## 3 Optimal OBC Cooling Plate Design and Results
### Optimization and Numerical Validation  
To establish the objective functions for optimization, the expected correlation equations of the response values and variables were derived using the regression analysis method; these functions have been listed in Table 5. For our regression analysis, the stepwise regression analysis method was used [30]. We performed ANOVA analysis of valid variables for stepwise regression analysis of the six \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**Fluid Flow Volume Distribution [Ratio]**} & \multirow{2}{*}{**SD**} & \multicolumn{2}{c}{**Distance Error**} & \multirow{2}{*}{**Product**} \\  & & & & & & & **Function Value** & & **Function Value** \\ \cline{2-9}  & \(R_{1}^{\prime}\) & \(R_{2}^{\prime}\) & \(R_{3}^{\prime}\) & \(R_{4}^{\prime}\) & \(R_{5}^{\prime}\) & \(R_{6}^{\prime}\) & \(\sigma\) & **E(R)\({}_{dist}\)** & **E(R)\({}_{mod}\)** \\ \hline
1 & 0.494 & 1.139 & 1.100 & 1.077 & 1.083 & 1.107 & 0.249 & 0.556 & 0.201 \\
2 & 0.343 & 1.077 & 1.081 & 1.105 & 1.166 & 1.228 & 0.327 & 0.731 & 0.368 \\
3 & 1.840 & 0.716 & 0.924 & 0.862 & 0.831 & 0.826 & 0.417 & 0.933 & 0.28 \\
4 & 1.829 & 0.619 & 0.904 & 0.875 & 0.877 & 0.895 & 0.420 & 0.940 & 0.297 \\
5 & 0.093 & 0.881 & 1.062 & 1.222 & 1.336 & 1.405 & 0.483 & 1.080 & 0.800 \\
6 & 0.077 & 0.729 & 0.989 & 1.239 & 1.425 & 1.540 & 0.539 & 1.206 & 0.849 \\
7 & 1.669 & 0.499 & 0.888 & 0.949 & 0.987 & 1.008 & 0.378 & 0.845 & 0.302 \\
8 & 1.653 & 0.330 & 0.844 & 0.978 & 1.070 & 1.124 & 0.429 & 0.959 & 0.458 \\
9 & 1.341 & 0.894 & 0.980 & 0.942 & 0.918 & 0.924 & 0.170 & 0.379 & 0.061 \\
10 & 1.214 & 0.728 & 0.934 & 0.989 & 1.037 & 1.098 & 0.164 & 0.368 & 0.07 \\
11 & 0.668 & 0.524 & 0.262 & 1.154 & 1.574 & 1.819 & 0.617 & 1.380 & 0.697 \\
12 & 1.872 & 0.37 & 0.833 & 0.924 & 0.976 & 1.025 & 0.488 & 1.092 & 0.467 \\
13 & 1.594 & 1.047 & 1.003 & 0.848 & 0.765 & 0.744 & 0.316 & 0.706 & 0.192 \\
14 & 1.079 & 0.633 & 0.937 & 1.052 & 1.125 & 1.173 & 0.196 & 0.439 & 0.112 \\
15 & 1.247 & 0.774 & 0.949 & 0.977 & 1.004 & 1.048 & 0.153 & 0.343 & 0.058 \\ \hline \hline \end{tabular} SD: Standard Deviation.  
\end{table} 
Table 4: Response values of the response surface experimental design model based on the analysis results. Figure 5: Comparison of velocity and pressure distributions via flow analysis: (**a**) velocity [m/s] of the experimental design model in Case 11; (**b**) pressure [kPa] of the experimental design model in Case 11; (**c**) velocity [m/s] of the experimental design model in Case 15; (**d**) pressure [kPa] of the experimental design model in Case 15.  
response functions in Table 5 as well as the previously introduced system evaluation functions and the results are listed in Appendix A. Based on our results, it can be deduced that variable B is effective in the regression equation of every response function with a \(p\)-value of lower than 0.01 for every response except for \(R^{\prime}_{3}\) and \(E(R)_{dist}\); this suggests that design variable B is the most influential factor among the system characteristics, followed by C, and then the interaction between B and C. In contrast, variable A has the lowest influence in determining the flow. Furthermore, based on the regression model, multiple correlation coefficients (R\({}^{2}\)) for the mean variation measurement values, except for \(R^{\prime}_{3}\), have values higher than 0.8. Six different cases for optimization were constructed by combining different product and error distance functions for the volume flow ratio and response value of each channel described in Section 3.1. First, optimization was performed based on the single-response objective function using the product function or error distance function to represent the system. In addition, the response surface optimization methodology was used for constructing a multi-response objective function by combining the volume flow ratio response prediction function, product function, and error distance function for each channel.