## 2 Materials and Methods
### Methods
#### 2.2.3 Three-Dimensional CFD Model  
(a) Simulations setup  
The numerical solution of the continuity, momentum, energy and species equations was computed using a CFD proprietary code (ANSYS Fluent 16), based on the finite volume method. In this work, a steady state, three-dimensional, viscous, turbulent and incompressible (since the maximum Mach number is below 0.3) flow was assumed. Pressure-velocity coupling was taken care of by the segregated, pressure-based solver, the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [33]. Summary details of spatial discretization are presented in Table 1. The convergence criteria were set to \(10^{-6}\) for the thermal energy and chemical residuals and \(10^{-4}\) for residuals from mass, momentum, turbulence kinetic energy and turbulence energy dissipation rate. Relevant physical and chemical quantities were monitored to assure convergence.  
The governing equations solved are the continuity equation (Equation (21)), momentum equation (Equation (22)), energy equation (Equation (23)) and conservation equation for chemical species (Equation (24)):  
\[\frac{\partial\rho}{\partial t}+\ \frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{21}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\ \frac{\partial}{\partial x_{i}} \big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{ \partial}{\partial x_{i}}\left[\mu\left(\frac{\partial u_{j}}{\partial x_{i}} +\frac{\partial u_{i}}{\partial x_{j}}\right)\right]+\frac{\partial}{\partial x _{i}}\left(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right)+\rho g_{i}+F_{i} \tag{22}\]  
\[\frac{\partial}{\partial t}\big{(}\rho c_{p}T\big{)}+\frac{\partial}{\partial x _{j}}\ \big{(}\rho u_{j}c_{p}T\big{)}-\frac{\partial}{\partial x_{j}}\ \bigg{(}K\frac{\partial T}{ \partial x_{j}}\bigg{)}=S_{T} \tag{23}\]  
\[\frac{\partial}{\partial t}(\rho Y_{m})+\frac{\partial}{\partial x_{j}}\ \big{(}\rho u_{j}Y_{m}\big{)}=-\frac{\partial}{\partial x_{j}}\ J_{m,j}+R_{m}+S_{m} \tag{24}\]  
Notice that for steady simulations, time-dependent terms become zero.  
(b) Turbulence  
A laminar regime was forced in the monolith because of small Reynolds numbers due to low velocity and very small hydraulic diameter inside channels.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Gradient & Least-squares cell-based \\ \hline Pressure & Standard \\ \hline Momentum & \\ Species conservation & Second order upwind \\ Turbulent kinetic energy & \\ \hline \end{tabular}
\end{table}
Table 1: Summary of the spatial discretization.  
In the other parts of the domain, Reynolds-averaged Navier-Stokes (RANS) approach was employed with the realizable \(k-\varepsilon\) model [34] selected. This model has the same turbulent kinetic energy equation as the standard \(k-\varepsilon\) model but holds an improved equation for \(\varepsilon\). Compared to standard \(k-\varepsilon\), it shows better performance for flows involving: planar and round jets (predicts round jet spreading correctly), boundary layers under strong adverse pressure gradients or separation, rotation, recirculation or/and strong streamline curvature [34].