## 4 Computation Models  
The mixture process of the fuel and air, ignition process and diffusion flame development are simulated with CONVERGE 3-D software. The n-heptane is widely used to study the diesel diffusion flame due to the similar cetane number and combustion heat to diesel [27; 28]. Therefore, n-heptane is chosen as the fuel for simulation in the present study. The volume of fluid (VOF) model is employed to simulate the multi-phase flow by tracking the volume of fluid within each fluid cell. RNG k-\(\epsilon\) turbulence model is used, and the transportation equations for the turbulent kinetic energy and turbulent dissipation are given by:  
\[\frac{\partial\rho k}{\partial t}+\frac{\partial\rho u_{i}k}{\partial x_{i}}= \tau_{ij}\frac{\partial u_{i}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}} \frac{\mu+\mu_{t}}{Pr_{k}}\frac{\partial k}{\partial x_{j}}-\rho\varepsilon+ \frac{C_{s}}{1.5}S_{s} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Density @15 \({}^{\circ}\)C** & **Viscosity@20 \({}^{\circ}\)C** & **Surface Tension @20** & **Calorific Value** \\ (kg/m\({}^{3}\)) & (mm\({}^{2}\)/s) & \({}^{\circ}\)C (kg/s\({}^{2}\)) & (MJ/kg) \\ \hline
840 & 3.6 & 25.6 & 43.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the diesel fuel. Figure 1: Schematic diagram of (**a**) the test system and (**b**) the temperature measuring points in the cross-section of the exhaust pipe. \[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}(\rho \epsilon u_{i})=\frac{\partial}{\partial x_{j}}\left(\frac{\mu+\mu_{t}}{Pr_{ \epsilon}}\frac{\partial\epsilon}{\partial x_{i}}\right)+c_{\epsilon 3}\rho\epsilon\frac{ \partial u_{i}}{\partial x_{i}}+\left(c_{\epsilon 1}\frac{\partial u_{i}}{\partial x_{i}} \tau_{ij}-c_{\epsilon 2}\rho\epsilon+c_{s}S_{s}\right)\frac{\epsilon}{k}+S-\rho R \tag{2}\]  
where \(\epsilon\) is the turbulent dissipation, \(c_{\epsilon 1}\), \(c_{\epsilon 2}\) and \(c_{\epsilon 3}\) are constants. The default values in CONVERGE are applied. The turbulent kinetic energy, \(\kappa\), is defined by the equation shown below:  
\[\kappa=\frac{1}{2}\widetilde{u_{i}u_{i}}^{\prime} \tag{3}\]  
The SAGE detailed chemistry solver is used to model the detailed chemical kinetics, which calculates the reaction rates for each elementary reaction. The multi-step reaction mechanism is given by:  
\[\sum_{m=1}^{M}\
u_{m,\gamma}^{\prime}\chi_{m}\Leftrightarrow\sum_{m=1}^{M}v_{ m,\gamma}^{\prime\prime}\chi_{m}\text{ for }\gamma=1,2,\dots,R \tag{4}\]  
where the \(\
u_{m,r}^{\prime}\) and \(v_{m,r}^{\prime\prime}\) are the stoichiometric coefficients for the reactants and products, respectively. For the \(r\)-th reaction, the rate of progress parameter \(q_{r}\) is given by:  
\[q_{r}=k_{fr}\prod_{m=1}^{M}[X_{m}]^{\
u_{m,r}^{\prime}}-k_{rr}\prod_{m=1}^{M}[ X_{m}]^{\
u_{m,r}^{\prime\prime}} \tag{5}\]  
where \(k_{fr}\) and \(k_{rr}\) are the forward and reverse rate coefficients, respectively. The governing equations for mass and energy are shown as follows:  
\[\frac{d[X_{m}]}{dt}=\dot{\omega}_{m} \tag{6}\]  
\[\frac{dT}{dt}=\frac{V\frac{dP}{dt}-\sum_{m}\left(\widetilde{h}_{m}\dot{\omega }_{m}\right)}{\sum_{m}\left([X_{m}]\overline{c}_{p,m}\right)} \tag{7}\]  
The equations above are extracted from the CONVERGE software. Figure 2 presents the structure of the intake manifold burner, the schematic of the model and the adaptive mesh evolution with time. The diameter of the fuel inlet pipe is 7.5 mm. The micro-channel between the heating rods and the metal housing of the intake manifold burner, which allows the flowing and gasifying of diesel, has a width of 1.75 mm. The base grid size is 4 mm, and the length of the combustion chamber is 500 mm. A fixed grid embedding is employed on the micro-channel, and the adaptive mesh refinement on temperature, species and velocity is employed. The minimum grid size is set to 0.5 mm. With the diesel flowing into the intake manifold burner, the grid number increases and continues to increase when it flows out of the flame shroud.