# Review on Mechanisms and Kinetics for Supercritical Water Oxidation Processes
## 3 Kinetics Models and Their Development
### Empirical Model
#### 3.1.3 Multi-Step Rate Expressions  
Multi-step rate expressions: Multi-step rate expressions divide the experimental data into several different reaction stages to better describe the SCWO phenomenon. There are various mathematical methods to establish multi-step rate expressions, such as multiple linear regression and multiple nonlinear regression. The common two-step kinetic model divides the SCWO process into the following two stages [88]: (1) a rapid initial reaction, in which reactive molecules are decomposed at a high rate to CO\({}_{2}\), H\({}_{2}\)O and a mixture of intermediates of a more refractory nature; (2) a slow subsequent reaction, in which the refractory compounds are gradually decomposed to more simple molecules (alcohols, organic acid, etc.) and eventually mineralized down to CO\({}_{2}\), H\({}_{2}\)O, N\({}_{2}\), and salts. Employing tubular reactors and using TOC and COD as the evaluation indices, [PERSON] et al. [88] proposed and developed two two-step first-order kinetic models to describe the SCWO kinetics of IPA under oxygen-rich conditions, and these two models had good predictions under oxygen-rich conditions. [PERSON] et al. [98] investigated the catalytic effect of Cr\({}^{3+}\)/ash for the degradation of naphthalene (Nap) in SCWO and established three-step kinetic equations and two-step kinetic equations for reaction systems with or without a catalyst. For System A (Nap+O\({}_{2}\)+H\({}_{2}\)O) without a catalyst, the oxidation of Nap was divided into three stages--a fast-slow-fast process: the fast stage, in which H\({}_{2}\)O\({}_{2}\) decomposed into H\({}_{2}\)O and O\({}_{2}\), the slow stage, in which oxidation occurred in saturated/subcritical water between 200 \({}^{\circ}\)C-120 atm and 360 \({}^{\circ}\)C-225 atm, and the final fast stage, in which rapid oxidation occurred in near-critical/supercritical water above 360 \({}^{\circ}\)C. For the System B (Nap+O\({}_{2}\)+Cr(NO\({}_{3}\))\({}_{3}\) + Al\({}_{2}\)Si\({}_{2}\)O\({}_{5}\)(OH)\({}_{4}\)+H\({}_{2}\)O), since the catalysis of Cr\({}^{3+}\)/ash and the oxidation of naphthalene in Stage 2 were faster in System B than in System A, the oxidation process in System B was divided into two stages. The kinetic equations of the two systems are shown in Equations (23)-(25) and Equations (26) and (27), respectively. Stage 1 of Nap-A  
\[-\frac{d[\text{COD}]}{d\tau}=-1.59\times 10^{2}\exp\left(-\frac{28.76}{RT} \right)(1-[\text{COD}])[\text{O}_{2}] \tag{23}\]  
Stage 2 of Nap-A  
\[-\frac{d[\text{COD}]}{d\tau}=-4.52\times 10^{-4}\exp\left(\frac{6.52}{RT} \right)(1-[\text{COD}])[\text{O}_{2}] \tag{24}\]  
Stage 3 of Nap-A(SCWO)  
\[-\frac{d[\text{COD}]}{d\tau}=-1\times 10^{8}\exp\left(-\frac{139.5}{RT} \right)(1-[\text{COD}])[\text{NT}]^{-0.86}[\text{O}_{2}]^{1}[\text{H}_{2} \text{O}]^{0.99} \tag{25}\]  
Stage 1 of Nap-B  
\[-\frac{d[\text{COD}]}{d\tau}=-1\times 10^{-2}\exp\left(-\frac{8.72}{RT} \right)(1-[\text{COD}])^{0.22}[\text{NT}]^{-0.78}[\text{O}_{2}]^{0.2} \tag{26}\]  
Stage 2 of Nap-B  
\[-\frac{d[\text{COD}]}{d\tau}=-7.27\times 10^{-1}\exp\left(-\frac{17.11}{RT} \right)(1-[\text{COD}])^{0.76}[\text{NT}]_{0}^{-0.24}[\text{O}_{2}]^{1.8}[ \text{H}_{2}\text{O}] \tag{27}\]  
The empirical models only reflected the overall trends of the experimental data due to the absence of the complex oxidation process in detail. When different algorithms were used to fit the experimental data, the results would also change accordingly. Table 2 compares the modeling approaches, advantages, and drawbacks of the above algorithms.