*2.2. Cylinder*

For the MVEM, the actual intermittent gas flowing process through the scavenging ports and exhaust valve is simplified as a flowing process through an equivalent orifice with fixed area under sub-sonic flow consideration. Consequently, the cylinder inflowing air mass flow rate .*msz* can be calculated with the following equation [5,8,9,12,13]:

$$\dot{m}\_{sz} = C\_z A\_z \frac{p\_s}{\sqrt{R\_s T\_s}} \sqrt{\frac{2k\_s}{k\_s - 1} \left[ \left(\frac{p\_c}{p\_s}\right)^{\frac{2}{\gamma\_s}} - \left(\frac{p\_c}{p\_s}\right)^{\frac{\gamma\_s + 1}{\gamma\_s}} \right]} \tag{1}$$

where *Cz* and *Az* are the flow coefficient and the equivalent orifice area, respectively; *Rs*, *Ts*, *ps* and γ*s* are the gas constant, temperature, pressure and specific heat ratio of the air in the scavenging manifold, respectively; *pe* is the exhaust manifold pressure.

As the exhaust valve lifting curve is not provided by the engine manufacturer and the orifice flow coefficient is also unknown, therefore, the product of *Cz* and *Az*(*CzAz*), is treated as a calibration parameter in this paper. *CzAz* can be approximated as a linear function of the brake power, that is *CzAz* = *kCA*0 + *kCA*1*Pb*.

As similar with the engine air inflowing process, the MVEM also treats the fuel injection as a continuous process. According to the mass conservation law, the mass flow rate of exhaust gas exiting the cylinders .*mze* can be calculated by adding .*msz* and the fuel injection rate .*mf* .

The energy released by the fuel burning cannot be fully exploited by the engine and converted to the mechanical energy directly, therefore, a portion of the thermal energy still remains in the exhaust gas. Consequently, based on the energy conservation law, the thermal energy of the exhaust gas exiting the cylinders can be written as [8,12,13]:

$$
\dot{m}\_{\overline{z}c}h\_{\overline{z}c} = \dot{m}\_{\overline{s}\overline{z}}c\_{p,\overline{s}}T\_{\overline{s}} + \zeta \eta\_{\text{comb}}\dot{m}\_{f}H\_{\text{LHV}} \tag{2}
$$

where *hze* is the specific enthalpy of exhaust gas; η*comb* is the combustion efficiency, which is a function of the air-fuel ratio (*A*/*F* = .*msz*/ .*mf*); *cp*,*<sup>s</sup>* is the constant pressure specific heat of the scavenging air; *HLHV* is the fuel lower heating value; ζ is the fuel chemical energy proportion in the exhaust gas, which can be fitted as a linear function of the brake power, that is ζ = *k*ζ0 + *k*ζ1*Pb*.

The mean indicated effective pressure *pi* is fitted as a linear function of the fuel index. According to the engine shop trial report, the mean friction effective pressure *pf* is always 1 bar for all the tested loading conditions, therefore, a constant value of *pf* is assumed in this paper. Having obtained *pi* and *pf*, the mean brake effective pressure *pb*can be calculated as *pb*= *pi*− *pf*.

Finally, the engine brake torque *Qb*, brake power *Pb* and brake specific fuel consumption (BSFC) can be derived as per the following equations:

$$Q\_b = \frac{n\_z V\_d \overline{p}\_b}{2\pi}, P\_b = \frac{\pi N\_c Q\_b}{30}, \text{BSFC} = \frac{\dot{m}\_f}{P\_b} \tag{3}$$

where *Vd* is the engine displacement volume of a single cylinder.
