## 2 Numerical Simulations
\begin{table}
\begin{tabular}{c c} \hline \hline
**Engine Specification** & **Value** \\ \hline brand and model & F1 AT-FIRE 1.2 8v \\ \(n\). of cylinders & 4 \\ Bore & 70.8 (mm) \\ Stroke & 78.86 (mm) \\ Rod to crank ratio & 3.27 \\ Compression ratio & 9.8 \\ Engine displacement & 1242 (cm\({}^{3}\)) \\ Engine maximum power & 43.1 (kW) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Internal Combustion Engine (ICE) specifications. set to roughly 60% of the gasoline counterpart, the A/F ratio must be set to \(\lambda\) = 2, and the engine must be supercharged in order to obtain MAP = 2 bar absolute. The engine model was then upgraded in order to implement supercharging (SC) in two different ways: using a turbocharger (TC) driven by an exhaust gas turbine or using a volumetric compressor (VC) driven directly by the engine. In the first case, the engine MAP is set to a desired value that, for the sake of simplicity, was kept constant as the engine speed varied; the engine back pressure (\(p_{b}\)) was evaluated supposing that the same mass flows through both the compressor and turbine and equating the two specific works with the following equation:  
\[p_{b}=\left(1\ -\ \frac{T_{MAN}\Big{(}\mathrm{MAP}^{\frac{k}{k}-1}\ -\ 1\Big{)}}{ \eta_{c}\eta_{t}T_{ExMAN}}\right)^{\frac{-k}{k-1}} \tag{2}\]  
where \(k\) is the isentropic coefficient (1.4 for air); \(T_{MAN}\) is the inlet manifold air temperature that is set to the ambient temperature (300 K) considering the presence of an intercooler; \(\eta_{c}\) and \(\eta_{t}\) are the compressor and turbine isentropic efficiencies set to 0.85 and 0.9, respectively; and \(T_{EXMAN}\) is the exhaust manifold gas temperature set to a first-try value and then verified by simulations with an iterative process. In the case of volumetric compressor supercharging (VCSC), as far as the compressor is driven directly by the engine, the corresponding FMEPc must be evaluated in order to obtain: \(\mathrm{BMEP}\) = \(\mathrm{IMEP}\ -\ \mathrm{FMEP}\ -\ \mathrm{FMEPc}\). Assuming, for the sake of simplicity, an adiabatic screw compressor, the same volumetric efficiency for both engine and compressor and the presence of an intercooler between compressor and engine, one obtains:  
\[\mathrm{FMEP_{c}}=\frac{k}{k\ -\ 1}\mathrm{MAP}\Big{(}\mathrm{MAP}^{\frac{k}{k} -\ 1}\ -\ 1\Big{)} \tag{3}\]  
The desired MAP value depends on the compressor displacement \(V_{c}\) and rotating speed \(n_{c}\):  
\[\mathrm{MAP}=\frac{2V_{c}n_{c}}{Vn} \tag{4}\]  
where \(V\) and \(n\) are the engine displacement and rotating speed, respectively. Three engine configurations were simulated: the naturally aspirated gasoline-fueled engine, the turbocharged (TC) hydrogen-fueled engine and the supercharged (SC), with volumetric compressor (VC) hydrogen-fueled engine. For all the configurations, the engine speed and MAP were varied obtaining many different operating conditions. For each operating condition, \(\lambda\) and SA were set as the optimal values (i.e., the maximum brake torque SA) or the knock-limited values (in the case of gasoline-fueled engines, those values were experimentally pre-determined [15; 16; 17]). Some preliminary simulations were performed in order to find the MAP value of the boosted hydrogen-fueled engine able to produce the same maximum power of the gasoline version (43.1 kW): for the TC version, this value was MAP = 2.1 bar absolute (in accordance with the above-mentioned considerations) while, for the VCSC version, the value was MAP = 2.7 bar absolute; the second MAP value was higher because of the higher energy needed to drive the compressor. Figure 1 shows the \(\lambda\) and SA maps used in the gasoline-fueled engine simulation for all the operating conditions. The SA value is expressed in crank angle degrees (CAD) before top dead center (BTDC). For both hydrogen configurations, the A/F ratio is fixed at \(\lambda\) = 2 for all the operating conditions, while the SA is set to the maximum brake torque value. Figure 2a,b shows the SA maps adopted for the TC and SC engine, respectively.