## 3 Optimization Design
### Single Objective Optimization of HE Based on Genetic Algorithm  
Optimization design is carried out for HE based on the genetic algorithm with MEGN as the fitness to improve the thermal performance of HE. The optimization design method is written into a program by MATLAB language. During the optimization design process, the initial population is randomly generated according to the range set for every variable. The number of individuals in the initial population is 40, and the maximum number of generations is taken as 300. In the optimization design program, the heat transfer rate of HE is kept as the same value as 50 MWt. Additionally, the penalty function is used to eliminate those individuals who do not meet the design requirements in the process of population evolution. For each generation in the evolution process, the best individual is selected based on the fitness of each individual. The parameters of the optimization model of HE and the comparison between the preliminary design and the optimized design are listed in Table 6. The fitness of the best individual in each generation changes with the number of populations, as shown in Figure 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Preliminary Design** & **Optimized Design** \\ \hline Tube—outside diameter (mm) & 20 & 10.2 \\ Tube—inside diameter (mm) & 13 & 8.8 \\ Length of tube (m) & 6 & 6 \\ Pitch of tubes (mm) & 24 & 12.9 \\ Shell-side diameter (m) & 1.29 & 0.987 \\ Number of tubes & 2269 & 4681 \\ Mass flow of liquid lead and SCO2 (\(\mathrm{k}\mathrm{e}\mathrm{s}^{-1}\)) & 2303.3/2052 & 2303.3/182.2 \\ Inlet/outlet temperatures of liquid lead (K) & 873.15/723.15 & 873.15/720.89 \\ Inlet/outlet temperatures of SCO2 (K) & 633.15/854.40 & 633.15/859.48 \\ Entropy generation number & 0.0854 & 0.0559 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between the preliminary HE and the optimized HE. \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Variable** & **Range** \\ \hline Tube—outside diameter (mm) & 10–40 \\ Pitch diameter ratio & 1.25–2 \\ Ratio of shell—inside diameter to the shell—inside diameter Calculated by Equation (1) & 1–2 \\ Length of tubes (m) & 3–6 \\ Layers of tubes & 20–40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ranges of design variables. As shown in Figure 3, the optimal solution basically does not change after 120 generations, and the EGN achieves a 50% drop, from 0.11 for the first generation to approximately 0.055. In order to analyze the performance of the optimization model of HE with the decreasing MEGN, the effectiveness (\(\varepsilon\)), the number of transfer units (NTU), and the total pumping power of HE is calculated. \(\varepsilon\)[42] and NTU represent the thermal performance of HE, which can be expressed as:  
\[\varepsilon=\frac{1-\exp\Bigl{\{}(-NTU)\Bigl{[}1+\frac{(inc)_{\min}}{(inc)_{ \max}}\Bigr{]}\Bigr{\}}}{1+\frac{(inc)_{\min}}{(inc)_{\max}}} \tag{20}\]  
\[NTU=\frac{kA}{(inc)_{\min}} \tag{21}\]  
The total pumping power represents the flow resistance of both the tube-side and shell-side in HE. The total pumping power can be expressed as [43]:  
\[\dot{W}=\frac{1}{1000\eta}\left(\frac{\dot{m}_{s}}{\rho_{s}}\Delta P_{s,sum}+ \frac{\dot{m}_{t}}{\rho_{t}}\Delta P_{t,sum}\right) \tag{22}\]  
where \(\eta\) is the pumping efficiency, and \(\Delta P_{s,sum}\) and \(\Delta P_{t,sum}\) are the sum of the pressure drop of shell-side and tube-side, respectively. Figure 4 shows the variation in the effectiveness, number of transfer units (NTU) and total pump power of HE with MEGN in the optimization design. Figure 3: Variation of the beat fitness value along with the number of generations. Figure 4: The effects of EGN on the different parameters of HE. (**a**) The effect of EGN on the effectiveness of HE. (**b**) The effect of EGN on the NTU of HE. (**c**) The effect of EGN on the total pump power of HE.