## 2 Materials and Methods
### Turbocharging  
Each of the two rows of cylinders has a turbocharger with a centrifugal compressor driven by a centripetal turbine from the exhaust gases and an intercooler. Equations (25) and (26) calculate \(MP_{\rm eng}\) developed by the internal combustion engine:  
\[MP_{\rm eng}=\frac{1}{3600}\times M_{\rm air}\times F\times\left(\frac{LHV}{ \rho_{\rm biogas}}\right)\times\eta_{\rm et} \tag{25}\]  
\[MP_{\rm eng}=V_{t}\times\frac{2n}{60\ n_{\rm stroke}}\times\rho_{\rm air}\times \eta_{\rm v}\times F\times\left(\frac{LHV}{\rho_{\rm biogas}}\right)\times\eta_ {\rm et} \tag{26}\]  
where \(\eta_{\rm et}\) is the engine effective efficiency. Equation (27) calculates the pressure ratio of the compressor (\(r_{\rm c}\)), neglecting the pressure losses in filter, intercooler and inlet manifold to cylinders. This equation considers the outlet pressure of the compressor (\(p_{1}\)) and the isentropic outlet pressure as equivalent (\(p_{1\rm S}\)):  
\[r_{\rm c}=\frac{p_{1}+p_{\rm atm-0}}{p_{0}}=\frac{p_{1\rm S}+p_{\rm atm-0}}{p _{0}} \tag{27}\]  
Equation (28) is used to calculate the temperature of the isentropic mixture at the compressor outlet (\(T_{\rm IS}\)), and the isentropic efficiency (\(\eta_{\rm is-C}\)) is calculated from Equation (29), considering the compressor outlet temperature (\(T_{1}\)) and the inlet temperature (\(T_{0}\)). \(\gamma\) is the adiabatic index:  
\[T_{\rm IS}=T_{0}\left(\frac{1}{r_{\rm c}}\right)^{\frac{(1-\gamma)}{\gamma}} \tag{28}\]  
\[\eta_{\rm is-C}=\frac{(T_{\rm IS}-T_{0})}{(T_{1}-T_{0})}\times 100 \tag{29}\]The thermal power of the compressor (\(TP_{\text{C}}\)) is estimated using the temperature difference between the outlet and inlet temperature as well as the specific heat value for the gas mixture (\(C_{\text{p-mix}}\)) as shown in Equation (30):  
\[TP_{\text{C}}=\frac{M_{\text{mix}}(1+F)}{3600}C_{\text{p-mix}}(T_{1}-T_{0}) \tag{30}\]  
In a similar way, the thermal power of the intercooler (\(TP_{\text{IC}}\)) is also estimated using Equation (31) with the outlet theoretical temperature (\(T_{\text{2S}}\)) included in the temperature difference term:  
\[TP_{\text{IC}}=\frac{M_{\text{mix}}(1+F)}{3600}C_{\text{p-mix}}(T_{1}-T_{ \text{2S}}) \tag{31}\]  
The efficiency of the intercooler (\(\eta_{\text{IC}}\)) is calculated as:  
\[\eta_{\text{IC}}=\frac{(T_{1}-T_{2})}{(T_{1}-T_{\text{2S}})}\times 100 \tag{32}\]  
where \(T_{2}\) is the temperature of the mixture at the intercooler outlet. The mechanical power of the turbine (\(MP_{\text{T}}\)) is calculated from Equation (33) using the temperature difference between the exhaust gases' inlet (\(T_{3}\)) and outlet (\(T_{4}\)) stream. The isentropic efficiency of the turbine (\(\eta_{\text{IS-T}}\)) is given by Equation (34) denoting the outlet isentropic temperature of turbine exhaust gases as (\(T_{\text{4S}}\)):  
\[MP_{\text{T}}=\frac{M_{\text{mix}}(1+F)}{3600}C_{\text{p-mix}}(T_{3}-T_{4}) \tag{33}\]  
\[\eta_{\text{is-T}}=\frac{(T_{3}-T_{4})}{(T_{3}-T_{\text{4S}})}\times 100 \tag{34}\]  
Equation (35) expresses the pressure relationship for the turbine under isentropic conditions, with \(p_{\text{4S}}\) denoting the pressure of the outlet exhaust gases' stream under isentropic conditions and \(p_{3}\) representing the pressure of the gas stream entering the turbine. The expansion of the exhaust gases in the turbine is assumed as an irreversible flow due to the adiabatic assumptions based on the small contact surface and the short residence time, considering also steady flow at the inlet and outlet zone. Compressors have three boundary zones, pumping, blocking and over speed zones whilst turbines have none. To achieve the necessary exhaust back-pressure, the turbine opposes to the flow of gases. For this reason, its effective passage area must be small enough, and this feature is called permeability. The turbine discharge pressure is obtained from summing up the reference pressure of the inlet gas mixture to the compressor (\(p_{0}\)) and the outlet back-pressure of exhaust gases (\(p_{\text{bp}}\)), see Equation (36).