## 2 Numerical Methods and Physical Models  
In this study, the FLUENT commercial code is used to simulate the reaction and fluid flow in a SRU thermal reactor. The SIMPLE algorithm by [PERSON] [10] is used to solve the governing equations. The discretizations of convection terms and diffusion terms are respectively performed using the power-law scheme and the central difference scheme. In terms of physical models, considering the accuracy and stability of the models and the evaluations of other researchers, the standard \(k\)-\(\varepsilon\) model [11], the P-1 radiation model [12] and the non-premixed combustion model with a \(\beta\)-type probability density function [13] are respectively used for the turbulence, radiation and combustion simulations. The standard wall functions [14] are used to resolve the flow quantities (velocity, the temperature, and the turbulence quantities) at the near-wall regions. For the steady-state three-dimensional flow field with the chemical reaction in this study, the governing equations for the continuity equation, momentum equation and turbulence equation (\(k\)-\(\varepsilon\) model) are described below. The governing equations for the energy equation, radiation equation, combustion equation and convergence criterion are described in a previous study by the author [15]. **Continuity equation**:  
\[\
abla\bullet\overset{\rightarrow}{v}=0 \tag{1}\]  
**Momentum equation**:  
In an inertial reference frame, the momentum equation [16] may be expressed as  
\[\
abla\bullet(\varrho\overset{\rightarrow}{v}\overset{\rightarrow}{v})=-\
abla p +\
abla\bullet\overline{\overline{\tau}}+\varrho\overset{\rightarrow}{g}+ \overset{\rightarrow}{F} \tag{2}\]  
where \(\overset{\rightarrow}{F}\) is the possible source term of the momentum equation which may include the body force caused by the interaction with dispersed phases or the extra force terms caused by physical models or numerical schemes. In Equation (2), the stress tensor \(\overline{\overline{\tau}}\) may be expressed as  
\[\overline{\overline{\tau}}=\mu\left[\
abla\overset{\rightarrow}{v}+\
abla \overset{\rightarrow}{v}^{\mathrm{T}}-\frac{2}{3}\
abla\bullet\overset{ \rightarrow}{v}I\right] \tag{3}\]  
where \(I\) is the unit tensor, the superscript T represents the transpose operator, and the third term on the right-hand side of the equal sign represents the volume dilation effect. By the long-time averaged method, Equation (2) may be converted to Reynolds-averaged Navier-Stokes (RANS) equations as follows:  
\[\frac{\partial}{\partial x_{j}}\left(\varrho u_{i}u_{j}\right)=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}-\frac{2 }{3}\xi_{ij}\frac{\partial u_{l}}{\partial x_{l}}\right)\right]+\frac{ \partial}{\partial x_{j}}\left(-\varrho\overline{u_{i}\prime u_{j}}\right) \tag{4}\]  
In Equation (4), the turbulent stresses (Reynolds stresses) may be related to the mean velocity gradient by the Boussinesq hypothesis as follows:  
\[-\varrho\overline{u_{i}\prime u_{j}}=\mu_{t}\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}\left( \varrho k+\mu_{t}\frac{\partial u_{k}}{\partial x_{k}}\right)\delta_{ij} \tag{5}\]  
In Equation (5), the turbulent (or eddy) viscosity may be expressed in terms of the turbulence kinetic energy (\(k\)) and turbulence dissipation rate (\(\epsilon\)) as:  
\[\mu_{t}=C_{\mu}\varrho\frac{k^{2}}{\epsilon} \tag{6}\]  
As mentioned above, \(k\) and \(\epsilon\) are modeled by the \(k\)-\(\epsilon\) model, as described below.