## 4 CFD Analysis of a Supercritical CO2 Heat Exchanger
### Flow Analysis  
The thermal characteristics of the SCO2 heat exchanger were analyzed using a commercial CFD code, CFX. The overall performance simulation of the SCO2 heat exchanger was difficult due to the computing power. Therefore, a design of the scaled SCO2 heat exchanger was created for the CFD analysis. Figure 4 shows the scaled SCO2 heat exchanger: Figure 4 indicates the overall three-dimensional geometry, and Figure 4 illustrates the staggered tube array used in the flow CFD analysis. In the SCO2 heat exchange chamber, three heat exchanger tubes with a staggered tube array were positioned. The center of the heat exchanger tube reflected the full-scale tube specification, while the half-scale tube specification was considered in the bottom and top of the tubes. The numbers of nodes and elements for the analysis were 9,555,606 and 10,698,900, respectively. Figure 5 shows the mesh distributions on the scaled SCO2 heat exchanger. Figure 5 illustrates the overall mesh distributions in the scaled SCO2 heat exchanger, Figure 5 indicates the mesh structure around the straight tube position, and Figure 5 shows the mesh formation near the tube bending location. Precise cells near the wall surface were considered to keep the turbulence effect. Several grid layers were applied around the heat exchanger tubes to consider the wall effect on the working fluids. For the turbulence model, a two-equation turbulence model of RNG k-\(\varepsilon\) was used: k is the turbulence kinetic energy, which is defined by the fluctuation in velocity, and \(\varepsilon\) is the turbulence eddy dissipation. The RNG k-\(\varepsilon\) model was improved from the standard k-\(\varepsilon\) model, and it was derived using the renormalization group theory. The RNG k-\(\varepsilon\) model has an additional term, which considers the eddy dissipation, average shear stress, and swirl effect. These features provide a more accurate turbulence model compared to the standard k-\(\varepsilon\) model. The transport equations for turbulence generation and dissipation are the same as those for the standard k-\(\varepsilon\) model, but the model constants differ [30]. Figure 4: Scaled SCO2 heat exchanger; (**a**) overall CFD analysis structure; (**b**) detailed view around heat exchanger tubes. For the continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\Big{(}\rho U_{j} \Big{)}=0 \tag{1}\]  
Figure 5: Mesh distributions on the scaled SCO\({}_{2}\) heat exchanger: (**a**) overall mesh distributions, (**b**) mesh distributions around the straight tube position, (**c**) mesh formation near the tube bending area. For the momentum equation:  
\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\Big{(} \rho U_{i}U_{j}\Big{)}=-\frac{\partial p^{\prime}}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\Bigg{(}\mu_{eff}\Big{(}\frac{\partial U_{i}}{ \partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\Big{)}\Bigg{)}+S_{M} \tag{2}\]  
For the transport equation for turbulence dissipation:  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial}{\partial x_{j}}( \rho U_{i}\varepsilon)=\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon RNG}}\Big{)}\frac{\partial\varepsilon}{ \partial x_{j}}\Bigg{]}+\frac{\varepsilon}{k}(C_{\varepsilon 1 RNG}P_{k}-C_{\varepsilon 2 RNG}\rho\varepsilon+C_{\varepsilon 1 RNG}P_{\varepsilon b}) \tag{3}\]  
where \(\rho\) is the density, p' is the modified pressure, \(\mu\) is the viscosity, S\({}_{M}\) is the sum of body forces and P\({}_{k}\) is the shear production of turbulence, while \(\sigma_{\varepsilon RNG}\), C\({}_{\varepsilon 1 RNG\text{ constant}}\), and C\({}_{\varepsilon 2\text{2 RNG constant}}\) are the RNG k-\(\varepsilon\) constants. The boundary conditions were based on the actual operation conditions of the SCO\({}_{2}\) heat exchanger.