# One-Dimensional Computation Method of Supercritical CO\({}_{2}\) Labyrinth Seal
## 1 Introduction  
At present, a large number of studies indicate that the supercritical CO\({}_{2}\)(SCO\({}_{2}\)) Brayton cycles may achieve much higher cycle efficiency than that of the Rankine cycle over a wide temperature range of heat sources with compact components, resulting in a significant decrease of generation costs [1; 2; 3]. The critical pressure of CO\({}_{2}\) is 7.38 MPa and the maximum pressure of a cycle may be up to 20 MPa, so the SCO\({}_{2}\) compressor's operational pressure is much higher than that of a traditional air compressor. Meanwhile, the viscosity of SCO\({}_{2}\) is close to that of air, and the density of SCO\({}_{2}\) is close to that of water, which may lead to a much thinner boundary layer and increase the leakage of the seal drastically. Hence, the leakage of SCO\({}_{2}\) in turbomachinery seals not only affects the cycle efficiency caused by parasitic losses, but also has paramount significance for evaluating the safety of the system under various working conditions [4]. The labyrinth seal, as one of the most commonly used non-contact seals, is widely employed in various rotary machines. There are different types of labyrinth seal, such as such as see-through, stepped, and staggered labyrinth seals. [PERSON] studied the see-through labyrinth seal used in SCO\({}_{2}\) flow both numerically and experimentally [5]. In the SNL (Sandia National Laboratory) SCO\({}_{2}\) Brayton cycle experiment loop, a stepped labyrinth seal was adopted for its compressor [6]. IET-CAS (Institute of Engineering Thermophysics, Chinese Academy of Sciences) developed a new kind of stepped-staggered labyrinth seal for its 1 MW SCO\({}_{2}\) compressor shaft end seal [7]. In terms of application, the labyrinth seals can be used as the major seal of the shaft end, the assisting seal of dry gas seals and the seal of the closed centrifugal impeller. Owing to the high power-density and the smallsize, tip clearance may lead to a serious decline in the performance, so the SCO\({}_{2}\) closed centrifugal impeller is to have a higher efficiency than that of semi-open impeller [8]. This means that the SCO\({}_{2}\) labyrinth seal may be significant in the future design of high-performance SCO\({}_{2}\) compressors. The labyrinth seal technology was firstly proposed by [PERSON] [9; 10] to solve the sealing problems in steam turbines. [PERSON]'s study [11] first described the internal flow through the labyrinth in detail. [PERSON]'s formula (Equation (1)) is the first computation formula to predict the leakage flow rate through the labyrinth seal, which is based on an isothermal process with negligible velocity in chamber, and all velocities are to be under sonic velocity in his original work [12]. \[\dot{m}=CA\frac{P_{0}}{\sqrt{RT_{0}}}\sqrt{\frac{1-\left(\frac{P_{n}}{P_{0}} \right)^{2}}{n-\ln(\frac{P_{n}}{P_{0}})}} \tag{1}\]  
where \(\dot{m}\) is the mass flow rate, kg/s, \(C\) is the flow discharge coefficient, \(A\) is the cross-sectional area of labyrinth seal, m\({}^{2}\), \(R\) is the ideal gas constant, \(P_{0}\) is the upstream pressure, MPa, \(T_{0}\) is the upstream temperature, K, \(P_{n}\) is the backpressure, MPa, and \(n\) is the teeth number. [PERSON] assumed that the pressure drops at sealing teeth could be neglected, using [PERSON]'s equation and the continuity equation [13]. Because it neglects the viscosity and compressibility of a fluid, it is only suitable for a small pressure difference on both sides of the sealing tooth or a large number of sealing teeth (n > 4). [PERSON]'s formula improves upon [PERSON]'s formula, and focuses on the impact of the kinetic energy transport effect on the seal. Therefore, a correction coefficient \(\gamma\) was defined. When \(\gamma\) is equal to 1, [PERSON]'s formula is reduced to [PERSON]'s formula [14]. [PERSON]'s formula is still [PERSON]'s formula in form [15]. A non-adiabatic wall surface and the kinetic energy transport effect are modified by using an empirical coefficient which is based on boundary layer theory.