# Enhanced Phosphorus Removal from Wastewater Using RSPRC and a Novel Reactor
## 3 Results and Discussion
### RSPRC Adsorption Kinetics  
When the adsorbent adsorbs the wastewater, its adsorption function is mainly realized by the chemical bond force, electrostatic attraction and Van der Waals force between the adsorbent surface molecules and the adsorbate molecules in the wastewater. Currently, pseudo-first order kinetic models and pseudo-second order kinetic models are often used to describe the dynamic behavior of solid-liquid static adsorption. The pseudo-first order kinetic equation can express the relationship between the rate of adsorption reaction and the environmental conditions (such as concentration) involved in the adsorption reaction or related to the adsorption reaction. This kinetic model is usually based on the solid adsorption equilibrium capacity. The differential is as follows:  
\[\frac{\mathrm{d_{qt}}}{\mathrm{dt}}=\mathrm{k_{1}}\big{(}\mathrm{q_{e}-q_{t}} \big{)} \tag{4}\]  
After integrating the differential equation, the pseudo-first order kinetic equation can be expressed as follows:  
\[\mathrm{ln}\big{(}\mathrm{q_{e}-q_{t}}\big{)}=\mathrm{ln}\mathrm{q_{e}-k_{1}t} \tag{5}\]  
where \(\mathrm{q_{e}}\) is the adsorption amount of the adsorbent surface to the solute under adsorption equilibrium (mg/g), \(\mathrm{q_{t}}\) is the adsorption amount of the adsorbent surface to the solute at the specified time (t) during the adsorption process (mg/g), and \(\mathrm{k_{1}}\) is the pseudo-first order kinetic equation adsorption rate constant. Figure 8: Effect of initial concentrations on the absorption of phosphorus. The pseudo-second order kinetic model is applicable to the solid-liquid static adsorption reaction in the presence of a saturated site, which can effectively represent the composite effect under the action of multiple adsorption mechanisms. Generally, the kinetic model is based on the adsorption equilibrium capacity. The differential is as follows:  
\[\frac{\mathrm{d_{qt}}}{\mathrm{dt}}=\mathrm{k_{2}}\Big{(}\mathrm{q_{e}-q_{t}} \Big{)} \tag{6}\]  
After integrating the differential equation, the pseudo-second order kinetic equation can be expressed as follows:  
\[\frac{\mathrm{t}}{\mathrm{q_{t}}}=\frac{1}{\mathrm{k_{2}}\mathrm{q_{e}}^{2}}+ \frac{1}{\mathrm{q_{e}}}\mathrm{t} \tag{7}\]  
where \(\mathrm{q_{e}}\) is the adsorption amount of the adsorbent surface to the solute under adsorption equilibrium (mg/g), \(\mathrm{q_{t}}\) is the adsorption amount of the adsorbent surface to the solute at the specified time (t) during the adsorption process (mg/g), and \(\mathrm{k_{2}}\) is the pseudo-second order kinetic equation adsorption rate constant. In general, there is a slight deviation when the single kinetic equation is used to reflect the actual reaction. Therefore, it is necessary to conduct a comparative study after fitting the pseudo-first order kinetic equation and the pseudo-second order kinetic equation. The relevant parameters of the pseudo-first order and pseudo-second order kinetic equations are presented in Table 5.  
The pseudo-first order kinetic model and the pseudo-second order kinetic model were utilized to analyze the adsorption kinetics of the adsorbent at different initial phosphorus concentrations. The results illustrate that the correlation coefficients of the pseudo-first order kinetic equation and the pseudo-second order kinetic equation are more than 0.9 and can better reflect the phosphorus adsorption process in different initial P concentrations. The fitting results of the pseudo-second order kinetic model are more accurate than the pseudo-first order kinetic model, because the correlation coefficients exceeded 0.98, while the correlation coefficient of the pseudo-first order kinetic model was only 0.9246 at an initial P concentration of 0.3 mg/L. The adsorption amount of the adsorbent surface to the solute under the adsorption equilibrium (\(\mathrm{q_{e}}\)) obtained by the pseudo-second order kinetic model was larger than that of the sample measured result (\(\mathrm{q_{exp}}\)).