## 6 Results and Discussion
### Thermal-Economic and Irreversibility Optimization  
The entropy generation analysis is from the point of view of energy balance to evaluate the performance of a heat exchanger. The entropy generation rate and total annual cost are the two objectives optimized by the multi-objective PSO to achieve the design parameters that produce the lowest possible total annual cost and the minimum entropy generation rate.  
The fresh air channel length \(L_{t}\), exhaust air channel length \(L_{\text{e}}\), apex angle \(\theta\), channel height \(H\), plate thickness \(\delta_{\text{plate}}\), and height \(L_{\text{h}}\) of the heat exchanger are the optimization parameters. The range of these parameters is listed in Table 3 in Section 3, and they are the same as those used in the thermal hydraulic performance optimization. The assumptions are made in Section 2, and the physical parameters held constant in Table 4 remained unchanged during the optimization. In addition, the parameters needed for the total annual cost evaluation are given in Table 1.  
The results of the Pareto-optimal points of the entropy generation rate and the total annual cost are shown in Figure 5. The entropy generation rate decreases with an increase in the total annual cost, which means if any one of smaller entropy generation rates replaces another it will always sacrifice quality of total annual cost. Similarly, five of the representative selected optimal design parameters are given in Table 7. A thinner plate thickness, larger channel length, and larger heat exchanger height tend to balance the entropy generation rate and total annual cost.  
A global sensitivity analysis results are shown in Figure 6. The bars representing the apex angle, channel height, and heat exchanger height are outstanding in the histogram.  
Figure 4: Graph of the variation of the \(j\) factor, \(f\) factor, and \(JF\) factor with three key variables, (**a**) channel height; (**b**) apex angle; and (**c**) heat exchanger height.  
Figure 5: Pareto-optimal points of the entropy generation rate and annual cost.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Points** & \(L_{t}\) **(m)** & \(L_{e}\) **(m)** & \(\theta\) **(rad)** & \(H\) **(m)** & \(\delta_{plate}\) **(m)** & \(L_{h}\) **(m)** & \
\begin{tabular}{c} **Entropy Generation** \\ **Rate (J) Kg\({}^{-1}\) K\({}^{-1}\))** \\ \end{tabular} &
\begin{tabular}{c} **Total Annual** \\ **Cost (S)** \\ \end{tabular} \\ \hline A & 0.4951 & 0.4926 & 2.4079 & 0.0088 & 0.00012 & 0.6979 & 4.096 \(\times\) 10\({}^{-6}\) & 93.17 \\ B & 0.4961 & 0.5000 & 1.9156 & 0.0081 & 0.00010 & 0.7000 & 3.822 \(\times\) 10\({}^{-6}\) & 121.19 \\ C & 0.4958 & 0.5000 & 1.5615 & 0.0072 & 0.00012 & 0.7000 & 3.562 \(\times\) 10\({}^{-6}\) & 171.02 \\ D & 0.5000 & 0.4964 & 1.3027 & 0.0067 & 0.00010 & 0.6946 & 3.389 \(\times\) 10\({}^{-6}\) & 232.79 \\ E & 0.5000 & 0.5000 & 1.1138 & 0.0058 & 0.00013 & 0.6953 & 3.264 \(\times\) 10\({}^{-6}\) & 336.08 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Selected optimal design parameters of the entropy generation rate and annual cost.  
Figure 6: Global sensitivity of cross-corrugated triangular ducts heat exchanger design parameters on entropy and total annual cost.  
To effectively show how these key design parameters affect the entropy generation rate and total annual cost in this design, three charts are made that demonstrate the two objectives changing with the three key parameters. For the three parameters, only one input parameter is varied, while the other input parameters are fixed. Figure 7a-c indicates the entropy generation rate and total annual cost change with \(H\), \(\theta\), and \(L_{\text{h}}\), respectively. The entropy generation rate and total annual cost decrease with the increasing variables and tend to fixed values.  
Figure 7: _Cont._