3.3.2. Effect of the Liquid-to-Solid (L/S) Ratio

The effect of L/S ratio within the range of 5–15 mL·g−<sup>1</sup> on the leaching process was investigated at a leaching temperature of 60 ◦C. As shown in Figure 8b, when the L/S ratio was increased from 5 mL·g−<sup>1</sup> to 10 mL·g−1, the recovery of the Al increased significantly. However, when the L/S ratio was increased to 15 mL·g−1, the recovery of the Al became relatively stable. Therefore, the L/S ratio of the leaching process had a significant influence on the recovery of the Al within a certain range. Increasing the amount of leaching solution can help to reduce the slurry viscosity and increase the concentration difference of the aluminate ions on the solid–liquid interface, thus accelerating the dissolution of NaAlO2 in the roasting clinkers. Therefore, an L/S ratio of 10 mL·g−<sup>1</sup> was selected as the optimum leaching condition for further experimentation.

## *3.4. Analysis of the Leaching Kinetics*

The leaching process of the NaAlO2 in the roasting clinkers is a liquid-to-solid heterogeneous reaction system, which takes place at the phase interfaces. In order to obtain a more efficient process, the kinetics of the dissolution of NaAlO2, including the reaction rate constant, reaction order, activation energy and the rate-determining step, should be

thoroughly evaluated. For the NaAlO2 dissolution in the roasting clinkers, the rate of reaction is given by Equation (14) [29].

$$-\frac{d\mathbb{C}\_{Al}}{dt} = k\,\mathbb{C}\_{Al}{}^{n}\tag{14}$$

where *CAl* is the content of NaAlO2 in the roasting clinkers after leaching for time *t*, *t* is the leaching time, *n* is the reaction order and *k* is the reaction rate constant, s<sup>−</sup>1.

The experimental results shown in Figure 8a were used to plot the different reaction orders (*n* = 0, 1, and 2) in Equation (14) and the corresponding results are shown in Figure 8c–e), respectively. The confidence level of the fitted straight line was high for the first order reaction (*n* = 1), while the linear relationship exhibited strong agreement in the second order reaction (*n* = 2), except for a slight deviation at 363.15 K. It has been shown that many reactions can be satisfactorily correlated using both the first and second order reactions. Therefore, it was not prudent to rely only on the linear correlation of the reaction equation to determine the reaction order. The half-life method was used to effectively determine the reaction order of the dissolution of NaAlO2 in the roasting clinkers (Equation (15)):

$$\chi\_{Al} = \frac{\mathbb{C}\_{Al,0} - \mathbb{C}\_{Al}}{\mathbb{C}\_{Al,0}} \tag{15}$$

The logarithmic form of Equation (15) is expressed as Equation (16):

$$k = \frac{1}{t} \ln \frac{1}{1 - \chi\_{Al}} \tag{16}$$

where *χAl* is the dissolution rate of NaAlO2 in the roasting clinkers at the time *t*, and *CAl*, 0 is the initial NaAlO2 content in the roasting clinkers.

When the dissolution rate of NaAlO2 was 50% (*χAl* = 0.5), the half-life of the first order reaction *t*1/2 is given by Equation (17), whereas the half-life of the second order reaction *t* 1/2 is written as Equation (18):

$$t\_{1/2} = \frac{\ln 2}{k} \tag{17}$$

$$t\_{1/2}' = \frac{1}{k \, \mathbb{C}\_{Al,0}}\tag{18}$$

The half-life results for first and second order reactions at different leaching temperatures are shown in Figure 8f. When the reaction order was one (n = 1), the half-life of the dissolution of NaAlO2 within the leaching temperature of 303.15–363.15 K was 7.67–4.05 min. However, when the reaction order was two (n = 2), the half-life within the temperature of 303.15–363.15 K was 6.15–1.82 min. Combined with the experimental results shown in Figure 8a, the half-life showed high reliability in first order reaction, indicating that the dissolution of the NaAlO2 was proportional to the reactant concentration.

The temperature dependence of the leaching reaction rate constant is expressed using the Arrhenius correlation (Equation (19)):

$$k = \mathbf{A} \cdot \exp\left(-E\_a / \mathbf{R}T\right) \tag{19}$$

where *k* is the overall reaction constant, A is the pre-exponential factor (min−1), R is the ideal gas constant (8.3145 J·mol−1·K<sup>−</sup>1) and *Ea* (J·mol<sup>−</sup>1) is the apparent activation energy of the dissolution of NaAlO2. The Arrhenius correlation can be rewritten in logarithmic form as Equation (20):

$$
\ln k = \ln \text{A} - \frac{E\_d}{\text{RT}} \tag{20}
$$

In general, when the activation energy is 40–300 kJ·mol−1, the reaction is controlled by the interface. However, when the activation energy is 8–20 kJ·mol−1, the reaction is controlled by the diffusion process [30]. Finally, the activation energy (*Ea*) calculated from the slope of Figure <sup>9</sup> was found to be 9.69 kJ·mol<sup>−</sup>1, which is lower than the value of 20 kJ·mol<sup>−</sup>1, thus confirming that the dissolution of NaAlO2 was controlled by the diffusion process. This result is also similar to the result of the dissolution of NaAlO2 mentioned in He's study, where the activation energy was reported to be 11.4010 kJ·mol−<sup>1</sup> [30].

**Figure 9.** Relationship between the leaching rate constant *k* and temperature *T*.
