## 5 Description of Modeling Approach  
The engine used in this paper is outlined in a 1D model by employing the GT-Power commercial code. The whole engine is represented in sub-components, including cylinders, intake/exhaust pipes and turbocharging system. In particular, the flow inside the intake/exhaust ducts is reproduced by solving the conservation equations of mass (1), energy (2) and momentum (3) reported below:  
\[\mathrm{d}\mathrm{m}/\mathrm{d}t=\Sigma\mathrm{i}\mathrm{n}, \tag{1}\]  
Figure 6: MFB 10% in Cyl #1 and Cyl #2 at different operating conditions as a function of EGR rate.  
Figure 7: MFB 10–50% in Cyl #1 and Cyl #2 at different operating conditions as a function of EGR rate.  
\[\mathrm{d}(\mathrm{m}\cdot\mathrm{e})/\mathrm{dt} =-\mathrm{p}\cdot\mathrm{d}V/\mathrm{dt}+\Sigma(\mathrm{\dot{m}h})- \mathrm{d}Q_{\mathrm{w}}/\mathrm{dt}, \tag{2}\] \[\mathrm{dm}/\mathrm{dt} =\{-\mathrm{dp}\cdot\mathrm{A}+\Sigma(\mathrm{\dot{m}\cdot u})-4 \cdot\mathrm{C}_{\mathrm{f}}\cdot\mathrm{p}\cdot\mathrm{u}\cdot\mathrm{l}\mid \mathrm{u}\mid\mathrm{\Lambda}\cdot\mathrm{dx}/2\cdot\mathrm{D}-\mathrm{A} \cdot\mathrm{C}_{\mathrm{p}}\cdot\mathrm{p}\cdot\mathrm{u}\cdot\mathrm{l}\mid \mathrm{u}\mid/2\}/\mathrm{dx}, \tag{3}\]  
where \(\mathrm{\dot{m}}\), \(\mathrm{m}\), \(\mathrm{V}\), \(\mathrm{p}\), \(\mathrm{e}\), \(\mathrm{h}\) and \(\mathrm{u}\) are mass flow rate, mass, volume, pressure, internal energy per unit mass, enthalpy per unit mass and velocity at boundary. \(\mathrm{Q}_{\mathrm{w}}\) is the heat exchanged through the walls, \(\mathrm{A}\) is the flow area, \(\mathrm{D}\) is the equivalent diameter. \(\mathrm{C}_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{f}}\) are the pressure and friction losses coefficients, while \(\mathrm{dx}\) and \(\mathrm{dp}\) represent the discretization length and the pressure differential across \(\mathrm{dx}\). Each cylinder is treated as a 0D volume. In this case, the scalar variables (e.g., pressure, temperature, internal energy, etc.) are assumed to be uniform over the volume. However, during the combustion process the cylinder volume is divided in two zones (burned and unburned regions), where the mass and energy conservation equations are solved. In the closed valve period of single cylinder (i.e., from IVC up to EVO), the energy equation is detailed for burned and unburned zones as reported by relations (4) and (5):  
\[\mathrm{d}(\mathrm{m}_{\mathrm{u}}\cdot\mathrm{e}_{\mathrm{u}})/\mathrm{dt} =-\mathrm{p}\cdot\mathrm{d}V_{\mathrm{u}}/\mathrm{dt}-\mathrm{d}Q_{ \mathrm{w,u}}/\mathrm{dt}-\mathrm{h}_{\mathrm{u}}\cdot\mathrm{dm}_{\mathrm{b} }/\mathrm{dt}, \tag{4}\] \[\mathrm{d}(\mathrm{m}_{\mathrm{b}}\cdot\mathrm{e}_{\mathrm{b}})/ \mathrm{dt} =-\mathrm{p}\cdot\mathrm{d}V_{\mathrm{b}}/\mathrm{dt}-\mathrm{d}Q_{ \mathrm{w,b}}/\mathrm{dt}+\mathrm{h}_{\mathrm{u}}\cdot\mathrm{dm}_{\mathrm{b} }/\mathrm{dt}, \tag{5}\]  
where \(\mathrm{dm}_{\mathrm{b}}/\mathrm{dt}\) represents the burning rate term evaluated through the turbulent combustion sub-model discussed in the following.  
The turbocharging operation is simulated by a \"map-based\" approach. Turbine and compressor maps, supplied by the manufacturer, are employed within the standard turbine and compressor objects, respectively. Flow permeability of the single cylinder head is taken into account through the measured steady flow coefficients, both under forward and reverse flow conditions. Standard injector objects, available in the 1D code, are adopted for gasoline injection. During port injections, it is assumed that 30% of the total liquid mass vaporizes immediately upon the injection event, as advised for a typical liquid injector in GT-Power user manual; details about the spray evolution and wall film formation are not taken into account. In order to enhance the model capability, it is integrated with refined phenomenological in-cylinder sub-models of turbulence, combustion, heat transfer and pollutant emissions. A detailed description of these sub-models is reported in the next subsection.