# Engine Optimization Model for Accurate Prediction of Friction Model in Marine Dual-Fuel Engine
## 3 Engine Optimization Model
### Description of WAVE Model  
Firstly, the dual-fuel engine model is constructed in WAVE using the various elements available in the software, which combines flow, mechanical, and control components, as shown in Figure 2. The engine model features two turbochargers, one intercooler, twelve cylinders, and intake and exhaust manifolds. Similar to modern diesel engines, each cylinder is equipped with four valves--two for intake, two for exhaust--along with one injector in the liquid fuel mode, as the main focus of this study. The dimension of each part is collected from the project guide of the manufacturer [55; 56]. The compressor and turbine maps for each turbocharger, as shown in Figure 3, are defined and rescaled from existing maps provided by the software to fit the engine's needs for airflow at a given pressure. The \"filling and emptying\" method is applied to control turbocharger performance and behavior using the energy and momentum equations recommended by [PERSON] and [PERSON] [57]. This approach treats each manifold or cylinder as a separate thermodynamic control volume, with its dynamics governed by energy and continuity equations as functions of time or crank angle. Regarding the fuel injection system, the key parameters, including injection timing, profile, pressure, duration, and the injection rate within the cylinder, are considered during the simulation. The rectangular injection rate profile is selected as it is the common rate profile used due to its benefits in achieving a more stable combustion process as well as emissions reduction [58]. Cylinder processes are computed using the first law of thermodynamics [8], incorporating the heat release rate (HRR) during combustion [59; 60], and heat transfer based on the empirical formula by [PERSON] [61]. The total heat added during the combustion process, dQ, as described in Equation (5), is computed for each crank angle. This calculation includes the heat released from fuel combustion, Qcomb, and the heat exchange, dQw, with the cylinder walls:  
\[\mathrm{dQ}=\mathrm{Q_{comb}}\mathrm{d}\mathrm{x_{b}}-\mathrm{dQ_{w}} \tag{5}\]  
\[\mathrm{Q_{comb}}=\mathrm{m_{fuel}}\mathrm{C}\mathrm{V} \tag{6}\]  
\[\mathrm{dQ_{w}}=\frac{\mathrm{hA_{w}}}{\omega}(\mathrm{T_{g}}-\mathrm{T_{w}}) \mathrm{d\theta} \tag{7}\]  
where \(\mathrm{m_{fuel}}\) is the amount of injected fuel, CV is the calorific value of the injected fuel, h is the heat transfer coefficient suggested by [PERSON], \(\mathrm{A_{w}}\) is the area of cylinder walls, \(\omega\) is the engine speed, \(\mathrm{T_{g}}\) is the burn gas temperature, and \(\mathrm{T_{w}}\) is the cylinder wall temperature. Figure 2: Layout of the 1D engine model implemented in WAVE. The burned mass fraction, \(\rm{x_{b}}\), is computed using the equation suggested by [PERSON] et al. (1992), as expressed in Equation (8). This formula has been improved by the Realis Simulation team to consider the three phases of combustion. It is well-suited for combustion in the diesel cycle and does not require calibration, except for the reference speed (BRPM), as in Equation (9), to control the burn duration, unlike the single- and multi-Wiebe functions ([PERSON], 2016). In some cases, the rated speed can be used as the reference speed ([PERSON], 2016; [PERSON], 2016). The double-Wiebe function is introduced as an alternative to the single-Wiebe function to capture different phases of the combustion process better. \[\rm{x_{b}}\!=\!\rm{p_{f}}\!\left\{\!1\!-\!\left[1-(0.75\tau)^{2}\right]^{5000} \right\}\!+\!\rm{d_{f}}\!\left\{\!1\!-\!\left[1-(\rm{cd_{3}}\tau)^{1.75}\right] ^{5000}\right\}\!+\!\rm{t_{f}}\!\left\{\!1\!-\!\left[1-(\rm{ct_{3}}\tau)^{2.5} \right]^{5000}\right\} \tag{8}\]  
where \(\rm{p_{f}}\), \(\rm{d_{f}}\), and \(\rm{t_{f}}\) are the mass fractions of the premix, diffusion, and tail burn curves, respectively.