# One-Dimensional Computation Method of Supercritical CO\({}_{2}\) Labyrinth Seal
## 3 Internal Flow Property of SCO\({}_{2}\) Labyrinth Seal
### Flow Discharge Coefficient C of SCO\({}_{2}\) Labyrinth Seal
#### 3.1.2 The Correlation of Flow Discharge Coefficient C in SCO\({}_{2}\) Labyrinth Seal  
[PERSON] et al. carried out a detailed study on the flow discharge coefficient of traditional labyrinth seals and obtained the experimental correlation (Equation (13)), in which the application range is 250 \(<\)_Re_ \(<\) 15,000, and Equation (13) is much more suitable for uncompressible fluids, including water and air at a high pressure ratio [20]. \[C=\frac{0.7757-0.002051\Big{(}\frac{w}{c}\Big{)}}{\Big{(}1+\frac{44.86\big{(} \frac{w}{c}\big{)}}{Re}\Big{)}^{0.2157}} \tag{13}\]  
Though this equation cannot be directly used for the SCO\({}_{2}\) labyrinth seal for its extremely high _Re_ number, as shown in Table 2, it can be expected that the \(C\) of SCO\({}_{2}\) labyrinth seals may have a similar form as Equation (13). [PERSON] et al. investigated the discharge coefficient \(C\) of nozzle in high _Re_ number, and It was found that the growth rate of discharge coefficient \(C\) is less than 2% when the _Re_ number increases from \(10^{5}\) to \(10^{7}\)[21], which may be suitable for SCO\({}_{2}\) labyrinth seals with the same _Re_ number range. Besides, it is difficult to consider the impact of all factors for the experimental data of SCO\({}_{2}\) labyrinth seal is limited. So, it is logical to ignore the influence of _Re_ number on the flow discharge coefficient \(C\) when the _Re_ number is \(10^{5}\)-\(10^{7}\). Considering the _Re_ number of SCO\({}_{2}\) labyrinth seals is far less than \(10^{7}\), the criterion, _Re_\(>1\times 10^{5}\), should be carefully checked when the 1-D method is used. [PERSON]'s single round-hole leakage experiment used a high-pressure tank as the upstream and used a low-pressure tank as the downstream [4]. The initial condition of high-pressure tank is 13.5 MPa and 160 \({}^{\circ}\)C and the initial condition of low-pressure tank is normal temperature and pressure. As the valve between high-pressure tank and low-pressure tank is opened, the huge pressure difference causes critical flow in the beginning. As the pressure of different tanks gradually balances, the leakage flow turns into non-critical flow in the end and the whole experiment lasted about 210 s as is shown in Figure 6. The mass flux was obtained by calculating the mass difference of the tank for every second. So, the uncertainty is severely affected by the measure of temperature and pressure, which leads to an average uncertainty of \(\pm\)3694.7 kg/m\({}^{2}\). Applying the definition of \(C\) to analyze the experimental results in Figure 6, the variation curve of \(C\) with pressure ratio can be obtained in Figure 7. The \(C\) increases with the decrease of the pressure ratio when the leakage flow is under critical condition (pressure ratio \(<\) 0.6), and \(C\) basically keeps unchanged on the area of non-critical flow. The experimental results indicated that when outlet pressure is equivalent to atmospheric pressure, the critical pressure ratio at the outlet reaches the maximum, so does the flow discharge coefficient \(C\). The single round-hole has the diameter (_c_) of 1.5 mm and length (_w_) of 5.0 mm, so the _w_/_c_ is 10/3 and a preliminary experimental correlation of \(C\) can be obtained, as is shown in Equation (14), which may not be very accurate due to the uncertainty of measurement. \[C=\left\{ \begin{array}{ll}0.96-0.08(\frac{P_{out}}{P_{c}}),\frac{P_{out}}{P_ {c}}\leq 1&,\frac{w}{c}=\frac{10}{3}\\ 0.88,\frac{P_{out}}{P_{c}}>1&\end{array} \right. \tag{14}\] \begin{table}
\begin{tabular}{c c c c} \hline  & **Position I** & **Position II** & **Position III** \\ \hline SCO\({}_{2}\) & 0.718 & 0.786 & 0.862 \\ Air & 0.68 & 0.67 & 0.66 \\ \hline \end{tabular}
\end{table} 
Table 3: Flow discharge coefficient of SCO\({}_{2}\) and air at different positions.