## 5 Description of Modeling Approach
### In-Cylinder Sub-Models and Tuning: Combustion, Turbulence and Pollutant Emissions
A single set of tuning constants was identified for the considered operating points, and tuning is kept fixed regardless the engine operating conditions. A particular attention is devoted to the modeling of the pollutant emissions. To this aim, \"in-house developed\" sub-models for the estimation of the main regulated cylinder-out emissions, namely HC, CO and thermal NO, are implemented into the 1D code. The propagation of the cylinder-out noxious species within the exhaust system up to the inlet section of TWC is also considered in the 1D model. In particular, CO concentrations are computed by solving, in the burned gas zone, a simplified chemical kinetic sub-model based on two-step reactions as reported in Reference [24]. The CO mechanism comprises the following high-temperature CO oxidation reactions (12) and (13):  
\[\text{CO}+\text{OH}\leftrightarrow\text{CO}_{2}+\text{H}, \tag{12}\]  
\[\text{CO}+\text{O}_{2}\leftrightarrow\text{CO}_{2}+\text{O}, \tag{13}\]  
For thermal NO emission, a refined approach is here adopted, where the semi-detailed chemical kinetics scheme proposed by [PERSON] [25] is applied to the burned gas zone to compute the NO concentrations in each cylinder. The kinetic mechanism includes 5 elements, 185 species and 937 reactions. This methodology, although computationally more expensive than the classic Zeldovich scheme, seems to be better in predicting the NO emissions of the single cylinder at varying the thermodynamic conditions, including pressure, temperature, inert content and mixture quality. HC production is simulated by the crevices model and a wall-flame quenching correlation, combined with a thermal-based HC oxidization equation during the expansion stroke. This means that the mixing-based oxidation contribution is neglected by assuming a perfect and instantaneous mixing of unburned charge from crevices with the hot burned gases. In the filling/emptying crevices model, HC species are assumed to accumulate/be released during the in-cylinder pressure rise/decrease phase in/from an arbitrary assigned constant volume, V\({}_{\text{crev}}\), as a fraction of the combustion chamber volume at TDC [26]:  
\[\text{x}_{\text{crev}},\,\%=100\cdot\text{V}_{\text{crev}}/\text{V}_{\text{ cyt-TDC}}, \tag{14}\]  
The volume V\({}_{\text{crev}}\) schematzes the crevices in the combustion chamber where the flame front extinguishes and it is controlled in the model by the input parameter, x\({}_{\text{crev}}\). The temperature in crevices volume is considered to be the same as the cylinder wall. A constant flow coefficient is attributed to the orifice of the crevices volume both for incoming and outgoing mass flows. Therefore, the pressure evolution within the crevice volume is computed by combining the related mass and gas temperature profiles. For the flame wall-quenching contribution, a simple correlation is here assumed which furnishes an estimation of the total HC arising from the extinguished flame at cylinder walls as a function of the flame quenching distance. This last parameter is assumed inversely proportional to the laminar flame speed, since the flame reaches the wall in laminar conditions. Flame quenching distance varies during the combustion evolution, and, in this study, it is taken towards the end of combustion process (i.e., 90% of mass fraction burned). Indeed, it is quite reasonable to assume that the HC generated by the flame-quenching during the initial combustion phase rapidly diffuse in the burned zone and oxidize. The correlation employed in this study for the quenching-related unburned HC resembles the simple model proposed in Reference [27], and it is shown in Equation (15):  
\[\text{HC}_{\text{quench}}=\text{H}_{1}\cdot\delta_{\text{L}}+\text{H}_{2}, \tag{15}\]  
where \(\delta_{\text{L}}\) is the laminar flame thickness, while H\({}_{1}\) and H\({}_{2}\) are sub-model tuning constants. Finally, for the oxidation of the overall HC level during the expansion stroke the one-step kinetic equation proposed in Reference [28] is adopted.