# Nozzle Design of Plug-and-Play Passive Pre-Chamber Ignition Systems for Natural Gas Engines
## 2 Methodology and Specification
### Numerical Model and Calibration  
Based on the boundary conditions provided by the one-dimensional simulation software GT-POWER, this study used the three-dimensional computational fluid dynamics software CONVERGE to describe the flow, heat transfer, and combustion phenomena. The spatial discretization of the transport equations adopted the second-order upwind numerical scheme. The time integration took the implicit format, with the time step deter \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Engine type & In-line six-cylinder, four-stroke \\ Intake system & Turbocharged, intercooler, cooled EGR \\ Bore & 136 mm \\ Stroke & 167 mm \\ Displacement & 14.6 L \\ Compression ratio & 12.5 \\ Rated power/speed & 390 kW/1700 rpm \\ Maximum torque/speed & 2500 N\(\cdot\)m/950\(\cdot\)1350 rpm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test engine specifications. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cases** & **Speed (rpm)** & **Torque (N\(\cdot\)m)** & **Conditions** & **Power (kW)** & **BMEP (MPa)** & **BTE (\%)** \\ \hline
1 & 900 & 1000 & Low speed, low load & 94 & 0.87 & 36.0 \\
2 & 1100 & 2000 & Normal power & 230 & 1.73 & 40.3 \\
3 & 1100 & 2500 & Low speed, high load & 288 & 2.16 & 40.3 \\
4 & 1700 & 2190 & Rated power & 390 & 1.90 & 39.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bench test operating conditions. Figure 2: Schematic of the experimental setup. mined by the Courant-Friedrichs-Lewy condition. The PISO algorithm was used for the pressure-velocity coupling. On the other hand, the RNG k-\(\epsilon\) model was used for turbulence simulation because it is effective and can reproduce statistically averaged flow patterns of the LES model (Table 4). The ignition process adopted the source/sink model, releasing 40 mJ of ignition energy in each cycle. Figure 3 shows the computational domain, of which the hybrid meshing process used adaptive refinement to capture the temperature and velocity gradients. The grid independence verification shows that, when the elemental size was below 4 mm, the cylinder pressure became consistent. The G-Equation modeled the flame dynamics, and the laminar flame velocity was corrected to match the experimental cylinder pressure and heat release rate curves. The governing equations of mass, species, momentum, and energy transport are given as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho v)=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho Y_{i})+\
abla\cdot(\rho vY_{i})=\
abla\cdot( \rho D\
abla Y_{i})+\dot{\rho_{i}} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho v)+\
abla\cdot(\rho vv)=-\
abla p+\
abla\cdot \boldsymbol{\tau}-\frac{2}{3}\
abla(\rho k) \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho e)+\
abla\cdot(\rho ve)=-\
abla\cdot\dot{ \boldsymbol{q_{s}}}-p\
abla\cdot\boldsymbol{v}+\rho e+\dot{q_{V}} \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Phenomenon** & **Sub-Model** \\ \hline Turbulence & RNG k-\(\epsilon\) model \\ Boundary layer & Enhanced wall treatment \\ Multiphase flow & Species transport model \\ Ignition & Source/sink model \\ CH\({}_{4}\) combustion & Opt. GRI-Mech 3.0 \\ NO\({}_{x}\) emission & Extended Zeldovich mechanism \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sub-models for physical and chemical phenomena. Figure 3: Computational domain, grid independence verification, and experimental calibration. where \(\rho\), \(v\), \(e\), and \(Y_{i}\) represent density, velocity, internal energy, and mass fraction of species \(i\), respectively. \(p\), \(\tau\), \(k\), and \(\varepsilon\) represent pressure, stress tensor, turbulence kinetic energy, and turbulence energy dissipation, respectively. \(\dot{q_{s}}\) and \(\dot{q_{V}}\) represent the surface heat flux and volume heat source, respectively. \(\dot{p_{i}}\) and \(D\) represent the chemical mass source term and mass diffusion coefficient, respectively.