## 6 Conclusions  
This study used the CFD method and the GRI-Mech 3.0 mechanism to numerically investigate the combustion and heat transfer characteristics of PMIFJs, which were verified through experimental results. The orthogonal array was used to generate initial sample points for \(\phi\), \(Re\), and \(H/d\) combinations. The KM and GA were used to create a model to predict unknown points and determine the optimized solution to the problem. Maximizing RMSE was used as an infilled criterion to improve the prediction accuracy of the KM. In addition, the effects of three operating parameters on PMIFJs are further analyzed. The following conclusions can be drawn from the results:  
1. The KM with good prediction ability is obtained through twenty sample points (sixteen initial points and four infilled points). Compared with six checking points more sensitive to \(\overline{Nu}\), the maximum relative errors are all within 1%. Moreover, this method reduces the simulated times of CFD by 75.3%. 2. From the response surface plots, it is known that \(\overline{Nu}\) of the PMIFJ shows an upward trend with the increase in \(\phi\), the increase in \(Re\), and the decrease in \(H/d\). Moreover, the effect of \(\phi\) is more significant with increasing \(Re\). 3. The sensitivity analysis points out that the ranking of operating parameters affecting global heat transfer performance is \(Re>\phi\gg H/d\), which means that the inlet velocity is the main key parameter, followed by the fuel-to-air ratio. 4. It is found that the height of the premixed cone and the scope of the high temperature post-flame region will significantly affect the behavior of local heat transfer. 5. The parameter combination to determine the maximum global heat transfer performance through GA is \(\phi=1.2\), \(Re=1200\), and \(H/d=1.5\), which means that the PMIFJ is at slightly rich-fuel and high-velocity conditions and its premixed cone slightly hits the impingement plate. Figure 16: The contours of 2D OH radical of the PMIFJ at \(Re=\)1200 and \(H/d=2.0\) against different \(\phi=\) (**a**) 0.9, (**b**) 1.0, (**c**) 1.1, and (**d**) 1.2. The main contribution of this paper is to provide an efficient prediction and optimization method framework for the heat transfer performance in the field of impinging flame jets. Although the proposed method applies to the heat transfer performance of premixed impinging flame jets, similar strategies can be effectively applied to combustion systems such as combustors, boilers, and turbine engines to systematically analyze the effects of each design variable (operating parameter) on thermal efficiency. In addition, this method is not limited to heat transfer performance but can also be used to predict and optimize pollutant emissions, such as NOx, CO, and particulate matter (PM). Currently, only a single objective function is modeled and optimized. The multiple objectives optimization of combustion systems is becoming increasingly crucial as combustion technology moves toward becoming more efficient and cleaner. However, multiple objectives are usually relative and contradictory. For example, enhancing the efficiency of the combustion system will raise NOx emissions due to the increase in thermal NO. Therefore, the multiple-objective optimization can be transformed into a single-objective optimization problem through the weighting function, or the multi-objective non-dominated sorting genetic algorithm (NSGA-II) can be introduced to determine the optimal solution that maximizes the balance of each objective. Conceptualization, X.-X.C. ; Methodology, R.-B.C. and C.-Y.W. ; Software, X.-X.C. ; Validation, X.-X.C., R.-B.C. and C.-Y.W. ; Formal Analysis, X.-X.C. ; Investigation, X.-X.C. ; Resources, R.-B.C. and C.-Y.W. ; Data Curation, X.-X.C. ; Writing--Original Draft Preparation, X.-X.C. ; Writing--Review and Editing, R.-B.C. and C.-Y.W. ; Visualization, X.-X.C. ; Supervision, C.-Y.W. ; Project Administration, R.-B.C. and C.-Y.W. Funding Acquisition, C.-Y.W.