# Review on Mechanisms and Kinetics for Supercritical Water Oxidation Processes
## 3 Kinetics Models and Their Development
### Empirical Model
#### 3.1.2 Multiple Nonlinear Regression Methods  
Relative to linear regression methods, multiple nonlinear regression methods can precisely fit the experiment data and describe the impact of experiment factors at the expense of higher computations. For example, [PERSON] et al. [70] used the POWELL algorithm to establish the kinetic model of supercritical water oxidation of methane at 580-652 \({}^{\circ}\)C and 245.8 bar, with the kinetic parameters being in the 95% confidence interval. Additionally, since the Runge-Kutta algorithm can reproduce the global characteristics of the experimental data under the small amount of calculation and Response Surface Methodology (RSM) can describe the reciprocal effect of parameters and find the optimal conditions, both of them were widely used in SCWO studies. Runge-Kutta algorithm expressions: [PERSON] et al. [84] compared the kinetic results of multiple linear regression expressions and Runge-Kutta algorithm expressions, thus validating that the Runge-Kutta algorithm was a more efficient method to get real waste kinetic parameters for the scale-up SCWO unit. The oxygen order in the kinetic equations must be considered for better simulation of commercial SCWO devices, thus further optimizing the oxygen consumption and minimizing the operating costs. [PERSON] et al. [86] established a kinetic model for oxygen concentration dependence in the supercritical water oxidation of cutting fluid wastewater. The oxygen order was 0.58 in the kinetic equation fitted by the Runge-Kutta algorithm, indicating the oxygen concentration had a significant effect on the degradation of cutting fluid wastewater and should be taken into account when establishing kinetic models. Their kinetic model was then used for simulating large scale SCWO processes to evaluate the feasibility of energy recovery in commercial SCWO devices [87]. Additionally, [PERSON] et al. [88] studied the SCWO kinetics of IPA through the Runge-Kutta algorithm and found that the oxygen order in the kinetic model was 0.774. With high oxygen content, the kinetic model obtained better prediction results at the initial rapid reaction than at the end of the reaction. Response Surface Methodology (RSM): RSM, a method to find the optimal experimental conditions, explores the relations between several explanatory variables and one or more response variables through multiple quadratic regressions. Different from the above empirical model methods that the kinetic models were obtained by establishing and solving power-law equations, RSM obtained the quantitative relations between the objective function and the process variables through polynomial fitting. Various studies on supercritical water oxidation showed that the oxidation reaction rate was sensitive to many process variables, such as temperature, oxidation coefficient, reaction time, reactant concentration and initial pH value of solution [89; 90; 91]. It is necessary to optimize the process variables to obtain optimal conditions since parameters significantly affect the oxidation efficiency. The parameters, however, interact with each other, and the single-factor experiments (i.e., changing one factor while keeping the others constant) that have been used in many studies are insufficient for accurate process optimization [92; 93; 94]. In this case, RSM can be used to evaluate the correlation degree of parameters and obtain the optimal conditions, which can effectively reduce the number of experiments, thus saving experiment time and consumptions. [PERSON] et al. [95] investigated the partial oxidation of landfill leachate in SCW at 450-550 \({}^{\circ}\)C, 23-27 MPa, 300-900 s and OC (oxidation coefficient) = 0.1-0.4, and then established the quadratic polynomial models with TOC removal rate (TRE), H\({}_{2}\) yield (GY\({}_{\text{H}2}\)), and carbon recovery rate (CR) as response values to evaluate the effect of the parameters on these values and predict experimental results.