# Design and Numerical Analysis of Recuperator for a Liquid Carbon Dioxide Energy Storage System
## 3 Numerical Calculation Method and Validation of Numerical Method
### Comparison of Computational Results with Experimental Results  
The SST k-\(\omega\) model, standard k-\(\epsilon\) model, and the RNG k-\(\epsilon\) model had been applied in previous studies of supercritical CO\({}_{2}\)PCHE. Therefore, these four turbulence models need to be validated in this study. The results obtained from these four turbulence models contrast with the experimental correlations provided by [PERSON] et al. [22], [PERSON] et al. [23], [PERSON] et al. [27], and [PERSON] et al. [28], as well as the correlations fitted based on numerical results by [PERSON] et al. [25]. The specific empirical formulas and their applicable ranges are shown in Table 5.  
The results were processed using the same method as [PERSON] et al. [25], obtaining the friction coefficients and \(Nu\) under different operating conditions. Figures 5 and 6 provide a comparative analysis of the calculated friction factors and \(Nu\) with the experimental correlations provided by [PERSON] et al. [22], [PERSON] et al. [23], [PERSON] et al. [27], and [PERSON] et al. [28], as well as the correlation equations adapted by [PERSON] et al. [25]. The friction factors and \(Nu\) were calculated for Reynolds numbers (\(Re\)) on the  
Figure 4: Temperature variation of carbon dioxide along the length on the cold and hot sides calculated by [PERSON] et al. [26]. cold side ranging from 4200 to 55,000 and from 1800 to 24,000 on the hot side. It should be noted that [PERSON] experimental correlation equations consist of two segments for the applicable Reynolds number range, which are represented by two separate lines. On the other hand, the experimental correlations by [PERSON] et al. and [PERSON] et al. and the numerical correlation equation by [PERSON] et al. [25] were all single range. The upper and lower limits of the correlation equations were represented by solid and dashed lines with the same color. Figure 5 reveals that four turbulence models can accurately compute the friction factors of the PCHE. The numerical calculation results for the cold side were in general accordance with the experimental findings of [PERSON] et al. The numerical results for the hot side were in line with the experimental findings by [PERSON] et al. and [PERSON] et al. Moreover, the numerical calculation results for both channels exhibited excellent conformity with those  
Figure 5: Comparison of numerical calculation results of the friction factors on both channels with the experimental correlation equation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Researcher** & **Local h or Nu** & **Friction Factor\(f\)** & **Applicability** \\ \hline [PERSON] et al. [22] & \(h_{h}=2.52\text{Re}^{0.681}\) & \(f_{h}=(-1.402\times 10^{-6}\pm 0.087\times 10^{-6})\text{Re}+(0.04495\pm 0.00038)\) & \(2800\leq\text{Re}\leq 5800\) \\ \cline{2-3} [PERSON] et al. [22] & \(h_{c}=5.49\text{Re}^{0.625}\) & \(f_{c}=(-1.545\times 10^{-6}\pm 0.099\times 10^{-6})\text{Re}+(0.09318\pm 0.00090)\) & \(6200\leq\text{Re}\leq 12,100\) \\ \hline [PERSON] et al. [23] & \(\text{Nu}=(0.1696\pm 0.0144)\text{Re}^{0.629\pm 0.009}\times\) & \(f_{h}=(0.3390\pm 0.0285)\text{Re}^{-0.158\pm 0.009}\) & \(3500\leq\text{Re}\leq 22,000\) \\ Angle 52\({}^{\circ}\) & \(\text{Pr}^{0.317\pm 0.014}\) & \(f_{c}=(0.3749\pm 0.1293)\text{Re}^{-0.154\pm 0.036}\) & \(0.75\leq\text{Pr}\leq 2.2\) \\ \hline [PERSON] et al. [27] & \(h_{h}=0.2104\text{Re}+44.160\) & \(f_{h}=-2.0\times 10^{-6}\text{Re}+0.0467\) & \(2400\leq\text{Re}\leq 6000\) \\ Angle 32.5\({}^{\circ}\) & \(h_{c}=0.1106\text{Re}+15.943\) & \(f_{c}=-2.0\times 10^{-6}\text{Re}+0.1023\) & \(5000\leq\text{Re}\leq 13,000\) \\ \hline [PERSON] et al. [28] & \(\text{Nu}=0.4\text{Re}^{0.60}\text{Pr}^{1/3}(2D/\! )^{0.75}\) & \(f=2.0\text{Re}^{-0.40}\cdot(2D/\! )^{0.75}\) & \(600\leq\text{Re}<3000\) \\  & \(\text{Nu}=0.4\text{Re}^{0.64}\text{Pr}^{1/3}(2D/\! )^{0.75}\) & \(f=4.8\text{Re}^{-0.36}\cdot(2D/\! )^{1.50}\) & \(10^{4}<\text{Re}<10^{5}\) \\  & \(D\) channel width, \(l\) channel pitch & & & \\ \hline [PERSON] et al.