## 2 Experimental Investigation
### Test Bench Data  
The experimental tests were performed in two ways. For each test, the AFR engine control unit (ECU) was disabled, and the AFR was superimposed by means of injection quantity at a fixed air mass flow. The experimental tests in SS conditions (15 engine operative points) were carried out with \(\lambda\) sweep from 0.90 to 1.10. However, the real value of AFR can be slightly different from the value derived from ECU, taking into account the variable measurement dynamics of the chemical species analyzers and the behavior of the installed \(\lambda\) sensors. Several analyses have been carried out in order to identify a reference \(\lambda\) value to describe discrepancies in exhaust gas composition. In fact, the engine was equipped with two types of sensors that provided AFR measurements. The first is the Smart NOx Sensor (SNS) 120 by ContinentalTM, consisting of a ceramic sensor element made of zirconia (ZrO2) and an electronic control unit. The second is the Universal Lambda Sensor (ZFAS-U) by NGKTM, made of two zirconia (ZrO2) substrate elements, one is the O2 pumping cell (Ip cell), the other is the O2 detecting cell (Vs cell) heated by a ceramic heater, which is supplied by a very small current. The SNS sensor has a very rapid response to composition changes and is very precise at stoichiometric conditions, but it presents major uncertainties when the exhaust gas is in rich or lean conditions because of the linear correlation between oxygen concentration of residual gas and AFR. On the other hand, for the ZFAS-U sensor, very precise at stoichiometric conditions, even a slight variation in the exhaust gas composition typical of NG engines, can considerably affect the sensed AFR value, which leads to oscillating values. For these reasons, it is not reliable in transient operations but remains suitable for SS conditions. \(\lambda\) values can also be calculated starting from the classic ratio between air and fuel flow intake rate and fuel and stoichiometric dosage. However, since the main constituents of the CNG mixture do not have a specific concentration throughout the duration of the test, but is included within a range of appropriate values, the value of \(\alpha_{st}\) is changeable. Moreover, especially in low-load conditions, deviations and measurement uncertainties of air and fuel flow increase and interfere with the result of the ratio. Therefore, the most reliable method to calculate \(\lambda\) for SS conditions was identified as an analytical relation present in regulation 49 [14] from test best bench data:  
\[\lambda_{i}=\frac{\left(100-\frac{c_{\text{CO}_{d}}\cdot 10^{-4}}{2}-c_{ HC_{w}}\cdot 10^{-4}\right)+\left(\frac{4}{4}\cdot\frac{1-\frac{2\times c_{ \text{CO}_{d}}\cdot 10^{-4}}{35\times c_{\text{CO}_{d}}}}-\frac{\kappa}{2}-\frac{ \kappa}{2}\right)\star\left(c_{\text{CO}_{2d}}+c_{\text{CO}_{d}}\cdot 10^{-4} \right)}{4.764\star\left(1+\frac{\alpha}{4}-\frac{\kappa}{2}+\gamma\right) \star\left(c_{\text{CO}_{2d}}+c_{\text{CO}_{d}}\cdot 10^{-4}+c_{\text{HC}_{w}}\cdot 10^{-4} \right)}}, \tag{1}\]  
where:  
\(\lambda_{i}\) is the instantaneous \"relative air-to-fuel ratio\";  
\(c_{\text{CO}_{2d}}\) is the dry CO2 concentration, in percentage;  
\(c_{\text{CO}_{d}}\) is the dry CO concentration, ppm;  
\(c_{\text{HC}_{w}}\) is the wet HC concentration, ppm;  
\(\alpha\) is the molar hydrogen ratio (H/C);  
\(\beta\) is the molar carbon ratio (C/C);  
\(\gamma\) is the molar sulfur ratio (S/C);  
\(\delta\) is the molar nitrogen ratio (N/C); and  
\(\varepsilon\) is the molar oxygen ratio (O/C);  
The usage of Equation (1) for the \(\lambda\) calculation allows to disregard the use of sensors and their mentioned inaccuracies, as this calculation is directly based on the species' concentrations actually measured in the engine exhaust. Not surprisingly, despite the assumptions that will be fully described in the next paragraph, this value is completely in line with what is calculated by the model.