# Design and Numerical Analysis of Recuperator for a Liquid Carbon Dioxide Energy Storage System
## 4 Numerical Model Verification of the PCHE and Mesh Independence
### Mesh Independence  
The study of grid independence is implemented to guarantee the reliability of numerical simulations. Thus, the numerical solutions should be studied within a relatively wide range of grid resolutions to make sure that the obtained results are not affected by the grid node. In this section, three sets of grids with different density distributions were used to explore the impacts of grid density on the simulation of the PCHE. The calculation meshes are exhibited in Figure 8, where the orange region indicates the cold fluid field, the green zone indicates the hot fluid field, and the purple zone indicates the solid field. An O-grid was applied near the walls to improve grid quality. Since the SST k-\(\upomega\) turbulence model needs the value \(y^{+}\) near the wall to be inferior to 2, the distance from the initial row of fluid domain cells to the wall was set to 0.0003 mm for all three sets of grids. It ensured that \(y^{+}\leq 1\). Additionally, all three sets of grids had more than 15 grid nodes arranged in the vertical wall direction within the boundary layer. The number of grid nodes for the three sets of grids is 3.341 million, 5.367 million, and 19.191 million, respectively. The grids are densified in all three directions at the same scale. Figure 8: Grids used for calculations. Figure 7: The structure of the PCHE. The calculations were performed by ANSYS CFX with the SST k-\(\upomega\) model, and the overall solution accuracy was second order. The boundary conditions for assessing mesh independence were set according to the design operating conditions. The inlet conditions were specified with mass flow rates and the outlet boundary condition was given average static pressures. The temperature of inlets was set to 298.2 K and 674.3 K, respectively. The outlet average static pressures were specified as 25 MPa for the cold side and 6.5 MPa for the hot side. The fluid-solid interface was defined as a coupled heat transfer boundary. The four sides of the model were set as periodic boundaries, and the rest of the solid walls were regarded as adiabatic walls. The fluid domain adopted the total energy model, and the thermophysical properties of carbon dioxide were obtained from two 2000 \(\times\) 3000 tables derived from NIST. The pressure range for the thermophysical properties table of the cold side CO\({}_{2}\) is 23-27 MPa, and for the hot side, it is 4.5-8.5 MPa. The temperature range is 225-850 K. The solid domain utilized a thermal energy model that neglected density variations during the solution process, and \"Steel\" was selected from the material library, whose thermal conductivity is 16.2 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). Three sets of grids with 3.341 million, 5.367 million, and 19.191 million grid nodes were used to simulate the flow and thermal characteristics of PCHE. The formulas for processing the calculation results are shown in Equations (13)-(15). The results of the simulations are illustrated in Figures 9-11. The blue arrow represents the flow direction of the cold CO\({}_{2}\), while the red arrow indicates the flow direction of the hot CO\({}_{2}\). Figure 9 presents the friction factors distribution of CO\({}_{2}\) along the PCHE both on the cold side and hot side for different grid numbers. It can be observed that the distribution of friction factors was uniform. As the number of grids varies from 3.341 million to 19.191 million, the distribution of friction factors decreased along the PCHE, and the values of the friction factors at each node were closed. Furthermore, the results obtained using 5.367 million grids were very close to those obtained using 19.191 million grids, while the value deviation was slightly larger when using 3.341 million grids. Figure 10 shows the temperature profiles of CO\({}_{2}\) simulated by different grid numbers. It is evident that the temperatures simulated by the three grids are exactly the same. Figure 11 illustrates \(Nu\) distributions of CO\({}_{2}\) along the PCHE both on the cold side and hot side for different grid numbers.