## 3 Research Status of Numerical Simulation
### Flow-Reaction  
In the field of SCWO, plug flow reactor (PFR) and continuous stirred tank reactor (CSTR, or perfectly stirred reactor, PSR) are used to represent the flow in tubular and volumetric reactors. In essence, both PFR and PSR belong to zero-dimensional flow-reaction model. [PERSON] et al. respectively simulated the flow of the core tube, the upper volume of the reactor and the cooling part of the reactor by using PFR, PSR, PFR flow models, and then incorporated the overall reaction kinetic model obtained by experimental fitting and the Peng-Robinson EoS describing the variation of density and enthalpy with temperature, to accurately predict the temperature distribution of premixed flame [34]. [PERSON] et al. numerically studied the laminar premixed hydrothermal flame of ethanol in one-dimensional tube flow [91]. After weighing the pros and cons of some EoS and overall reaction kinetic models, they selected the EoS of [PERSON] et al. and the ethanol overall reaction kinetic model of [PERSON] et al. [92] based on experimental fitting data. Moreover, equations obtained by [PERSON] et al. [93] and [PERSON] et al. [94; 95] were used to estimate transport properties such as viscosity, thermal conductivity and diffusion coefficient. For 4.3 wt% ethanol, they investigated the effects of preheating temperature (350 \({}^{\circ}\)C, 400 \({}^{\circ}\)C), flow velocity (1.0 m/s, 2.0 m/s, 3.0 m/s) and wall heat dissipation coefficient (0.0 W/m\({}^{2}\)/K, 2500 W/m\({}^{2}\)/K, 25,000 W/m\({}^{2}\)/K) on the flame temperature distribution in the tube. The results showed that the temperature rise at the preheating temperature of 350 \({}^{\circ}\)C (\(\Delta\)T = 80 \({}^{\circ}\)C) was about 200 \({}^{\circ}\)C lower than that at preheating temperature of 400 \({}^{\circ}\)C (\(\Delta\)T = 280 \({}^{\circ}\)C)because of the significant specific heat capacity in the transcritical region, and that the increase in flow velocity led to the thickening of the reaction zone. [PERSON] et al. [43] and [PERSON] et al. [29] have respectively carried out two-dimensional axisymmetric numerical simulation for the WCHB-3. They all adopted the Peng-Robinson EoS to calculate the density, specific heat capacity and enthalpy of the components in the supercritical state. The Peng-Robinson EoS is written as:  
\[ \begin{split}& P=\frac{RT}{v^{\prime}-b_{c}}-\frac{a_{m}}{v^{ \prime}(v^{\prime}+b_{c})+b_{c}(v^{\prime}-b_{c})}\\ & where\ v^{\prime}=v+VT,\\ & a_{c}=0.45724\frac{R^{2}T_{c}^{2}}{p_{c}^{2}}\\ & b_{c}=0.077796\frac{RT_{c}}{P_{c}},\\ &\alpha=\left[1+\kappa\big{(}1-\sqrt{T_{r}}\big{)}\right]^{2}\end{split}  \tag{1}\]  
\[ \begin{split}& a_{c}=0.45724\frac{R^{2}T_{c}^{2}}{p_{c}^{2}}\\ & b_{c}=0.077796\frac{RT_{c}}{P_{c}},\\ &\alpha=\left[1+\kappa\big{(}1-\sqrt{T_{r}}\big{)}\right]^{2} \end{split}  \tag{2}\]  
where \(VT\) is the constant volume translation, \(T_{c}\) is the critical temperature, \(P_{c}\) is the critical pressure. \(a_{c}\) and \(b_{c}\) represent the intermolecular attractive and repulsive forces, respectively. In order to remain brevity, the introduction of other coefficients like \(b_{i}\) would not be given here. While [PERSON] adopted [PERSON]'s model [93] to get the viscosity and thermal conductivity and [PERSON]'s model [96] to obtain the mass diffusion coefficient, [PERSON] adopted [PERSON]'s model to calculate the viscosity and thermal conductivity and [PERSON]'s model [97] to get the mass diffusion coefficient. In terms of [PERSON]'s model, the formula of dynamic viscosity is written as:  
\[\eta=\eta^{*}\eta^{0}=1.0\times 10^{-7}\Bigg{(}\frac{1}{G_{2}}+E_{6}y+\frac{ \Omega v}{F_{cm}\sqrt{T_{m}}}\eta^{*}\Bigg{)}\frac{40.785F_{cm}\sqrt{M_{m}T}}{ V_{cm}^{2/3}\Omega v} \tag{3}\]  
where the \(\eta^{0}\) is low-pressure viscosity, \(k_{m}\) is a correction factor for polar substance and \(\Omega_{p}\) is the collision integral as a function of temperature. For the sake of brevity, the introduction of other coefficients would not be repeated here.