## 2 Materials and Methods
### CFD-FVM Model (Step 2.1)
Only the full knowledge of all simulation parameters would theoretically result in a perfectly-tuned numerical model. In any other case, there is a need to align the model output(s) with that of the experiments by tuning the unknown parameters. Consequently, a set of unknown parameters has been chosen to act as independent variables to be inquired in their effect on the temperature \(T_{1}\), \(T_{2}\), \(T_{3}\) (controlled outputs) and, hence, to be used as tuning parameters for the CFD-FVM model. For all other parameters, the default values proposed by software were kept as valid. Hence, tuning parameters were chosen both according to a literature review and starting from some considerations on the experimental setup used in the first phase (Table 4):  
* Radiation and convection (natural or forced) coefficients affect the heat exchange inside and outside the exhaust ducts, whereas material conductivity is responsible for heat conduction through the solid components [42; 46];
* The exhaust gas mass flow and the inlet gas temperature influence the amount of discharge thermal power, the fluid dynamics, and the boundary layers inside the pipes influence the heat transfer [43; 45; 49];
* Thermal contact resistance might affect the heat exchange between different components and between the thermocouples and the metal surfaces where they lay. Once these parameters have been individuated, viable values (reference, upper, lower) were found for each of them, obtaining a set of values used to generate a first simulation plan, i.e., as a starting point for the CFD-FVM model tuning procedure:  
* The reference value for the EG mass flow rate \(\dot{m}_{ex}\) was calculated, as a first approximation, using the following formula (\(\lambda=AFR/AFR_{s}\) mean air-fuel equivalence ratio, measured experimentally in the operative point P\({}_{0}\); \(AFR_{s}\) air-fuel ratio under stoichiometric conditions, equal to 14.5 as indicated in [60] for diesel oil; \(\dot{m}_{fuel}\) average fuel consumption in kg\(\cdot\)s\({}^{-1}\), measured experimentally in the operative point P\({}_{0}\)): \[\dot{m}_{ex}=(\lambda\cdot AFR_{s}+1)\cdot\dot{m}_{fuel}\] (3) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Quantity** & **Abbreviation** & **Units** & **Ref. Values** & **Lower Values** & **Upper Values** \\ \hline Inlet exhaust gas mass flow rate & \(\dot{m}_{ex}\) & kg\(\cdot\)s\({}^{-1}\) & 0.004 & 0.002 & 0.008 \\ Inlet exhaust gas temperature & \(T\) & K & 750 & 600 & 900 \\ Superficial convective exchange & \(H_{h}\) & W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) & 10 & 5 & 50 \\ coefficient (engine head) & \(H_{ex}\) & W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) & 10 & 5 & 50 \\ Superficial convective exchange & \(H_{ex}\) & - & 0.65 & 0.30 & 1.00 \\ Engine head emissivity & \(e_{h}\) & - & 0.65 & 0.30 & 1.00 \\ Exhaust duct surface emissivity & \(e_{ex}\) & - & 0.65 & 0.30 & 1.00 \\ Head material conductivity & \(C_{h}\) & W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\) & 15 & 5 & 100 \\ Contact thermal resistance & \(C_{r}\) & m\({}^{2}\cdot\)K\(\cdot\)W\({}^{-1}\) & 0.00058 & 0.00010 & 0.00120 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reference, upper and lower values for the 8 parameters used in the CFD-FVM model tuning procedure. The lower and upper values were calculated, respectively, by dividing and multiplying by two the reference value so as to have a wide range of investigation having the same magnitude order of the reference value;
* The reference value for the EGT at the exhaust pipe inlet (i.e., the approximate temperature immediately after the exhaust valve), namely \(T\), has been set at 750 K, a realistic value considering that the gas temperature at the end of the adiabatic expansion phase is typically between 550 and 650 \({}^{\circ}\)C (823.15-923.15 K) and the temperature of the EGT at the pipe outlet can be 220-250 \({}^{\circ}\)C (593.15-623.15 K).