## 4 Numerical Simulation on Simplified Model of Preliminary Design HE and Optimized Design HE
### Physical Model  
It is difficult to directly use the prototype parameters for numerical simulation due to the large size of the preliminary model and the optimization model (point B) of HEs. Therefore, the simplified models of these two models are constructed based on the similarity principal method, which are called Model A and Model B. According to the similarity design method, the boundary conditions of the simplified model should remain the same as for the prototype, and several dimensionless criteria on the same point should also be equal. These dimensionless criteria reflect the influence of different parameters on the physical phenomena in HE. In the actual application, those parameters that are significantly related to the most concerned physical phenomena are preferentially guaranteed to be similar, while other weakly correlated parameters are ignored. In this study, attention is mainly focused on the heat transfer performance of HE so that the temperature distribution and velocity distribution in the simplified models should be consistent with those in the prototype. A simplified model of HE is designed based on the same MATLAB program in Section 2 and the information of Model A and Model B is listed in Table 9.  
It can be seen from Table 9 that the simplified models consist of seven heat exchange tubes, and the inlet and outlet temperatures on the shell-side and tube-side are consistent with the prototype. The mass flowrate of the fluid in the simplified model is proportionally reduced to ensure that the flow velocity of lead and SCO\({}_{2}\) is the same as that in the prototype. Because the velocity distribution and temperature distribution of the simplified model need to be similar to the prototype, the length of the heat exchange tube in the simplified model is almost the same as the prototype, which is not suitable for numerical simulation. After consideration, this study only takes five HEUs of model A and model B as the sample to simulate. The selection of HEUs is based on the relative change rate of MEGN of each HEU. The relative change rate of MEGN refers to the change in MEGN of each HEU in model B compared with that of each HEU in model A. A positive value means that the MEGN of the HEU decreases after optimization design, while a negative value is the opposite. The relative change rate of MEGN of each HEU is shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Model A** & **Model B** \\ \hline Tube—outside diameter (mm) & 20 & 13.4 \\ Tube—inside diameter (mm) & 13 & 11.6 \\ Length of tube (m) & 5.9 & 5.9 \\ Pitch of tubes (mm) & 24 & 17.2 \\ Shell-side diameter (m) & 0.072 & 0.051 \\ Number of tubes & 7 & 7 \\ Mass flow of liquid lead and SCO\({}_{2}\) (kg\(\cdot\)s\({}^{-1}\)) & 7.106/0.633 & 3.445/0.277 \\ Inlet temperatures of liquid lead (K) & 873.15 & 873.15 \\ Inlet temperatures of SCO\({}_{2}\) (K) & 633.15 & 633.15 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Information of the small-scale models of HE.  
According to the results from Figure 6, five HEUs close to the outlet of SCO\({}_{2}\) are selected as the research object, because the MEGN of these heat exchanger units is reduced the most, which is more conducive to explain the impact of optimization design. The diagram of HE is shown in Figure 7. The five HEUs from Model A and Model B are named sample 1 and sample 2, respectively.  
Figure 6: Relative change of \(N_{s1}\) in each HEU.  
Figure 7: Diagram of HE and the tube layout.