*2.3. Turbocharger*

The power absorbed by the compressor can be written as:

$$P\_{\mathcal{L}} = \dot{m}\_{\mathcal{L}} c\_{p,a} (T\_{\mathcal{c},out} - T\_{\mathcal{c},in}) \tag{4}$$

where .*mc* is the compressor mass flow rate; *cp*,*<sup>a</sup>* is the air constant pressure specific heat; *Tc*,*in* and *Tc*,*out* are the temperature of air entering and exiting the compressor, respectively.

The compressor mass flow rate depends on not only the turbocharger rotational speed but also the pressure ratio across it with the latter can be written as:

$$
\Pi\_c = \frac{p\_s + \Delta p\_{ac} - \Delta p\_{bl}}{p\_{amb} - \Delta p\_{af}} \tag{5}
$$

where *ps*, Δ*pac*, Δ*pbl*, *pamb* and Δ*pa f* are the scavenging manifold pressure, air cooler pressure drop, auxiliary blower pressure increase, ambient pressure and compressor air filter pressure drop, respectively. In this paper, Δ*pa f* is fitted as a second-order polynomial of .*mc*. Modeling approaches for Δ*pac* and Δ*pbl* will be introduced in Section 2.5.

The temperature of air exiting the compressor can be calculated by the following equation, which is derived based on the definition of compressor isentropic efficiency:

\_

$$T\_{\mathfrak{c\\_out}} = T\_{\mathfrak{c\\_in}} (1 + (\Pi\_{\mathfrak{c\\_in}}^{(\gamma\_a - 1)/\gamma\_a} - 1) / \eta\_{\mathfrak{c\\_in}}) \tag{6}$$

where γ*a* is the air specific heat ratio; η*c* is the isentropic efficiency.

The power generated by the turbine can be written as:

$$P\_t = \dot{m}\_t c\_{p, \mathcal{E}} (T\_{t, \dot{m}} - T\_{t, \text{out}}) \tag{7}$$

where .*mt* is the turbine mass flow rate; *cp*,*<sup>e</sup>* is the exhaust gas constant pressure specific heat; *Tt*,*in* and *Tt*,*out* are the temperature of exhaust gas entering and exiting the turbine.

In this paper, the turbine is simplified as a nozzle and the exhaust gas mass flow rate flowing through it can be calculated based on the assumption of one-dimensional isentropic adiabatic flow with the input data including the gas thermodynamic properties in the exhaust manifold, the pressure ratio across the turbine as well as the turbine equivalent flow area and flow coefficient.

The pressure ratio Π*t* is calculated by the following equation:

$$
\Pi\_t = \frac{p\_{amb} + p\_{t,back}}{p\_c} \tag{8}
$$

where *pt*,*back* is the turbine back-pressure and it is fitted as an exponential function of the turbine mass flow rate in this paper.

The turbine equivalent flow area *At* can be computed as per the following equation:

$$A\_{l} = \sqrt{\frac{\overline{\left(A\_{D} \cdot A\_{S}\right)^{2}}}{A\_{D}^{2} + A\_{S}^{2}}} \tag{9}$$

where *AD* and *AS* are the flow area of the turbine impeller and nozzle ring, respectively.

For the turbocharger turbine investigated in this paper, its flow coefficient *Ct* and isentropic efficiency η*t* only depend on the expansion ratio Γ*t* (Γ*t* = 1/Π*t*) and are not influenced by the turbocharger rotational speed. Therefore, *Ct* and η*t* are modeled by using look-up table in this paper based on the turbine performance map.

The temperature of exhaust gas exiting the turbine can be calculated based on the definition of turbine isentropic efficiency as the following equation:

$$T\_{t,out} = T\_{t,in} \left( 1 - \eta\_t \left( 1 - \Pi\_t^{\left( \gamma\_t - 1 \right) / \gamma\_t} \right) \right) \tag{10}$$

where γ*e* is the specific heat ratio of exhaust gas.

Finally, the turbocharger shaft angular speed ω*tc* can be calculated by integrating the following equation, which is derived based on the angular momentum conservation law:

$$\frac{d\omega\_{\rm tc}}{dt} = \frac{1}{f\_{\rm tc}} (\frac{P\_{\rm t}}{\omega\_{\rm tc}} \eta\_{\rm tc,m} - \frac{P\_{\rm c}}{\omega\_{\rm tc}}) \tag{11}$$

where *Jtc* is the moment of inertia of the turbocharger rotating part; η*tc*,*<sup>m</sup>* is the turbocharger mechanical efficiency.

### *2.4. Scavenging and Exhaust Manifolds*

The scavenging and exhaust manifolds are treated as control volumes in the MVEM. By applying the mass conservation law, the mass changing rate in the manifold can be computed as:

$$\frac{dm}{dt} = \dot{m}\_{\text{in}} - \dot{m}\_{\text{out}} \tag{12}$$

where *k*ζ0 and .*mout* are the mass flow rate entering and exiting the manifold, respectively.

The temperature changing rate in the manifold can be derived by applying the energy conservation law. The differential equation governing the temperature changing rate in the manifold can be written as:

$$\frac{dT}{dt} = \frac{\dot{m}\_{\text{in}}\mathbf{c}\_{\upsilon}(T\_{\text{in}} - T) + R(T\_{\text{in}}\dot{m}\_{\text{in}} - T\dot{m}\_{\text{out}}) + \dot{Q}\_{\text{ht}}}{m\mathbf{c}\_{\upsilon}}\tag{13}$$

where *Tin* is the gas temperature entering the manifold; *Qht* is the heat dissipation; *cv* is the gas constant volume specific heat.

.

Due to the negligible temperature difference between the scavenging air and the surrounding, the heat dissipation in the scavenging manifold is neglected, whereas the heat dissipation in the exhaust manifold must be taken into account because the exhaust gas temperature is much higher than the surrounding. . *Qht* is computed as it was done in Theotokatos and Tzelepis by using the overall heat transfer coefficient and heat transfer area [12].

Having obtained the stored mass and gas temperature by integrating Equations (12) and (13), the gas pressure in the manifold can be derived by using the ideal gas state equation.

### *2.5. Air cooler, Auxiliary Blower and Wastegate*

The air temperature exiting the air cooler *Tac*,*out* can be written as:

$$T\_{\rm ac,out} = T\_{\rm ac,in} - \eta\_{\rm ac} (T\_{\rm ac,in} - T\_{\rm cw}) \tag{14}$$

where *Tac*,*in* is the air temperature entering the air cooler; η*ac* is the cooling efficiency; *Tcw* is the cooling water temperature, which is assumed to be constant and equals to 300 K in this paper.

In this paper, the air cooler cooling efficiency and the pressure drop is fitted as a second-order polynomial of the air mass flow rate flowing through it.

Auxiliary blower of the centrifugal type, which is driven by an induction motor running at fixed rotational speed, is commonly adopted for marine large scale two-stroke diesel engines. Consequently, the auxiliary blower can be modeled as a centrifugal compressor that runs at fixed rotational speed. Following this idea, the pressure increase across the blower Δ*pbl* and blower efficiency η*bl* thus only depend on the air volume flow rate *Vbl*. Therefore, Δ*pbl* and η*bl* are fitted as a second and third order polynomial of . *Vbl*, respectively, based on the auxiliary blower performance map in this paper. Having obtained Δ*pbl* and η*bl*, the air temperature exiting the blower can be computed by using Equation (6), which is originally used to calculate the air temperature existing the compressor.

.

To prevent the compressor from entering into the unstable surging area, many marine large scale two-stroke diesel engines are equipped with a wastegate in recent years, which is a bypass configuration in parallel with the turbine. In this paper, the wastegate is simplified as an ideal nozzle and the gas mass flow rate is calculated based on the assumption of one-dimensional isentropic adiabatic flow. It should be noted that the wastegate shares the same upstream and downstream condition with the turbine.
