## 4 Numerical Simulation on Simplified Model of Preliminary Design HE and Optimized Design HE
### Numerical Simulation Results  
Based on the optimization design and the numerical model of HE, the flow and heat transfer characteristics were studied first. Figure 10 shows the flow velocity and temperature distribution of lead and SCO\({}_{2}\) in sample 2.  
The flow velocity of lead is much less than the flow velocity of SCO\({}_{2}\) in sample 2 from the result in Figure 10a. It can be seen from the velocity contours of liquid lead and SCO\({}_{2}\) that the flow velocity of SCO\({}_{2}\) changes a little along the flow direction while the flow velocity of lead descends a little along the flow direction. It can be seen from Figure 10b  
Figure 10: Pressure and temperature distribution of lead and SCO\({}_{2}\) in sample 2. (**a-1**) Flow velocity of lead in sample 2. (**a-2**) Flow velocity of SCO\({}_{2}\) in sample 2. (**b**) Temperature distribution of lead and SCO\({}_{2}\) in sample 2.  
that the temperature of SCO\({}_{2}\) in the heat exchange tube increases gradually along the flow direction, and the temperature in the center of the tube increases the slowest. The temperature of liquid lead decreases faster in the area close to the heat exchange tube. In addition, from the temperature contours of the four HEUs outlet sections, it can be found that the temperature gradient near the wall inside the tube and near the wall outside the tube is relatively large.  
For a quantitative study, specific temperature distribution of SCO\({}_{2}\) and lead are collected. The axial temperature distribution of SCO\({}_{2}\) and liquid lead are shown in Figure 11. The temperature of SCO\({}_{2}\) increases 25.1 K in sample 2. The temperature of lead illustrates a uniform upward trend along the flow direction with a 18.2 K drop in temperature in sample 2. The temperature changing curves of lead and SCO\({}_{2}\) are approximately parallel, except for the area near the inlet of SCO\({}_{2}\).  
The pressure distributions of lead and SCO\({}_{2}\) along the flow direction are shown in Figure 12. The operating pressure of liquid lead is 0.1 MPa and its pressure loss in sample 2 is 1.12 kPa. The operating pressure of SCO\({}_{2}\) is 20 MPa, and the pressure drop of it in sample 2 is 1.19 kPa. The flow velocity of liquid lead is much less than SCO\({}_{2}\), but the pressure drop is near to SCO\({}_{2}\), which is due to the high density and high viscosity of liquid lead.  
Figure 11: Temperature distribution of lead and SCO\({}_{2}\) in sample 2.  
Figure 12: Pressure distribution of lead and SCO\({}_{2}\) in sample 2.  
In order to further illustrate the influence of MEGN on the thermal and hydraulic performance of HE and evaluate the comprehensive performance of it, the field synergy analysis was carried out for the heat transfer performance and flow resistance in sample 1 and sample 2. The field synergy angle commonly used in field synergy analysis is the angle between the velocity vector and the temperature gradient, as shown in the following equation:  
\[\beta=\arccos\frac{\mathbf{U}\cdot\
abla T}{|\mathbf{U}|\cdot|\
abla T|} \tag{26}\]  
where \(\mathbf{U}\) is the velocity vector; and \(\
abla T\) is the temperature gradient.  
The cosine of the field synergy angle is called the field synergy number, which can also be used to evaluate the synergy of the velocity vector and temperature gradient. A larger field synergy number means a better thermal-hydraulic performance of HE. Figure 13 shows the average field synergy number in the heat exchange tubes.  
It can be seen from Figure 13 that the average field synergy number of SCO\({}_{2}\) in the heat exchange tube of sample 2 is greater than that of sample 1 on the whole. Such results indicate that the synergy of the velocity vector and temperature gradient in sample 2 are better, i.e., that sample 2 has better comprehensive performance.