## 3 Optimization Design  
### Genetic Algorithm  
There are different geometric factors which would have an impact on the performance of a heat exchanger such as the tube diameter and shell-side diameter. The effect of different factors are difficult to quantitatively analyze due to their completed change. Each factor is analyzed individually for the optimized thermal-hydraulic performance of a heat exchanger with the other factors constant and so much time will be used in the traditional optimization \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Total Heat Transfer Rate** \\ \hline Condition in Table 1 & 50 MWt \\ Verification result & 48.97 MWt \\ Relative error & 2.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Heat transfer rate validation result. \begin{table}
\begin{tabular}{c c} \hline \hline
**Property of Liquid Lead** & **Correlations** \\ \hline Density (kg:m\({}^{-3}\)) & \(\rho=11367-1.1944\cdot T\) \\ Viscosity (Pa:s) & \(\mu=4.55\times 10^{-4}\)exp(1069/T) \\ Thermal conductivity (W:m\({}^{-1}\cdot\)K\({}^{-1}\)) & \(\lambda=9.2+0.011\cdot T\) \\ Heat capacity (J:kg\({}^{-1}\cdot\)K\({}^{-1}\)) & \(C_{p}=175.1-4.961\times 10^{-2}\cdot T+1.985\times 10^{-5}\cdot T^{2}- 1.524\times 10^{6}\cdot T^{-2}-2.099\times 10^{-9}\cdot T^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Property correlation of liquid lead. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Value** \\ \hline Tube—outside diameter (mm) & 20 \\ Tube—inside diameter (mm) & 13 \\ Length of tube (m) & 6 \\ Pitch of tubes (mm) & 24 \\ Shell-side diameter (m) & 1.29 \\ Number of tubes & 2269 \\ Mass flow of liquid lead and SCO\({}_{2}\) (kg:s\({}^{-1}\)) & 2303.3/205.2 \\ Inlet/outlet temperatures of liquid lead (K) & 873.15/723.15 \\ Inlet/outlet temperatures of SCO\({}_{2}\) (K) & 633.15/831.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Preliminary design results of HE. method. In order to solve this problem, the genetic algorithm is used in the optimization design. The genetic algorithm can estimate several factors at the same time to solve the optimal solution problem based on the biological evolution theory. Applying the genetic algorithm to the optimal design of HE can quantitatively analyze different variables at the same time and save a lot of time. Figure 2 illustrates the flow chart of genetic algorithm. As shown in Figure 2, the parameters of each individual are transferred to the fitness function as input after the initial population is generated. The objective function of each HEU and the HE is calculated through the subsection model mentioned before and the thermal-hydraulic model of the heat exchanger. The conditions in the fitness function are set to eliminate unqualified individuals, which is realized in the algorithm through penalty function. After evaluating the fitness of each individual, the population is updated through selection, crossover and mutation until the optimal individual that meets the conditions is found or the maximum number of generations is reached. Different related variables are associated with the fitness (objective function) to estimate the performance of HE. The first generation is randomly created and the next generation is generated by selection, crossover and mutation. Eventually, the heat exchanger with the best fitness will be proposed after the continuous alternation of the generation. The modified entropy generation number is the fitness in the present study. Entropy generation is caused by irreversible loss in the process of flow and heat transfer. It is used to evaluate the irreversible loss in the heat exchanger. The irreversible loss of the heat exchanger occurred in two parts: the irreversible loss from flow resistance and the irreversible loss from heat transfer with limited temperature difference.