## 3 Materials and Methods
### Discrete Phase  
In the discrete phase, each particle is tracked by the Lagrangian approach. The force balance on the particles is described as:  
\[\frac{du_{p}}{dt}=F_{D}(u-u_{p})+\frac{g_{x}(\rho_{p}-\rho)}{\rho_{p}}, \tag{8}\]  
where \(F_{D}(u-u_{p})\) is the drag force per unit particle mass:  
\[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}R_{e}}{24}, \tag{9}\]  
\[Re=\frac{\rho d_{p}\big{|}u_{p}-u\big{|}}{\mu}, \tag{10}\]  
where \(u\) is the fluid phase velocity, \(u_{p}\) is the particle velocity, \(\mu\) is the molecular viscosity of the fluid, \(\rho\) is the fluid density, \(\rho_{p}\) is the particle density, and \(d_{p}\) is the diameter of the particle. \(Re\) is the relative Reynolds number calculated from the slip velocity between the fluid phase velocity and the particle velocity. The particles are tracked by the mean and fluctuating velocity as:  
\[u*=\overline{u}+u^{\prime}, \tag{11}\]  
where the fluctuating velocity in Equation (11) is defined by the stochastic discrete random walk model. The burning rate of the char particles is adopted from the study of [PERSON] [31]. The particle surface reaction rate is calculated as:  
\[\overline{R}=A_{p}\eta YR, \tag{12}\]  
\[R=R_{kin}\bigg{(}p_{n}-\frac{R}{D}\bigg{)}^{N}, \tag{13}\]  
where \(A_{p}\) is the particle surface area (m\({}^{2}\)), \(Y\) is the mass fraction of surface species in the particle, \(\eta\) is the effectiveness factor (dimensionless), \(R\) is the rate of particle surface species reaction per unit area (kg/m\({}^{2}\)/s), \(p_{n}\) is the bulk concentration of the gas phase species (kg/m\({}^{3}\)), \(D\) is the diffusion rate coefficient for the reaction, \(R_{kin}\) is the kinetic rate of reaction (units vary), and \(N\) is the apparent order of reaction. In this study, D is 5E-12 and N is 0.5. The kinetic rate of reaction \(r\) is defined as:  
\[R_{kin}=AT^{n}e^{-(E/RT)}, \tag{14}\]  
where the \(n\) (temperature exponent) is 1. The Rosin-Rammler distribution [32] is employed for the simulation. The water within the coal slurry is assumed to atomized and evaporated when injecting into the gasifier. The rate of vaporization is of the droplet is calculated as:  
\[\frac{dm_{p}}{dt}=\pi d^{2}k_{c}(C_{s}-C_{\infty}), \tag{15}\]  
where \(k_{c}\) is the mass transfer coefficient and \(C_{s}\) is the concentration of the vapor at the particle's surface. \(C_{\infty}\) is the vapor concentration of the bulk flow. The empirical correlation by [33; 34] is employed to evaluated the value of \(k_{c}\) as:  
\[Sh_{d}=\frac{k_{c}d}{D}=2.0+0.6 Re_{d}^{0.5}Sc^{0.33}, \tag{16}\]  
where \(Sh\) is the Sherwood number, \(Sc\) is the Schmidt number (defined as \(v\)/\(D\)), \(D\) is the diffusion coefficient of vapor in the bulk flow. \(Re_{d}\) is the Reynolds number, \(D\) is the droplet diameter. When the droplet temperature reaches the boiling point, the evaporation rate [35] is evaluated as:  
\[\frac{dm_{d}}{dt}=\pi d^{2}\bigg{(}\frac{\lambda}{d}\bigg{)}\Big{(}2.0+0.46\text{ Re}_{d}^{0.5}\Big{)}\ln\Big{(}1+C_{p}(T_{\infty}-T)/h_{fg}\Big{)}/C_{p}, \tag{17}\]  
The convection, radiation, devolatilization, and surface reactions are included in the energy equation as:  
\[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}R_{e}}{24}, \tag{9}\]0 \[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}R_{e}}{24}, \tag{9}\]1  
where \(m_{p}\) and \(T_{p}\) are the particle mass and particle temperature, respectively. \(h_{fg}\) is the latent heat, \(H_{vac}\) is the heat release due to surface reaction, and \(f_{h}\) is the fraction of due to the heat absorption of particle. \(\theta_{R}\) is the radiating temperature, defined as:  
\[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}R_{e}}{24}, \tag{9}\]2  
and \(G\) is the incident radiation, which is related to the radiative intensity as:  
\[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}R_{e}}{24}, \tag{9}\]3  
The Pi model [36] is employed for the radiation in the gasifier.