## 2 Tools and Methods
### Theoretical Tools
#### 2.5.1 0D Thermodynamic Model  
The methodology [13] applied in the thermodynamic characterization was performed with CALMEC-SICICLO [15]. The main assumptions for the thermodynamic model are:  
* Chamber pressure and temperature are assumed to be spatially uniform.
* Three species are considered: air, fuel vapor, and stoichiometric combustion products.
* The chamber volume deformation due to pressure and inertia is calculated by means of a simple deformation model [20]. However, for this study, the chamber volume deformation was measured and imposed in the model.  
Figure 4: Scheme of the thermodynamic characterization model.  
* The specific heat of the gas depends on both the temperature and composition [21].
* Heat transfer is calculated with the modified Woschni-like model [22], where the convective heat transfer is calculated using Equation (1), simplified for a motored condition: \[h=CD^{-0.2}p^{0.8}T^{-0.53}[C_{W1}c_{m}+C_{W2}c_{u}]^{0.8}\] (1)
* Finally, a filling and emptying model is normally used to calculate the trapped mass at IVC (m\({}_{c}\)) in the original methodology, however, for this study, the trapped mass was calculated by a GT-Power model.  
where \(c_{m}\) is the mean piston speed; \(c_{u}\) is the instantaneous tangential velocity of the gas chamber adjusted using CFD calculations [15]; \(p\) and \(T\) are instantaneous pressure and temperature respectively; \(C\) is a constant whose value is 0.012; \(D\) is the cylinder diameter; and \(C_{W1}\) and \(C_{W2}\) are constants to be adjusted for each engine. Effective compression ratio (CR), TDC position, and \(C_{W1}\) must be adjusted simultaneously during the calculation.  
Even though the measured piston position trace was developed from \(-50\) aTDC to 50 aTDC for both TDC in the cycle, the instantaneous volume curve was calculated quantifying only the maximum deformation at firing TDC. This takes into account that the impact of mechanical deformation near the TDC is more significant in the volume change because the volume is the smallest and the deformation is the largest. It is a good approximation to assume that the deformation at the TDC can be scaled with the forces during the rest of the cycle. In the case of this study, these forces come from in-cylinder pressure.  
Thus, the instantaneous volume in the model is calculated by Equation (2). It is the sum of the combustion chamber volume (\(V_{CC}\)), the volume displaced by the piston (\(V_{d,int}\)) (function of crank angle), and the volume changes caused by the mechanical deformations (\(\Delta V_{def}\)). The first term is a function of CR and \(V_{d,inst}\) is calculated taking into account the engine geometry.  
\[V=V_{CC}+V_{d,inst}+\Delta V_{def} \tag{2}\]  
Mechanical deformation is calculated by means of Equation (3). The constant \(K_{def}\) is adjusted in order for the model to obtain the same volume deformation at TDC that was measured in the experiments. \(\Delta V_{p}\) is the volume variation due to in-cylinder pressure for each crank angle degree.  
\[\Delta V_{def}=K_{def}\big{[}\Delta V_{p}\big{]} \tag{3}\]  
Trapped mass, as shown in Equation (4), is considered as the sum of fresh air mass (\(m_{a}\)), residual mass (\(m_{res}\)) from the previous cycle, and the short circuit mass (\(m_{SC}\)) that occurs during the valve overlap event. As above-mentioned, this part will be input data on the model and will be estimated by GT-Power.  
\[m_{c}=m_{a}+m_{res}+m_{SC} \tag{4}\]