## 2 Methodologies
### Evaluation of Combustion Parameters  
Figure 2 shows an example of in-cylinder pressure with the relevant first derivative (dp/d\(\theta\)), which was calculated after proper filtering of the pressure signal. The actual timing of the three injections is also presented. Injection timings were kept constant at their standard levels in each operating point, as well as pilot and pre-injected quantities (Tables 3-5). As zero crank angle degree corresponded to the beginning of the intake stroke, combustion TDC occurred at 360 crank angle degrees. From the pressure trace, two maximum values are apparent, the first related to the compression stroke and to the combustion of pilot and pre-injections, the second to the main combustion. This behaviour, observed in all the tested conditions, is typical of automotive diesel engines since Euro 2/Euro 3 phases of the legislation on exhaust emissions, due to the shift of main injection timing towards or after the top dead centre in order to reduce NO\({}_{\mathrm{X}}\). For the statistical analysis, the second maximum level was always considered.  
Referring to the first derivative of pressure signal, both maximum values (dp/d\(\theta\))\({}_{\mathrm{MAX}\,1}\) and (dp/d\(\theta\))\({}_{\mathrm{MAX}\,2}\) were considered, together with a further level (dp/d\(\theta\))\({}_{\mathrm{MAX}}\), derived from the pressure derivative of the compression-expansion signal in the unfired engine, at the same intake conditions.  
A combustion noise indicator I\({}_{\mathrm{n}}\) was therefore evaluated, applying Equation (8) [30]:  
\[\mathrm{I}_{\mathrm{n}}=\mathrm{n}/\mathrm{n}_{\mathrm{idle}}\times[(\mathrm{ dp}/\mathrm{d}\theta)_{\mathrm{MAX}\,1}+(\mathrm{dp}/\mathrm{d}\theta)_{\mathrm{ MAX}\,2}]/(\mathrm{dp}/\mathrm{d}\theta)_{\mathrm{MAX}}. \tag{8}\]  
where n is the engine speed of the tested condition and n\({}_{\mathrm{idle}}\) is the engine speed in the idling mode.  
Figure 3 presents an example of the rate of heat release with two relevant parameters obtained from this curve, i.e., the maximum value of heat flux (RoHR\({}_{\mathrm{MAX}}\)) and the heat released during the premixed phase of the main combustion (HR\({}_{\mathrm{premix}}\)). The second quantity was estimated following the procedure shown in Reference [31], calculating the heat released from the start of the main combustion (corresponding to the lowest level of heat release following the combustion of the pilot and pre-injections) till to the crank angle at which RoHR\({}_{\mathrm{MAX}}\) occurred. Always referring to the premixed phase of the main combustion, another parameter was obtained, through the calculation of the first derivative of the rate of heat release and the evaluation of the relevant maximum value (dRoHR/d\(\theta\))\({}_{\mathrm{MAX}}\). This quantity allowed us to further assess how the combustion process developed in the first period, following the ignition of the fuel injected during the main event.  
Figure 2: Example of an in-cylinder pressure diagram, the curve of the pressure first derivative and related parameters.  
Figure 4 shows the mass fraction burned, obtained by the integration of heat release rate, and the crank angles corresponding to the release of 10%, 50% and 90% of the total heat. \(\theta_{50}\) represents the centre of combustion and is useful for its control [32]. The difference (\(\theta_{90}\)-\(\theta_{10}\)) is commonly assumed as an index of combustion duration, as the initial and final phases may be affected by significant uncertainties in the estimation of actual burning start and end angles.