the gas-liquid interfaces. Eq. (8) to Eq. (11) are defined as the net chemical reaction rate model of the main components.

And next, we would derive an implied mathematical relation of the total amount of the gaseous TiCl4, gaseous TiCl3, and gaseous Mg in the prototype Kroll reduction reactor. The primitive implied mathematical relation states that

$$x({\text{TiC}}{{\text{l}}_4}):x({\text{TiC}}{{\text{l}}_3}):x({\text{Mg}})=\frac{{m({\text{TiC}}{{\text{l}}_4})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_4})}}:\frac{{m({\text{TiC}}{{\text{l}}_3})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_3})}}:\frac{{m({\text{Mg}})}}{{{\text{M}}({\text{Mg}})}}$$

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where M is the relative molecular mass, in the unit of kg·mol-1, and *x* is the molar fraction of the gaseous component near the gas-liquid interfaces of the TiCl4-Mg system. This statement is equivalent to

$$\begin{gathered} m({\text{TiC}}{{\text{l}}_3})=\frac{{{\text{M}}({\text{TiC}}{{\text{l}}_3})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_4})}} \times \frac{{x({\text{TiC}}{{\text{l}}_3})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4}) \\ =0.8132 \times \frac{{x({\text{TiC}}{{\text{l}}_3})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4}) \\ \end{gathered}$$

13a

$$\begin{gathered} m({\text{Mg}})=\frac{{{\text{M}}({\text{Mg}})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_4})}} \times \frac{{x({\text{Mg}})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4}) \\ =0.1281 \times \frac{{x({\text{Mg}})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4}) \\ \end{gathered}$$

13b

Thus, we have

$$\frac{{dm({\text{TiC}}{{\text{l}}_3})}}{{d\tau }}=\frac{d}{{d\tau }}\left[ {0.8132 \times \frac{{x({\text{TiC}}{{\text{l}}_3})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4})} \right]$$

14a

$$\frac{{dm({\text{Mg}})}}{{d\tau }}=\frac{d}{{d\tau }}\left[ {0.1281 \times \frac{{x({\text{Mg}})}}{{x({\text{TiC}}{{\text{l}}_4})}} \times m({\text{TiC}}{{\text{l}}_4})} \right]$$

14b

The molar fraction of gaseous component near the gas-liquid interfaces had determined in another Kroll reduction experimental work, and the details could be found in Fig. S1, Table S5, and Fig. S2 in the Supplementary Material. We also assume that the gaseous TiCl4 has a well-uniformity mass density. Thus, the total amount of gaseous TiCl4 at each time is given by

$$m({\text{TiC}}{{\text{l}}_4})=\frac{{L - l}}{{v({\text{TiC}}{{\text{l}}_4})}}$$

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where *L* is the total height of the prototype Kroll reduction reactor, about 5.20 m in this work. *l* is the height of the liquid Mg level, and the *v*(TiCl4) is the specific volume of gaseous TiCl4 that can be calculated by its virial equation of state in our previous work [12].

## 2.2.2 Reaction rate modeling of independent reaction

We define and symbol the reaction rate of the independent reactions (2) to (6), as *r*0, *r*1, *r*2, *r*3, and *r*4, in the unit of mol·m-2·h-1, respectively. And those reaction rates formed the reaction rate model of the independent reactions. The component disappearance and appearance rates [28] bridge the net chemical reaction rate model of the main components and the reaction rate model of the independent reactions by the following liner relationships

$$\frac{{r({\text{TiC}}{{\text{l}}_4})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_4})}}={r_1}+{r_2}$$

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$$\frac{{r({\text{Mg}})}}{{{\text{M}}({\text{Mg}})}}= - {r_0}+2{r_1}$$

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$$\frac{{r({\text{TiC}}{{\text{l}}_3})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_3})}}= - {r_1} - {r_2}+{r_3}$$

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$$\frac{{r({\text{TiC}}{{\text{l}}_2})}}{{{\text{M}}({\text{TiC}}{{\text{l}}_2})}}= - {r_3}+{r_4}$$

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Because the solubility of TiCl2 in liquid MgCl2 is very large, about 40% at 1173.0 K, a solution of TiCl2/MgCl2 salt is formed near the gas-liquid interfaces [15]. Meanwhile, with the powerful flows of the liquid Mg and MgCl2 with the high temperature moving at a rate of 0.2 ~ 0.7 m·s-1 [18], the solid TiCl2 is transported to the active sites that are newly formed by Ti sponge. Therefore, we assumed that the diffusion and reduction of TiCl2 within the liquid MgCl2 are not the rate-determining steps of magnesiothermic reduction of TiCl4 near the liquid-gas interfaces in the prototype Kroll reduction reactor [29]. And the net chemical reaction of solid TiCl2 near the gas-liquid interfaces, in Eq. (11), is set to *zero* for its fast disappearance rate and diffusion via the quasi-steady-state approximation methodology, resulting in

$$r({\text{TiC}}{{\text{l}}_2})= - \frac{{dm({\text{TiC}}{{\text{l}}_2})}}{{d\tau }}=0$$

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That is to say that the external mass transfer resistance is negligible to effectively evaluate the overall chemical reaction rate.

## 2.2.3 Kinetic modeling of magnesium reduction of TiCl4

We, therefore, conclude that for this complex reaction system of magnesiothermic reduction of TiCl4, the overall chemical reaction rate equation cannot be inferred from the stoichiometric equation, but must be determined experimentally. Let the overall chemical reaction rate equation be the functional relationship between the net chemical reaction rate of the Ti sponge product *r*(Ti) and the feeding rate of TiCl4 at a constant temperature and a constant pressure, which is given by

$$r({\text{Ti}})=\frac{{dm({\text{Ti}})}}{{d\tau }}=k({T_{{\text{in}}}})f(\Delta p)q_{m}^{\alpha }$$

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where *m*(Ti) is the total amount of generated Ti sponge, in the unit of kg·m-2, *k*(*T*in) and *f*(Δ*p*) are the reaction rate constants that respectively be related to the temperature and pressure of the gas-liquid interfaces in the prototype Kroll reduction reactor, *T*in is the temperature of the gas-liquid interfaces in the unit of K, and Δ*p* is the gauge pressure of the gas-liquid interfaces in the prototype reactor, in the unit of kPa. The power *α* is called the overall reaction order of magnesiothermic reduction of TiCl4.

According to the net chemical reaction rate model for the main components and the reaction rate model of the independent reactions, the overall chemical reaction rate equation is equivalent to

$$\begin{gathered} r({\text{Ti}})=\frac{{dm({\text{Ti}})}}{{d\tau }}={\text{M}}({\text{Ti}}) \times {r_4}={\text{M}}({\text{Ti}}) \times {r_3} \\ =0.2525 \times \left[ {{q_{\text{m}}} - \frac{{dm({\text{TiC}}{{\text{l}}_{\text{4}}})}}{{d\tau }}} \right] - 0.3105 \times \frac{{dm({\text{TiC}}{{\text{l}}_3})}}{{d\tau }} \\ \end{gathered}$$

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## 2.2.4 Arrhenius equation for kinetic model

A large portion of the field of the chemical kinetic modeling can be displayed by, or investigated in terms of the Arrhenius equation that related the reaction rate constant *k*(*T*in) to the temperature *T*in, and we have [30]

$$k({T_{{\text{in}}}})=\Lambda \exp \left( { - \frac{{{E_a}}}{{R{T_{{\text{in}}}}}}} \right)$$

23a

where *R* is the gas constant, *E**a* is the activation energy with dimensions of kJ·mol-1, and Λ is the pre-exponential factor, which has the unit of *k*(*T*in). The Arrhenius equation implies that Λ and *E**a* are temperature dependent, but it is reasonable to accept that they are essentially independent of temperature in this usual experimental work. It is commonly applied in the linearized form [31]

$$\ln \left[ {k({T_{{\text{in}}}})} \right]= - \frac{{{E_a}}}{R}\frac{1}{{{T_{{\text{in}}}}}}+\ln \Lambda$$

23b

The goal of kinetic modeling is to calculate the reaction rate constants or to establish the chemical reaction rate equation, this is equivalent to calculate *E**a* and Λ, which can yield from the slope and the intercept of the straight line in an Arrhenius plot of ln[*k*(*T*in)] against 1/*T*in.