# One-Dimensional Computation Method of Supercritical CO\({}_{2}\) Labyrinth Seal
## 5 Conclusions  
Based on the internal flow characteristics of the labyrinth seal, the 1-D flow hypothesis and the thermodynamic process hypothesis are presented. The 1-D computation method of the labyrinth seal suitable for a real fluid (including a two-phase flow) is proposed.  
The SCO\({}_{2}\) sealing flow has the \(Re\) number that is two orders of magnitude higher than that of air, and is turbulent flow. The flow discharge coefficient is obviously larger. At the sealing chamber, the kinetic energy transport effect of the supercritical CO\({}_{2}\) working fluid is much more obvious,  
Figure 26: The effect of labyrinth seal type on sealing efficiency.  
Figure 25: The effect of teeth number Z on sealing efficiency.  
and the residual kinetic energy coefficient is larger. The influence of Re number on characteristic parameters is assumed to be negligible under the condition \(10^{5}<Re<10^{7}\). Based on the experimental, numerical results, and traditional theory, the experimental correlation discharge coefficient \(C\) and the residual kinetic energy coefficient \(\theta\) used in SCO\({}_{2}\) labyrinth seals are proposed. The two-phase critical flow was studied, and it is not suitable to use the \(a_{v}\) and \(a_{l}\) in the NIST model to calculate the speed of sound of the two-phase flow in the labyrinth seal, which leads to computational error in the calculation of two-phase critical flow. The speed of sound of two-phase flow is corrected and the problem of critical flow \(Ma\) number is perfectly solved.  
In this paper, to verify the validity of 1-D method, the results of three types of labyrinths including round-hole, see-through and staggered labyrinth seals are compared, which shows that the computational results of 1-D method are consistent with the experimental results for different types of labyrinth seals and under different inlet conditions, and it is also proven that the flow discharge coefficient and kinetic residual coefficient of 1-D method may not be affected by \(Re\) number (\(Re>10^{5}\)) to some extent.  
The 1-D method has been used to discuss the influence of various factors on the sealing performance, and the conclusions are as follows:  
1. The sealing efficiency increases as the pressure ratio increases and increases slowly when the pressure ratio is far from the critical pressure.
2. The minimum values of the sealing efficiency exist and are at the inlet temperatures near the pseudo-critical temperature.
3. As the teeth number continues to increase, the growth rate of the sealing efficiency slows down.
4. The staggered labyrinth seal has more advantages on the inlet condition of gas-like SCO\({}_{2}\) improving the sealing efficiency by 22% compared with the see-through labyrinth seal.  
Conceptualization, Y.Z. and Y.J.; Methodology, Y.Z.; Software, Y.Z.; Validation, Y.Z., Y.G. and H.C.; Investigation, Y.Z.; Writing-Original Draft Preparation, Y.Z.; Writing-Review & Editing, Y.J. and C.G.; Project Administration & Funding Acquisition, Y.J. and S.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding or This research was funded by the Strategic Priority Research Program of the Chinese Academy of Sciences, grant number [XDA21010201]. And the APC was funded by Major Special Programs of Science and Technology in Hebei Province (20284502Z) and Industrial Innovation and Entrepreneurship Team Project of Hebei Province (205A4501D).  
The authors declared that there is no conflict of interest.