## 2 Materials and Methods  
The three-dimensional Lagrangian-Eulerian multiphase transient model was used to simulate the flow pattern of three-phase gas-liquid-solid flow. The present numerical study includes the continuity and momentum conservation equations, which were applied for each phase. The governing equations of continuity (1) and momentum (2) balance for the liquid phase can be written as follows:  
\[\
abla\cdot(\rho u)=0 \tag{1}\]  
\[\rho\frac{\partial u}{\partial t}+\rho(u\cdot\
abla)u=\
abla\cdot\left[-pI+ \mu(\
abla u+(\
abla u)^{T})\right] \tag{2}\]  
where: \(\rho\), \(\mu\), and \(u\) are the density (kg/m\({}^{3}\)), viscosity (m\({}^{2}\cdot\)s\({}^{-1}\)), and fluid velocity (m/s), respectively, \(p\) is the pressure (Pa), and \(I\) is the unit vector of the phase. The energy conservation equation is calculated from Formula (3):  
\[\rho C_{p}\bigg{(}\frac{\partial T}{\partial t}+(u\cdot\
abla)T\bigg{)}+\
abla \cdot(-k\
abla T)=Ua(T_{ext}-T) \tag{3}\]  
The gas phase is described by the continuity equation and the volume-averaged Navier-Stokes equations, where the gas phase density is calculated using the Peng-Robinson equation of state. The gas particle drag is calculated using the Beetstra correlation. The calculation of gas phase mass fraction is determined by Equation (4):  
\[\frac{\partial(\varepsilon_{q}\rho_{q}u_{q})}{\partial t}+\
abla\cdot\left( \varepsilon_{q}q_{S}u_{q}w_{q}\right)=-\
abla\cdot\left(\varepsilon_{q}\rho_ {q}q_{m}\right)+S_{m} \tag{4}\]  
where: \(\varepsilon_{q}\) is the gas holdup, \(\rho\) is the density (kg/m\({}^{3}\)), \(u\) is the velocity (m/s), \(w_{q}\) is the liquid mass fraction, and \(S_{m}\) is the source term for gas particle mass transfer. The gas phase thermal energy balance is represented by Equation (5) below:  
\[c_{p,g}\bigg{[}\frac{\partial(\varepsilon_{q}\rho_{q}T_{q})}{\partial t}+ \
abla\cdot\left(\varepsilon_{q}q_{S}u_{q}T_{q}\right)=-\
abla\cdot\left( \varepsilon_{q}q_{T}\right)+Q_{p}\bigg{]} \tag{5}\]  
where: \(T_{q}\) is the temperature (K), \(c_{p,g}\) is the specific heat of the gas (J/mol\(\cdot\)K), and \(q_{T}\) and \(Q_{p}\) are the source terms for the interphase with the particle. The basic transport reaction for concentration of species is written as follows:  
\[\frac{\partial}{\partial t}(\rho\omega_{i})+\
abla\cdot\left(\rho\omega_{i} u\right)=-\
abla\cdot J_{i}+R_{i} \tag{6}\]  
where: \(\omega_{i}\) is the mass fraction of the \(k\)-th component in the phase \(i\), \(J_{i}\) is the diffusive flux of the \(k\)-th component (kg/m\({}^{2}\)s), and \(R_{i}\) is the production of \(i\) as a result of a chemical reaction. In this study, the average mixture diffusion model was used, which was selected for the diffusive stream and was calculated as follows:  
\[J_{i}=-\bigg{(}\rho D_{i}^{m}\
abla\omega_{i}+\rho\omega_{i}D_{i}^{m}\frac{ \
abla M_{n}}{M_{n}}\bigg{)} \tag{7}\]  
where: \(\omega_{i}\) is the mass fraction of the \(k\)-th component in the phase \(i\), \(J_{i}\) is the diffusive flux of the \(k\)-th component (kg/m\({}^{2}\)s), \(R_{i}\) is the production of \(i\) as a result of a chemical reaction, and \(D_{i}^{m}\) is the multi-component diffusion coefficient. The solid phase is represented by particles, which are regarded as the discrete phase, where the Lagrange method is employed to describe particle motion control. The momentum balance equation is as follows:  
\[\frac{\partial u_{p}}{\partial t}=-\frac{\
abla p}{\rho_{p}}+F_{D}\Big{(}u_{ f}-u_{p}\Big{)}+\frac{\left(\rho_{p}-\rho_{f}\right)}{\rho_{p}}g-\frac{\
abla \cdot\frac{=}{\tau}}{\rho_{p}} \tag{8}\]where: \(u_{p}\) is the particle velocity (m/s), \(\rho_{p}\) is the particle density (kg/m\({}^{3}\)), \(\overline{\tau}\) is the solid tensor caused by the particle interaction, \(F_{D}\Big{(}u_{f}-u_{p}\Big{)}\) is the particle acceleration caused by resistance, and \(\frac{\
abla p}{\rho_{p}}\) is the acceleration of particles due to the pressure gradient.