*4.1. Cylinder Pressure Analytic Model*

### (1) Compression Phase (EVC to SOC)

The actual compression process can be approximated by a polytropic process with a polytropic index of *ncomp*. Based on this assumption, the instantaneous in-cylinder pressure *pcomp* and temperature *Tcomp* during this phase can be calculated as:

$$p\_{\rm comp}(\theta) = p\_{\rm rev}(\frac{V\_{\rm env}}{V(\theta)})^{n\_{\rm comp}} \tag{27}$$

$$T\_{\rm comp}(\theta) = T\_{\rm rev}(\frac{V\_{\rm env}}{V(\theta)})^{n\_{\rm comp} - 1} \tag{28}$$

where *pevc*, *Vevc* and *Tevc* are the pressure, volume and temperature at EVC, respectively, and the EVC is treated as the reference point of the compression polytropic process.

As there is no mass exchange between the cylinder and surrounding during this phase, the in-cylinder trapped mass *mtrap* can be calculated by using ideal gas state equation.

(2) Expansion Phase (EOC to EVO)

The expansion phase is also treated as a polytropic process. The calculation method of instantaneous in-cylinder pressure *p*exp and temperature *<sup>T</sup>*exp during this phase is also as similar with the compression phase. The main difference lies in the determination of polytropic index and the pressure and temperature of the reference point.

The temperature at the reference point of expansion phase can be estimated by adding an additional temperature increment based on *Tcomp*(0◦):

$$
\Delta T = \frac{m\_{f, \text{cycle}} H\_{\text{LHV} \parallel \text{comb}}}{c\_v (T\_{\text{comp}} (0^\circ)\_\prime \phi\_{\text{trap}}) (m\_{\text{trap}} + m\_{f, \text{cycle}})} \tag{29}
$$

$$T\_{\text{exp,ref}} = T\_{\text{comp}}(0^\circ) + \Delta T \tag{30}$$

The theoretical foundation of Equations (29) and (30) is the constant volume heating process in the ideal Otto cycle. Once *<sup>T</sup>*exp,*re f* is determined, *p*exp,*re f* can be calculated by using ideal gas state equation. (3)CombustionPhase

During the combustion phase, the in-cylinder instantaneous pressure *pcomb* can be calculated by interpolating between *pcomb* and *p*exp with the Wiebe function *fwiebe* selected as the interpolating function:

$$p\_{\rm comb} = (1 - f\_{\rm nucbe}) \cdot p\_{\rm comp} + f\_{\rm uvide} \cdot p\_{\rm exp} \tag{31}$$

### (4) Blow-down Phase (EVO to SPO)

As can be observed from Figure 14, the instantaneous in-cylinder pressure *pexh* during this phase presents linear variation trend roughly, which, thus, can be approximated by a straight line crossing the points of (<sup>θ</sup>*evo*, *pevo*) and (<sup>θ</sup>*spo*, *pspo*) as the following equation:

$$p\_{exh} = p\_{cro} + \frac{p\_{spo} - p\_{cro}}{\theta\_{spo} - \theta\_{cro}} (\theta - \theta\_{cro}) \tag{32}$$

### (5) Scavenging and Post Exhaust Phase (SPO to EVC)

As can be observed from Figure 14, the instantaneous in-cylinder *pscav*\_*pe* during this phase does not fluctuate significantly, which is always between the scavenging and exhaust manifold pressure. Consequently, *pscav*\_*pe* can be approximated by a straight line with a constant pressure value.
