## 5 Case Study
### Genetic Algorithm Optimization Analysis  
The optimization process was conducted on a PC equipped with an Intel Core i7-13700 CPU (16 cores, up to 5.20 GHz) and 16 GB of RAM. Using the mesh with approximately 1,231,434 cells as a reference, the physical simulation time for each case was about 31,800 s, with a fixed time step of 600 s. Depending on the configuration and convergence behavior, the total wall-clock computation time typically ranged from 3.5 to 4.5 h per simulation. A second-order implicit time integration scheme was used throughout, with a maximum of 200 inner iterations per time step. Convergence within each step was determined by monitoring residuals for continuity, momentum, and species transport equations, all of which were required to fall below 1 \(\times\) 10\({}^{-4}\). In addition to residuals, we also tracked the evolution of antioxidant concentration at key monitoring points to ensure the physical reliability of the solution. After six generations of GA evolution, the optimization yielded the following results:  
As illustrated in Figure 5,the optimization results show that the best fitness value rapidly decreases in early generations, from 33,609 s in generation 0 to 31,800 s in generation 3, and then stabilizes. This indicates effective convergence due to selection and crossover  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Flow Rate** & **Pressure P** & **Speed n** & **Efficiency** & **Installation** & **Motor** \\  & **Q (m\({}^{3}\)/h)** & **(MPa)** & **(rpm)** & **\(\eta\) (\%)** & **Height (m)** & **Power** \\ \hline
65 YHC3-25 & 25 & 0.6 & 950 & 65 & 5 & 5.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of 65 YHC3-25 diesel pump.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Property**} & **Specification** \\ \hline \multicolumn{1}{c}{Acidity (as KOH)/(mg/100 mL)} & \(\leq\)7 \\ \multicolumn{1}{c}{Kinematic viscosity (20 \({}^{\circ}\)C)/(mm\({}^{2}\)/s)} & 3.0–8.0 \\ \multicolumn{1}{c}{Cetane number} & \(\geq\)45 \\ \multicolumn{1}{c}{Oxidation stability (insolubles)/(mg/100 mL)} & \(\leq\)2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Property specifications.  
operations, concentrating the population near optimal regions. The average fitness also improves from 39,937 s to 36,840 s, reflecting overall population quality enhancement.  
As illustrated in Figure 6,early generations (e.g., generation 0) exhibited broad diversity in inlet velocity values, ranging from approximately \(-0.01\) to \(0.015\). As evolution proceeds, this range narrows significantly. By generation 6, most values cluster around zero, suggesting convergence and reduced diversity. Velocity distributions become increasingly focused over time. In the early stages, individuals exhibit varied combinations with a wide spread in fitness. In later stages, high-performing individuals tend to cluster around specific velocity ranges, indicating that the GA has identified effective combinations for optimal performance.  
Figure 5: Evolution of best and average fitness values across generations.  
Figure 6: Diversity analysis of inlet parameters.  
A similar trend is observed in the conditioning state parameters. Initially (generations 0-1), the combinations are scattered. By generation 5 and 6, the population converges on a narrow range of effective configurations, showing that the algorithm favors specific conditioning state patterns linked to higher performance. Less effective combinations are gradually eliminated through evolutionary operations.