# One-Dimensional Computation Method of Supercritical CO\({}_{2}\) Labyrinth Seal
## 1 Introduction
Equation (2) is the [PERSON] formula of a see-through labyrinth seal. \[\dot{m}=5.67C\frac{A}{\sqrt{RT_{0}}}\frac{P_{0}}{\sqrt{1-\xi}}\beta,\ \xi=\frac{8.52}{\frac{\xi-w}{c}+7.23},\beta=\sqrt{\frac{1-\left(\frac{P_{n}}{P_ {0}}\right)^{2}}{n-\ln(\frac{P_{n}}{P_{0}})}} \tag{2}\]  
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, \(s\) is the length of sealing chamber, mm, \(w\) is the width of sealing tooth, mm, and \(c\) is the clearance at sealing tooth, mm. \(P_{0}\) 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] proposed an iterative computation method, which applies to computation in both subcritical and critical working conditions [16]. However, all the above methods are based on the ideal gas thermodynamic process and can hardly satisfy the computation requirement of the labyrinth seal of the real fluid, in particular, supercritical fluid and two-phase fluid. Recently, along with the application of SCO\({}_{2}\) becoming the research hot spot, some researchers are working towards establishing a computational model for the SCO\({}_{2}\) labyrinth seal. [PERSON] et al. conducted an experimental study on the flow characteristics of the critical flow of SCO\({}_{2}\) under different single round hole structures, which can be seen as the single tooth labyrinth seal, and summarized empirical formulas (Equation (3)) [17]. Though the formulas are consistent with the single round hole experimental results, they are hardly used for the labyrinth's much more complex sealing structures. Additionally, the empirical formulas of nonlinear fitting applied for modifying the incompressible seal equation greatly limit the applicability of the model. \[G=\left(0.5463+0.0587\left(\frac{p_{0}}{p_{cr}}\right)^{2.07}\left(\frac{\rho _{0}}{\rho_{cr}}\right)^{-0.939}\right)\sqrt{2p_{0}(p_{0}-p_{c})} \tag{3}\]where G is mass flow flux, kg/(m\({}^{2}\)s); \(P_{0}\) is the upstream pressure, MPa; \(\rho_{0}\) is the upstream density, kg/m\({}^{3}\); \(P_{\sigma}\) is the critical pressure of CO\({}_{2}\), MPa; \(\rho_{\sigma}\) is the density at critical point of CO\({}_{2}\), kg/m\({}^{3}\); \(P_{c}\) is the choked pressure, MPa. [PERSON] et al. modified the traditional computation theory by replacing 8.52 with 1.42 in Equation (2) according to the experimental data and obtained the computational model of the SCO\({}_{2}\) labyrinth seal, in which CO\({}_{2}\) was considered as an approximately ideal gas with a compression factor of 1, resulting in a huge computing error for the two-phase outlet conditions of the model [4]. Therefore, the model still needs to be further improved, because the outlet condition of the SCO\({}_{2}\) compressor major seals are mostly two-phase state. [PERSON] proposed an isentropic computational model of SCO\({}_{2}\) in his dissertation, and introduced a homogeneous flow model to enable the model to compute two-phase outlet conditions [18]. This model is essentially a zero-dimensional model that only considers the influence of inlet and outlet conditions and sealing clearance while neglecting that of the internal sealing structure. In general, there is no accurate computational model that comprehensively considers both the physical property (including two-phase flow) changes and internal flow characteristics of the SCO\({}_{2}\) labyrinth seals. In this study, the flow is reasonably simplified and the equation of state of an ideal gas is not required for the whole calculation, which means that there is no need to correct the model for the compressibility of SCO\({}_{2}\), so the relevant empirical coefficients in the one-dimensional(1-D) method, including flow discharge coefficient and kinetic residual coefficient, can be more universally applicable.