#### Abstract

The numerical model of a staggered labyrinth seal working with supercritical carbon dioxide (S–CO_{2}) is established. The dynamic and static characteristics of the staggered labyrinth seal for different axial shifting distances of the rotor, various cavity geometries (heights/widths of the rotor convex plate, heights of the seal cavity), and seal clearances were investigated and compared with the conventional see-through labyrinth seal. The results show that the effective damping coefficient (*C*_{eff}) with positive axial shifting distance is higher than that with negative axial shifting distance. When the rotor with convex plate operates without axial shifting, the cross-coupled complex dynamic stiffness (*h*_{R}) of the staggered labyrinth seal shows little effect on the *C*_{eff}, and the average direct damping (*C*_{avg}) has a dominant influence on the *C*_{eff}. As the whirling frequency (Ω) is lower than 60 Hz, the *C*_{eff} decreases with increasing height of the rotor convex plate. For Ω < 140 Hz, the damping coefficient generally increases with the decreasing height of the seal cavity. For Ω < 160 Hz, the *C*_{eff} of the see-through labyrinth seal is about 107%–649% of the staggered labyrinth seal. Otherwise, the *C*_{eff} of the staggered labyrinth seal is about 105%–113% of the see-through labyrinth seal. The *C*_{eff} of the seal with the rotor convex plate width of 5.13 mm is relatively high, which is conducive to the stability of the system. The *C*_{eff} increases with the decreasing seal clearance. The *C*_{eff} of the seal with 0.4 mm clearance is about 116%–148% the seal with 0.6 mm. The leakage flow rate of the staggered labyrinth seal of the see-through labyrinth seal is increased by about 45.5%. The leakage flow rate of the staggered labyrinth seal decreases with the increasing convex plate height, the seal cavity height, and the decreasing seal clearance.

#### 1. Introduction

During the 1960s–1970s, supercritical carbon dioxide (S–CO_{2}) was first proposed as a working fluid in a Brayton cycle by Angelino [1], Feher [2], and Combs [3]. Compared with the traditional steam cycle, S–CO_{2} has many unique physical properties such as high cycle efficiency, high density, low viscosity, good performance in compression, heat transfer and stability, and nontoxic gas. It has been regarded as a promising working medium and is widely utilized for advanced turbomachines with the Brayton cycle [1, 2]. Meanwhile, the staggered labyrinth seal, which affects the leakage flow rate and system stability, is crucial for the efficient and safe operation of each turbomachine [3, 4]. For the staggered labyrinth seal working with S–CO_{2}, there is an urgent demand to evaluate its leakage performance and rotordynamic characteristics.

In recent years, many research institutions (Sandia National Laboratories [5], Bechtel Marine Propulsion Corporation and Bettis Atomic Power Laboratory [6, 7], Southwest Research Institute and General Electric [8–10], Tokyo Institute of Technology [11], Korea Advanced Institute of Science and Technology [12, 13], University of Central Florida [14], etc.) have carried out experimental investigations or field tests to study the performance of turbomachines including turbine expander and turbine compressor with S–CO_{2}. Zhang et al. [15] compared the one-dimensional design model under the three ideal gas, compressible, and incompressible S–CO_{2} physical property forms and used direct numerical simulation methods to perform three-dimensional numerical calculations. It is found that the error of the ideal gas model is up to 12%. Behafarid and Podowski [16, 17] performed aerodynamic analysis and optimized the design of the S–CO_{2} turbine designed by the Korea Advanced Institute of Science and Technology and modified the turbine blades to improve the aerodynamic efficiency of the turbine components in the system. In 2016, Kim et al. [18] presented a computational fluid dynamic (CFD) analysis of a supercritical carbon dioxide, and they employed two methods of the real gas property estimations including real gas equation and real gas property (RGP) file (a required table from NIST REFPROP). The results show that the Peng–Robinson equation of estate (PREoS) method inserts a significant error in the calculation of entropy and enthalpy, while the other thermodynamic properties such as the thermal conductivity and density are almost identical to those of RGP prediction. Implementing the RGP table method indicates a very good agreement with NIST REFPROP.

For the annular seal with S–CO_{2} as a working fluid, present researches mainly focus on its leakage performance. Odabaee et al. [19] proposed a turbine generator set with S–CO_{2} as the working medium. The turbine part communicates with the high-speed generator through the labyrinth seal and dry gas seal to realize zero fluid leakage. In 2012, Wang et al. [20] investigated the leakage of S–CO_{2} in rolling piston expander experimentally. The study compared four classic leakage models. The analysis shows that the laminar leakage model is suitable in the leakage simulation of expander. In 2013, Tian [21] performed a computational study to investigate the leakage through seals subject to large pressure differential using open-source CFD software OpenFOAM. A fluid property interpolation table program was implemented in the OpenFOAM. The results show that the carryover coefficients are independent of pressure drop across the seal and are only a function of seal geometries. In 2015, Pidaparti [22] presented a numerical study of S–CO_{2} flow in see-through labyrinth seals using OpenFOAM. They also constructed a test facility and measured the leakage rate and pressure drop of S–CO_{2} in the seal for various designs and conditions. The experimental and numerical results show a good agreement for a two-tooth labyrinth seal at two upstream conditions. Increasing the radial clearance and decreasing the cavity length tend to increase the leakage flow rate. There is an optimum cavity height resulting in a minimum leakage flow rate when other parameters are fixed. This trend holds true for the effect of blade quantities while holding the total length fixed. The final optimization designs and the corresponding leakage rates are different for the air and S–CO_{2}. In 2018, Yuan et al. [23] designed a novel test rig for traditional annular seals at the University of Virginia in the ROMAC laboratory with the goal of testing the sealing performance with S–CO_{2}. Bennett et al. [24] presented a numerical study of a novel stepped-staggered labyrinth seal with S–CO_{2} to evaluate its performance compared with the see-through labyrinth. The computational fluid dynamic calculation was carried out using the NUMECA commercial code, and the NIST REFPROP database was used for the computation of S–CO_{2} in the numerical investigation. The results show that the stepped-staggered labyrinth seals have better seal performance than the see-through labyrinth and avoid the assembly problem of axial interlocking labyrinth. The leakage flow rate increases with an increase in the radial clearance. The seal performance is worst as length/height flow rate is equal to 1, and the sealing performance shows a good symmetry of geometric topology with length/height flow rate. It may exist a best width/height flow rate that leads to the least leakage flow rate. In 2018, Zhu et al. [25] studied the design and sealing performance comparison of the 450 MWe S–CO_{2} turbine shaft end seal of GE Company. In 2018, Du et al. [26] used the air and S–CO_{2} model, respectively, to study the effect of working fluid on the spiral seal dry gas sealing performance under different inlet temperatures and pressures. In 2019, Du et al. [27] studied the effects of high-speed S–CO_{2} dry gas seals on actual effects and analyzed the effects of actual gas effects, inertial effects, and turbulence effects on film stiffness and leakage flow rate under different medium pressure and velocity conditions. The actual gas effect makes the film stiffness and leakage flow rate increase significantly. The turbulence effect increases the film stiffness and decreases the leakage flow rate, and the inertia effect is weak.

S–CO_{2} also has a potential threat to the system stability as the air and steam. Therefore, there is a pressing need to obtain the related data of the rotordynamic performance for the turbine with S–CO_{2}. So far, few publications or reports could be found to investigate the rotordynamic characteristics of the S–CO_{2} seal. In this study, a three-dimensional numerical model of a staggered labyrinth seal working with S–CO_{2} is established using the computational fluid dynamic method. An S–CO_{2}.rgp (real gas property) file based on the National Institute of Standard and Technology (NIST) Shen [28] was embedded in the software. A dynamic transient CFD model based on an infinitesimal theory proposed by Zhang et al. [29] was utilized to get the dynamic force coefficients for the labyrinth seal working with S–CO_{2}. The CFD model predicts the dynamic force coefficients for the seal operating under various rotor axial shifting distances, rotor convex plate heights/widths, seal cavity heights, and clearance conditions. The stability of the conventional labyrinth seal system was compared.

#### 2. Numerical Method

##### 2.1. Geometrical Model

In this study, a three-dimensional CFD numerical model of the staggered labyrinth seal, which is based on a turbine diaphragm seal, is established as the research object. Figures 1 and 2 show the two-dimensional model of the staggered labyrinth seal for this study. Table 1 gives the detailed geometric dimensions. To investigate the effect of the rotor axial shifting distance, rotor convex plate heights/widths, seal cavity heights, and clearances on the seal dynamic characteristics, the staggered labyrinth seal with five shifting distances (*z* = −4 mm, −2 mm, 0 mm, +2 mm, and +5 mm), four rotor convex plate heights (*a* = 0 mm, 1.5 mm, 2 mm, and 2.5 mm), three rotor convex plate widths (*b* = 4 mm, 5.13 mm, and 6.13 mm), three seal cavity heights (*h* = 5.5 mm, 7 mm, and 8.5 mm), and three clearances (*δ*_{2} = 0.4 mm, 0.5 mm, and 0.6 mm) are employed in the numerical simulation. Table 2 gives the specific calculation conditions. In Figure 1, the rotor is shifting along the axial direction, the negative sign is shifting along the negative direction of the rotor along the *z*-axis, and the positive sign is shifting along the positive direction of the rotor along the *z*-axis.

##### 2.2. Numerical Model

The present numerical simulation was conducted to solve the compressible RANS equations using a commercial software [29]. Table 3 lists the detailed numerical approaches for CFD analysis in this study. The working fluid is the supercritical carbon dioxide (real gas). The standard *k-ε* is used as the turbulence model, with the turbulence intensity of 5%. The value of *y*+ is controlled within ∼300. The walls of rotor and stator are defined to be adiabatic, smooth, and have no slip. The total pressure and temperature are defined at the inlet boundary, while the average static pressure is specified at the outlet. CEL is used to define the whirling orbit of the rotor.

To improve the computational accuracy, a structured grid is employed. The blade tip, where the flow changes dramatically, is properly meshed with more grids. The scalable wall function method is used to combine the wall physical quantity with the turbulent core area. After grid independence verification, the total number of grids of the model is determined to be about 3.32 × 10^{6} to 3.82 × 10^{6}. The detailed grid distribution is shown in Figure 3.

The ideal gas assumption is no longer applicable for the supercritical carbon dioxide. To calculate the real gas property accurately, this study generates S–CO_{2}.rgp (real gas property) physical property file based on NIST REFPROP [30] for CFD program call.

##### 2.3. Rotordynamic Coefficient Solution

In this study, the seal dynamic characteristic identification method based on an infinitesimal theory is applied to solve the rotordynamic characteristics of arbitrary elliptical orbits and eccentric positions under actual conditions.

Figure 4 gives a two-dimensional schematic diagram of the rotordynamic model, assuming the rotor whirling in an elliptical orbit at any eccentric position. In the system, *O* is the housing center, is the rotor center, *O*_{1} is the whirling center, and *ω* and Ω are the rotational speed and whirling frequency of the rotor. The new coordinate system *eO*_{1}*α* is established by taking the major and minor axis of the elliptical orbit as the coordinate axis direction, where *θ* is the counterclockwise rotation angle between *eO*_{1}*α* and the original coordinate system *xOy*. After that, the *e*-axis and *α*-axis coincide with the long and short semi-axis of the elliptical orbit, and a coordinate system (*e*, *α*) is established.where *m* and *n* are the length of the major and minor axis of the elliptical orbit.

In the *eO*_{1}*α*, the rotor velocity is as follows:

In the *eO*_{1}*α*, the rotor acceleration is as follows:

For a small whirling orbit of the rotor, the dynamic model can be simplified as follows:where , and . and are the flow-induced forces and static flow-induced forces in the *e* and *α* directions. , , and are rotor displacements, velocities, and accelerations in the *e* and *α* directions. , , and are the stiffness, damping, and mass inertia coefficients.

In the transient analysis, the seal reaction forces at *t* = 0 or *t* = *T*/4 in *e* and *α* directions can be stated as follows:

The stiffness and damping coefficients in the coordinate system can be stated as follows:

##### 2.4. Case Verification

To verify the calculation accuracy and reliability of the present numerical method, a prior numerical simulation is carried out based on the experimental labyrinth seal model working with S–CO_{2} [32]. As shown in Figure 5, the inlet pressure is set at 10 MPa, and the temperature is 45°C. The leakage flow rates of 13 different pressure ratios (0.3–0.9) were calculated and compared with the experimental results, and the calculation error is less than 8%. To verify the numerical calculation method, the experimental labyrinth seal and pocket damper seal model from Ertas et al. [19] are modeled. The results show that the numerical simulation in this study shows a pretty good prediction in the direct and cross-coupled stiffness, damping, and the effective damping of the labyrinth seal and the pocket damper seal. It verifies the reliability of the numerical method in identifying the dynamic force coefficient of the seal, as shown in Figure 6.

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#### 3. Results and Discussion

The direct complex dynamic stiffness coefficient *H*_{R}, the average direct damping coefficient *C*_{avg}, the cross-coupled complex dynamic stiffness coefficient *h*_{R}, and the effective damping coefficient *C*_{eff} are the main factors to measure the rotordynamic performance of the system. Ertas [33] experiments confirmed that *K*_{xy} = −*K*_{yx}. *H*_{R}, *h*_{R}, *C*_{avg}, and *C*_{eff} are defined as follows:

##### 3.1. Seal Leakage

Figure 7 shows the velocity vector with a different rotor axial shifting (*z* = −4 mm, −2 mm, 0 mm, +2 mm, and +5 mm). Compared with the original model (no shifting), for *z* = −2 mm, the turbulent dissipation effect of the main vortex in the seal cavity is enhanced, and the additional small vortex effect is increased, which increases the energy dissipation and reduces the leakage. For *z* = −4 mm, the fluid through effect increases, the turbulent flow in the seal cavity is significantly reduced, the energy dissipation is insufficient, no additional small vortex is formed, and the leakage increases. For *z* = +5 mm and +2 mm, the energy dissipation in the additional small vortex is relatively small, and the leakage is relatively increased.

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**(b)**

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Compared with the see-through labyrinth seal, the staggered labyrinth seal rotor has a convex plate structure, and the seal blades are different in height and staggered, which greatly increases the complexity of the flow field. It not only facilitates the formation of the whirling but also enhances the effect of the jet on the wall surface, which greatly increases the energy dissipation in the flow channel and reduces the leakage flow rate.

To compare the leakage performance under different calculation conditions, the relative leakage rate *r* is defined as follows:where *M*_{i} (*i* = 1, 2, 3, …, 14) is the calculated leakage flow rate under operating conditions and *M*_{0} is the leakage flow rate of the original model.

Table 4 gives a comparison of the labyrinth seal leakage flow rate under various calculation conditions. The see-through labyrinth seal has a higher leakage flow rate of about 45.5% than the staggered labyrinth seal. The leakage flow rate of the staggered labyrinth seal for z = -2 mm is about 6% lower than that of the original model, and the leakage flow rate of the staggered labyrinth seal for b = 5.13 mm is about 5% lower than that of the original model. The leakage flow rate of staggered labyrinth seal decreases with the increasing height of rotor convex plate, the height of seal cavity, and the decrease in seal clearance.

##### 3.2. Seal Dynamic Characteristic Analysis

###### 3.2.1. Effect of the Rotor Axial Shifting Distances on the Dynamic Force Coefficients

Figures 8 and 9 display the variation of the direct complex dynamic stiffness coefficient *H*_{R} and cross-coupled complex dynamic stiffness *h*_{R} with whirling frequency for different rotor axial shifting distances. For *z* = −4 mm, the direct complex dynamic stiffness coefficient *H*_{R} is relatively low in frequency dependence. For *z* = +2 mm, the *H*_{R} is relatively high in frequency dependence. For *z* = 0 mm, the cross-coupled complex dynamic stiffness coefficient *h*_{R} is close to zero. The *h*_{R} changes from negative to positive under shifting condition, and the frequency dependence is higher than that without shifting. For *z* = −4 mm, −2 mm, +2 mm, and +5 mm, the zero point of the *h*_{R} changes from negative to positive at the whirling frequency of approximately 160 Hz, 120 Hz, 200 Hz, and 140 Hz.

The average direct damping coefficient *C*_{avg} and the effective damping coefficient *C*_{eff} vary with whirling frequencies for various rotor axial shifting distances and are depicted in Figures 10 and Figure 11. The *C*_{avg} generally appears to increase with increasing whirling frequency, with minimal variation at rotor axial shifting of −4 mm. The *C*_{avg} of the rotor with positive axial shifting is relatively high. For *z* = 0 mm, the *C*_{eff} increases with the increasing whirling frequency. The cross-coupled complex dynamic stiffness coefficient *h*_{R} has little effect on the effective damping, and the average direct damping has a dominant influence on it. For *z* ≠ 0 mm, the *C*_{eff} decreases with the increasing whirling frequency. For low whirling frequencies (<100 Hz), the effective damping is relatively high, and the stability of the rotor system is relatively strong. As the whirling frequency increases, the effective damping tends to be stable.

###### 3.2.2. Effect of the Rotor Convex Plate Heights on the Dynamic Force Coefficients

Figures 12 and 13 depict the variation of the direct complex dynamic stiffness coefficient *H*_{R} and the cross-coupled complex dynamic stiffness coefficient *h*_{R} vs. whirling frequency for different rotor convex plate heights. For *a* = 2.5 mm, the direct complex dynamic stiffness coefficient *H*_{R} is negative, and the rest are positive. With the increase in *a*, the direct complex dynamic stiffness coefficient increases. The cross-coupled complex dynamic stiffness coefficient *h*_{R} is negative for *a* = 0 mm, and its absolute value decreases with increasing whirling frequency, which has a strong frequency dependence. When *a* ≠ 0 mm, the *h*_{R} is close to zero, showing frequency independence.

The average direct damping coefficient *C*_{avg} and the effective damping coefficient *C*_{eff} vary with whirling frequencies for various rotor convex plate heights and are depicted in Figures 14 and 15. For *a* = 2.5 mm, the *C*_{avg} frequency dependence is relatively strong. For *a* = 0 mm, the *C*_{avg} decreases with increasing whirling frequency, and it increases with increasing whirling frequency for *a* = 2.0 mm and 2.5 mm. When *a* = 1.5 mm, the *C*_{avg} has a relatively low amplitude with whirling frequency. When *a* = 0 mm, the *h*_{R} has a great influence on the effective damping coefficient *C*_{eff}. When *a* ≠ 0 mm, the *C*_{avg} has a dominant influence on it. For Ω > 100 Hz, the *C*_{eff} of *a* = 2.5 mm is higher than that of other different convex plate heights. When *a* ≠ 0 mm, the *C*_{eff} increases with the increasing whirling frequency. When *a* = 0 mm, the *C*_{eff} decreases greatly with the increasing whirling frequency. For Ω > 160 Hz, the *C*_{eff} of different convex plate heights tends to be stable. For Ω < 60 Hz and *a* ≠ 0 mm, the *C*_{eff} decreases with the increasing height of convex plate. For Ω > 60 Hz and *a* ≠ 0 mm, the *C*_{eff} increases with increasing height of rotor convex plate.

###### 3.2.3. Effect of the Seal Cavity Heights on the Dynamic Force Coefficients

Figures 16 and 17 show the variation of the direct complex dynamic stiffness coefficient *H*_{R} and cross-coupled complex dynamic stiffness coefficient *h*_{R} vs. whirling frequency for different heights of seal cavities. The *H*_{R} exhibits a quadratic nonlinear change with the whirling frequency increases. The absolute value of the *H*_{R} increases with the increasing height of the seal cavity when Ω < 240 Hz, and the effect is opposite when Ω > 240 Hz. As the whirling frequency increases, the direct complex dynamic stiffness coefficient first decreases and then increases. For *h* = 5.5 mm, the *h*_{R} decreases with the increasing whirling frequency, and it fluctuates around zero for *h* = 7.0 mm and 8.5 mm, and the frequency dependence is low.

Figures 18 and 19 depict the variation of the average direct damping coefficient *C*_{avg} and the effective damping coefficient *C*_{eff} vs. whirling frequency for different heights of seal cavities. The *h*_{R} has little effect on the *C*_{eff}, and the *C*_{avg} has a dominant influence on it. The *C*_{avg} and *C*_{eff} increase with the increasing whirling frequency, and the frequency dependence is strong. At low whirling frequencies (Ω < 140 Hz), the damping coefficient generally increases with the decreasing height of the sealing cavity. At high whirling frequencies (Ω > 140 Hz), the damping coefficient generally increases with the increasing height of the sealing cavity. As the whirling frequency increases, the stability of the rotor system is enhanced, which is conducive to the safe operation of the unit.

###### 3.2.4. Effect of the Rotor Convex Plate Widths on the Dynamic Force Coefficients

The convex plate width was enlarged based on the original model. Figures 20 and 21 show the direct complex dynamic stiffness coefficient *H*_{R} and cross-coupled complex dynamic stiffness coefficient *h*_{R} vs. whirling frequency for different widths of rotor convex plate. When *b* = 6.13 mm, 5.13 mm, and 4.00 mm, the *H*_{R} is positive, changes from negative to positive and negative, and exhibits quadratic nonlinear changes. For Ω < 200 Hz, with the increase in *b*, the direct complex dynamic stiffness coefficient gradually decreases. Among them, *b* = 6.13 mm is less dependent on the whirling frequency. The *h*_{R} has low-frequency dependence and has little effect on the *C*_{eff}.

The average direct damping coefficient *C*_{avg} tends to be consistent with the *C*_{eff}, as shown in Figures 22 and 23. For *b* = 4.00 mm, the damping coefficient increases with the increasing whirling frequency. For *b* = 5.13 mm, the damping coefficient increases first and then decreases with the increasing whirling frequency. For *b* = 6.13 mm, the damping coefficient first increases with the whirling frequency and then becomes stable. The *C*_{eff} with *b* = 5.13 mm is relatively high, which is conducive to the stability of the sealing system. For Ω < 100 Hz, the *C*_{eff} with *b* = 6.13 mm is higher than 4.00 mm. Otherwise, the *C*_{eff} with *b* = 4.00 mm is higher than that with 6.13 mm.

###### 3.2.5. Effect of Sealing Clearances on the Dynamic Force Coefficients

Figures 24 and 25 depict the direct complex dynamic stiffness coefficient *H*_{R} and cross-coupled complex dynamic stiffness coefficient *h*_{R} vs. whirling frequency for different sealing clearances. The *H*_{R} exhibits a quadratic nonlinear change with the whirling frequency increases. The absolute value of the *H*_{R} decreases with the increasing sealing clearance when Ω < 180 Hz, and the effect is opposite when Ω > 180 Hz, great frequency dependence on whirling frequency, and the system stability is high at high frequencies. The *h*_{R} of *δ*_{2} = 0.4 mm is changed from positive to negative. When *δ*_{2} = 0.5 mm and *δ*_{2} = 0.6 mm, the *h*_{R} changes stably with the whirling frequency, and the frequency dependence is low.

As shown in Figures 26 and 27, the *h*_{R} has little effect on the effective damping *C*_{eff}, and the *C*_{avg} has a dominant influence on it. The *C*_{eff} increases with the increasing whirling frequency and the decreasing sealing clearance. When the sealing clearance is 0.4 mm, the *C*_{eff} is about 116%–148% of that with the clearance of 0.6 mm, and the frequency dependence decreases with the increasing seal clearance.

###### 3.2.6. Effect of the Sealing Type on the Dynamic Force Coefficients

Figures 28 and 29 depict the direct complex dynamic stiffness coefficient *H*_{R} and cross-coupled complex dynamic stiffness coefficient *h*_{R} vs. whirling frequency for different seal structures. The *H*_{R} of the see-through labyrinth seal is negative, the *H*_{R} of the staggered labyrinth seal is positive, and the see-through labyrinth seal frequency dependence is relatively strong. The *h*_{R} of the staggered labyrinth seals tends to zero, and the *h*_{R} of the see-through labyrinth seal is negative, and its absolute value decreases with increasing whirling frequency, and the frequency dependence is relatively high.

The variations of average direct damping coefficient *C*_{avg} and effective damping coefficient *C*_{eff} vs. whirling frequencies for different seal structures are depicted in Figures 30 and 31, respectively. The *C*_{avg} and *C*_{eff} of the see-through labyrinth seals decrease with the increasing whirling frequency, while the *C*_{avg} and *C*_{eff} of the staggered labyrinth seal increase with the increasing whirling frequency. For Ω < 160 Hz, the *C*_{eff} of the see-through labyrinth seal is significantly higher than that of the staggered labyrinth seal, which is about 107%–649% of the staggered labyrinth seal. Otherwise, the *C*_{eff} of the staggered labyrinth seal is about 105%–113% of the see-through labyrinth seal. The sealing system possesses better stability.

#### 4. Conclusions

This study reports a comprehensive investigation on the leakage and rotordynamic performance of the staggered labyrinth seal working with supercritical carbon dioxide. A computational fluid dynamic method is employed to establish a fully three-dimensional numerical model for the staggered labyrinth seal. An identification method based on the infinitesimal theory is applied to obtain the dynamic force coefficients. The CFD model predicts the dynamic force coefficients for the seals operating under various rotor axial shifting distances, heights/widths of rotor convex plate, heights of seal cavity, and sealing clearance conditions. Several conclusions are summarized as follows.

The *C*_{eff} of the seal with positive axial shifting is higher than that with negative shifting, and the sealing system is more stable. When the rotor with convex plate operates without axial shifting, the *h*_{R} of the staggered labyrinth seal has low-frequency dependence and has little effect on the *C*_{eff}. The *C*_{avg} has a dominant influence on the *C*_{eff}.

For Ω < 60 Hz and *a* ≠ 0 mm, the *C*_{eff} decreases with the increasing height of convex plate. For Ω > 60 Hz and *a* ≠ 0 mm, the *C*_{eff} increases with increasing height of rotor convex plate. For Ω < 140 Hz, the damping coefficient generally increases with the decreasing seal cavity height. Otherwise, the damping coefficient generally increases with increasing seal cavity height. For Ω < 160 Hz, the *C*_{eff} of the see-through labyrinth seal is about 107%–649% of the staggered labyrinth seal. For Ω > 160 Hz, the *C*_{eff} of the staggered labyrinth seal is about 105%–113% of the see-through labyrinth seal. The *C*_{eff} for the seal with the rotor convex plate width of 5.13 mm is relatively high, which is conducive to the stability of the sealing system. The *C*_{eff} increases with the decreasing seal clearance. The *C*_{eff} of 0.4 mm clearance is about 116%–148% of 0.6 mm.

The see-through labyrinth seal has a higher leakage flow rate of about 45.5% than the staggered labyrinth seal. The rotor axial shifting is -2 mm, and the width of rotor convex is 5.13 mm, which is about 6% and 5% lower than that of the original model. The leakage flow rate of the staggered labyrinth seal decreases with the increasing height of rotor convex plate, the seal cavity height, and the decreasing seal clearance.

#### Nomenclature

m, n: | Amplitudes of excitation (mm) |

C_{avg}: | Average direct damping coefficient (N·s/m) |

C_{eff}: | Effective damping coefficient (N·s/m) |

H_{R}, h_{R}: | Direct and cross-coupled complex dynamic stiffness coefficient (kN/m) |

C_{xx}, C_{yy}, C_{xy}, C_{yx}: | Direct and cross-coupled damping coefficient (N·s/m) |

R: | Seal inner radius (mm) |

h: | Cavity depth (mm) |

h_{1}: | Low blade height (mm) |

h_{2}: | High blade height (mm) |

d_{1}: | Low blade root length (mm) |

d_{2}: | High blade root length (mm) |

d: | Blade thickness (mm) |

l_{1}: | Low blade 1 and low blade 2 distance (mm) |

l_{2}: | Low blade 2 and high blade distance (mm) |

l_{3}: | High blade and low blade 1 distance (mm) |

δ_{1}: | Low blade and rotor clearance distance (low blade and convex plate clearance distance) (mm) |

δ_{2}: | High blade and rotor clearance distance (mm) |

a: | Convex plate height (mm) |

z: | Rotor axial shifting distance (mm) |

b: | Convex plate width (mm) |

K_{xx}, K_{yy}, K_{xy}, K_{yx}: | Direct and cross-coupled stiffness coefficient (kN/m) |

M_{xx}, M_{yy}, M_{xy}, M_{yx}: | Direct and cross-coupled inertial coefficient (kg) |

P_{in}: | Supply pressure (MPa) |

P_{out}: | Discharge pressure (MPa) |

T: | Temperature (K) |

ω: | Rotational speed of the rotor (RPM) |

: | Whirling frequency (Hz) |

r: | Relative leakage rate (%) |

M_{i}: | Leakage flow rate under operating conditions (kg/s) |

M_{0}: | Leakage flow rate of the original model (kg/s). |

#### Data Availability

All data included in this study are available upon request to the corresponding author.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors are grateful for the grants from the National Natural Science Foundation of China (51875361).