## 6 Results and Discussion
### Thermal Hydraulic Performance Optimization
#### 6.1.2 Multi-Objective Optimization  
The \(j\) factor and \(f\) factor are two mutually conflicting indices to unilaterally evaluate the thermal hydraulic performance of a heat exchanger. There is no set of design parameters that simultaneously yields the maximum \(j\) factor and the minimum \(f\) factor. The PSO algorithm with multi-objective optimization is introduced to search the geometric design parameters to determine the appropriate trade-off of the two objectives. Five sets of representative selected parameters are listed in Table 6. Each of the optimal solutions is a trade-off of the two conflicting objective functions at an appropriate level. These results can be selected by the designer based on the project's limits and the available investment.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(L_{\text{f}}\) (m) & \(L_{\text{e}}\) (m) & \(\theta\) (rad) & \(H\) (m) & \(\delta_{\text{plate}}\) (m) & \(L_{\text{h}}\) (m) & \(j\) Factor & \(f\) Factor \\ \hline
0.5000 & 0.4755 & 0.3787 & 0.0021 & 0.0003 & 0.6730 & 0.0772 & 1.8965 \\
0.4933 & 0.4818 & 0.4630 & 0.0024 & 0.0003 & 0.6689 & 0.0665 & 1.6402 \\
0.4832 & 0.4768 & 0.5660 & 0.0025 & 0.0005 & 0.6611 & 0.0577 & 1.4264 \\
0.4536 & 0.4285 & 0.7252 & 0.0033 & 0.0005 & 0.6188 & 0.0440 & 1.0959 \\
0.4140 & 0.4061 & 0.8728 & 0.0044 & 0.0008 & 0.5522 & 0.0331 & 0.8302 \\ \hline \end{tabular}
\end{table}
Table 6: Selected optimal design parameters of \(j\) factor and \(f\) factor.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Parameters** & **Values** & **Physical Parameters** & **Values** \\ \hline \(c_{\text{pf}}\) & 1.005 kJ kg\({}^{-1}\) K\({}^{-1}\) & \(\dot{\mathrm{Q}}_{\text{f}}\) & 100 m\({}^{3}\) h\({}^{-1}\) \\ \(c_{\text{pe}}\) & 1.005 kJ kg\({}^{-1}\) K\({}^{-1}\) & \(\dot{\mathrm{Q}}_{\text{e}}\) & 100 m\({}^{3}\) h\({}^{-1}\) \\ \(Pr_{\text{f}}\) & 0.701 & \(\mu_{\text{f}}\) & 1.86 \(\times\) 10\({}^{-5}\) Pa s \\ \(Pr_{\text{e}}\) & 0.701 & \(\mu_{\text{e}}\) & 1.86 \(\times\) 10\({}^{-5}\) Pa s \\ \(P_{\text{fi}}\) & 1.014 \(\times\) 10\({}^{5}\) Pa & \(\rho_{\text{a}}\) & 1.165 Kg m\({}^{-3}\) \\ \(P_{\text{ei}}\) & 1.014 \(\times\) 10\({}^{5}\) Pa & \(\lambda_{\text{a}}\) & 0.0263 \(\times\) 10\({}^{-3}\) KW m\({}^{-1}\) K\({}^{-1}\) \\ \(T_{\text{fi}}\) & 308 K & \(\lambda_{\text{plate}}\) & 2.4 \(\times\) 10\({}^{-4}\) KW m\({}^{-1}\) K\({}^{-1}\) \\ \(T_{\text{ei}}\) & 300 K & \(R_{\text{a}}\) & 0.287 kJ kg\({}^{-1}\) K\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Physical parameters held constant during the optimal design of the heat exchanger.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(L_{\text{f}}\) (m) & \(L_{\text{e}}\) (m) & \(\theta\) (rad) & \(H\) (m) & \(\delta_{\text{plate}}\) (m) & \(L_{\text{h}}\) (m) & **Objectives** \\ \hline
0.3431 & 0.3468 & 0.3322 & 0.0058 & 0.0004 & 0.4086 & max \(j\) factor 3.7463 \(\times\) 10\({}^{-2}\) \\
0.1730 & 0.1730 & 2.6452 & 0.0079 & 0.0017 & 0.1299 & min \(f\) factor 1.6932 \(\times\) 10\({}^{-1}\) \\
0.3673 & 0.3662 & 0.7553 & 0.0032 & 0.0006 & 0.5299 & max \(JF\) factor 3.8407 \(\times\) 10\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimal results of the thermal hydraulic performance with single objective optimization.