# Engine Optimization Model for Accurate Prediction of Friction Model in Marine Dual-Fuel Engine
## 4 Results and Discussions
### Computation of the Terms of the Friction Model  
Following the optimization and validation of the engine performance parameters across various load conditions, the next step involves refining the empirical friction model. This is achieved by fitting the optimized simulation results to the Chen-Flynn correlation to ensure accurate and scalable FMEP prediction. To determine the coefficients of the Chen-Flynn correlation, the FMEP and the Pmax at each engine speed are incorporated into the friction tool developed by the Realis Simulation team. Using the standard correlation coefficients presented in Table 4 as a baseline, the  
Figure 6: (a) In-cylinder pressure diagram and (b) fuel burn rate for four different engine operating points. fitting process is executed with the friction tool to compute the percentage error between the calculated and predicted FMEP values for each engine speed. Initially, the percentage error remains within 7.2% across all speeds. However, to ensure high accuracy for subsequent simulations, it is necessary to minimize these errors further. To refine the fitting process, the second and fourth terms of the Chen-Flynn correlation are retained as per the original equation, with fixed values of 0.006 and 0.2, respectively, while the first and third terms are computed based on the data defined in the Excel tool. Bcf and Qcf are fixed to the standard literature values (0.006 and 0.2, respectively) due to their relatively low sensitivity to engine-specific variations and the absence of reliable experimental data to refine them. These terms predominantly capture baseline friction behavior associated with pressure and velocity effects that are consistent across engine types. In contrast, Acf and Ccf are treated as variable terms and optimized using the friction fitting tool, as they are more influenced by the specific design and operational characteristics of the engine (e.g., surface finish, lubrication, piston speed). This approach strikes a balance between modeling accuracy and practical constraints, ensuring that the refined coefficients accurately reflect the real behavior of the studied engine while maintaining computational efficiency. Adjustments are made by replacing the computed FMEP values with the predicted ones from the fitting model at engine speeds through multiple iterations. This iterative refinement is repeated across the speed range, starting with the points with a higher percentage of errors, until achieving a stable and improved R2 between the calculated and predicted FMEP curves. This method ensures enhanced accuracy and reliability of the correlation for future applications. Figure 7 illustrates the strong correlation between the newly computed values and the predicted FMEP values, demonstrating the effectiveness of the fitting process. The final R2 value between the two curves is 0.9999, indicating a nearly perfect match. The percentage error across the engine speeds ranges from 0.13% to \(-\)0.13%, reflecting a significant reduction in discrepancies and ensuring high accuracy of the model across varying operating conditions. Table 4 shows the new values of each term compared to the standard values. Comparable studies using empirical fitting techniques and numerical optimization also report R2 values above 0.98 for friction prediction models [71; 72], which supports the validity of our results given the controlled simulation environment and absence of experimental noise. After incorporating the updated coefficient values into the Chen-Flynn correlation, minor deviations, such as those presented in the computed results in Table 3, are to be expected. These discrepancies arise due to the sensitivity of the correlation to changes in its parameters, especially when applied to complex thermodynamic systems. Nevertheless, the overall trends remain consistent, and the observed variations fall within acceptable error margins for engineering applications. Additionally, the performance of the engine at various operating conditions along the engine load diagram will be calculated.