## 3 Optimization Design
### Multi-Objective Optimization of HE Based on NSGA-II  
It can be seen from the results in Section 3.2 that optimizing HE with MEGN as a single objective function will greatly increase the total pumping power which indicates that the pressure drops of fluids on both sides of HE will increase. This is an undesirable result in the optimization design. In order to balance the heat transfer performance and hydraulic performance of HE, this section also takes the total pumping power as the objective function, and uses NSGA-II to carry out the multi-objective optimization of HE.  
The program of NSGA-II is also written based on MATLAB language. The modified entropy generation number and total pumping power of the heat exchanger are the two objectives of optimization. The population is set to 80 and the genetic generation is 200 in the optimization. The offspring individuals are selected by tournament in the selection, and the polynomial mutation method is used during the mutation. The optimized variables and  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\zeta_{s}\) & \(\zeta_{t}\) \\ \hline Preliminary HE & 1 & 1 \\ Optimized HE & 1.04 & 1.47 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of the performance of preliminary HE and optimized HE.  
relative ranges are the same as those of single-objective optimization. Figure 5 illustrates the pareto front which results from optimization.  
It can be seen from Figure 5 that the total pumping power decreases with the increase in the MEGN on the Pareto front. Select three representative points--A, B and C--as the analysis object, and pareto point A is similar to the model from single-objective optimization. The variable values of HE corresponding to pareto points A, B and C are listed in Table 8.  
It can be seen from Table 8 that the outer diameter of the heat exchange tube plays a major role in the multi-objective optimization to the objectives. The smaller outer diameter of the heat exchange tube will bring less irreversible loss to the heat exchanger. Although the other three design variables were almost the same for points A, B and C, they also provide an important reference significance for the design of HE. Compared to point C, the HE of point A shows better thermal performance where \(\varepsilon\) increases by 5.84% and NTU increases by 77.84%. However, with the decrease in MEGN of point A, the total pumping power of HE also greatly increases, which represents a large increase in the flow resistance of the fluid in the heat exchanger.  
Furthermore, it can be found that the heat exchanger corresponding to point B has both good thermal performance and a relatively small flow resistance by comparing the variable values in the HE models corresponding to points A, B and C. In terms of thermal-hydraulic performance optimization, point A and point C are not the most suitable options, and  
Figure 5: Pareto points of multi-objective optimization.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Point A** & **Point B** & **Point C** \\ \hline Outer diameter of tube/mm & 10.1 & 13.4 & 20.8 \\ Pitch diameter ratio & 1.25 & 1.28 & 1.25 \\ Length of tubes/m & 6 & 6 & 6 \\ Ratio of shell—inside diameter to the shell—inside diameter Calculated by Equation (1) & 1 & 1 & 1 \\ Layer of tubes & 40 & 40 & 40 \\ MEGN & 0.0556 & 0.0601 & 0.079 \\ Pumping power/kW & 73.5 & 17.1 & 2.1 \\ \(\varepsilon\) & 0.598 & 0.587 & 0.565 \\ NTU & 5.651 & 4.492 & 3.127 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Variable values of pareto points A, B and C.  
point B is a better choice because it has the best comprehensive performance. The multi-objective results show that the multi-objective optimization design of HE aiming to reduce the modified entropy generation number and total pumping power can optimize both the thermal and hydraulic performance of HE. The results of multi-objective optimization provide various options of HE in practical application.