Improving the rotordynamic stability of short labyrinth seals using positive preswirl
Wanfu Zhang^{1} , Qianlei Gu^{2} , Hao Cao^{3} , Yingfei Wang^{4} , Lu Yin^{5}
^{1, 2, 4, 5}School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, P. R. China
^{1}Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, Shanghai, 200093, P. R. China
^{3}State Grid Hunan Electric Power Company Limited Research Institute, Changsha, 410007, P. R. China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 22, Issue 6, 2020, p. 12951308.
https://doi.org/10.21595/jve.2020.21131
Received 29 October 2019; received in revised form 22 March 2020; accepted 14 April 2020; published 30 September 2020
JVE Conferences
Introducing a negative preswirl at the upstream of annular gas seals has been considered as an effective way to improve the system stability. This paper demonstrates a stability enhancement approach for a short labyrinth seal using positive preswirls. The static and dynamic characteristics of the labyrinth seal with various blade numbers (5, 10, 15), inlet preswirl ratios (–0.3, –0.15, 0, 0.15, 0.3) were studied. Results show that the inlet preswirl ratio has a dramatic effect on the circumferential location of the highpressure spot for each seal cavity, particularly for the first cavity. The inlet preswirl ratio has opposite effects on the system stability due to the difference of highpressure spot locations between the first cavity and the others. An increasing positive inlet preswirl could improve the system stability for the labyrinth seal with fewer blades (e.g. 5 blades). Its characteristics is mainly dominated by the first seal cavity. For the labyrinth seal with 10 blades, the system characteristics shows slight dependency on the inlet preswirl ratio. For the labyrinth seal with more blades (e.g. 15 blades), the negative inlet preswirl still increases the system stability, which agrees with the conventional conclusion. The paper provides a deeper understanding on the stability improvement of the labyrinth seal.
 A stability enhancement approach for a short labyrinth seal could be realized by using positive preswirls.
 Inlet preswirl ratios has opposite effects on the system stability due to the difference of high pressure spot locations between the first cavity and the others.
 For the labyrinth seal with some centain blades, the system characteristics shows slight dependency on the inlet preswirl ratio.
Keywords: labyrinth seal, preswirl, computational fluid dynamics, rotordynamic coefficient.
1. Introduction
Labyrinth seals are used extensively in turbomachines to minimize the leakage flow rate by separating high pressure fluid from entering into a lowpressure region. Moreover, the rotor stability is threatened by the labyrinth seal due to the driving forces generated in the seal cavity [1]. Proposing effective approaches to improve the dynamic stability of labyrinth seals is crucial for the safety and stable operation of turbomachines [2].
Instability phenomenon resulting from labyrinth seals was reported firstly by Den Hartog [3] in 1950s. Then Thomas [4] presented a physical explanation of the occurrence of selfexcited vibrations through clearance flows at the blade tips and interstage glands. In 1965, Alford [5] put forward the occurrence of a circumferential variation of static pressure acting on the cylindrical surface of a rotor particularly within labyrinth seals, which results in the aerodynamic exciting force on jet turbo compressors and turbines. It was concluded that unequal radial clearances at the entry and discharge have a significant effect on the excitation. However, the rotational motion of the rotor was neglected in the stability analysis. In 1974, Rosenberg et al. [6] indicated that lateral forces can be produced due to the inlet preswirl and rotating surface in case of parallel clearances when an eccentricity exists with or without the occurrence of spiral flow. Until 1970s1980s, lots of analytical work [711] and experimental measurements [1214] were consecutively carried out to predict the leakage flow rate, circumferential flow, pressure distribution qualitatively.
To suppress the circumferential velocity component and improve the rotordynamic characteristics of labyrinth seals, two methods are generally used. The first approach is setting up swirl brakes at the upstream of the seal. The second one is utilized by means of shunt injection. Circumferential flows [12, 15, 16] due to swirl brakes, shunt injection, circumferential entry velocity, and rotational shaft surface at the seal inlet have a significant effect on the system stability. It can be usually categorized into zero, positive, negative preswirl in terms of the circumferential entry direction.
In 1980, Benckert and Wachter [12] firstly reported that the inlet flow direction of the seal in the same direction of shaft rotation would destabilize the rotorseal system, while the reversing preswirl at the seal inlet can effectively improve the stability by offsetting the circumferential flow. This phenomenon has drawn many research attentions on the negative preswirl. In 1986, Kirk [17] firstly studied the effect of the reverse jet (shunt injection) on unstable compressors and successfully overcame the subsynchronous vibration of centrifugal compressors. In 1990s, Memmott [18, 19] employed a shunt injection method to solve problems of subsynchronous vibrations of highparameter centrifugal compressors. Shunt injection has proven to be dynamically beneficial in many cases [20], but there have been some cases in which they were not satisfactory. In 1999, Soto and Childs [21] experimentally studied the effect of shunt injection on the dynamic characteristics of a 20tooth labyrinth seal by removing the fourth tooth. The effects of pressure ratio, rotational speed and injection direction (radial and against rotation) on seal stability were compared. It is found that the jet against rotation can effectively reduce the crosscoupled stiffness coefficient under various operating conditions and improve the effective damping of the labyrinth seal. However, a 100 % increase in the leakage flow rate was found for measured cases due to the presence of shunt injection. In 2003, Kim et al. [22] developed a quasi threedimensional CFD perturbation model for a 19tooth labyrinth seal with shunt injection at any axial location. The axial location of the injection appears to have only a slight effect on rotordynamic stability for the considered cases. In 2007, Gans [23] presented the effects of reversedswirl brakes in a combined labyrinthbrush seal. The reversedswirl brakes are intended to induce fluid rotation against shaft rotation and reduce the destabilizing effects of a combined labyrinthbrush seal. In 2012, Brown and Childs [24] investigated the rotordynamic coefficients for a holepattern seal with negative preswirl. Results show that the negative preswirl could reverse the sign of crosscoupled stiffness coefficients and significantly increase the effective damping coefficient. In 2016, Childs et al. [25] studied rotordynamic performance of negativeswirl brakes for a 16tooth labyrinth seal using both prediction and experimental measurements. The negativeswirl brake produces a change in the sign of the crosscoupled stiffness coefficient with an appreciable magnitude, and the stability of forward precessional modes would be enhanced. In 2018, Untaroiu et al. [15] studied on the geometry of negativeswirl brake in order to suppress the circumferential flow and improve the stability of a 16tooth labyrinth seal more effectively.
Published literatures demonstrated that the prerotation into a labyrinth seal in the direction of the rotor rotation generally creates a positive crosscoupled stiffness that would destabilize a rotor. Similarly, the prerotation against the direction of shaft rotation produces a negative crosscoupled stiffness coefficient that would stabilize the rotor. Note that the circumferential pressure distribution in the seal cavity could be different from that in the oil film of journal bearings. A highpressure spot always appears in the circumferential convergentclearance for a journal bearings. However, the highpressure spot in the labyrinth seal may vary for each seal cavity. Some static instability phenomena [26, 27] indicate the highpressure spot appears in the circumferential divergentclearance of the seal cavity. Rare publications study the detailed pressure distribution for each seal cavity and discuss the contribution of each seal cavity on the overall stability, particularly for a short seal including few teeth working with preswirls. It is of interest to further study the rotordynamic performance of the labyrinth seal considering different preswirls.
This research investigates the effects of two negative preswirls, zero preswirl, two positive preswirls on the seal reaction force of each seal cavity and rotordynamic coefficient for a short labyrinth seal. Threedimensional flow in the short labyrinth seal is simulated by solving the Reynoldsaveraged NavierStokes (RANS) equations in a commercial software [28]. The circumferential pressure distribution for each cavity is analyzed for various whirling frequencies. The static and dynamic force coefficients are further investigated to evaluate the stability characteristics.
2. Rotordynamic model of the labyrinth seal
Fig. 1 depicts the rotordynamic model for annular gas seals. In the coordinate system ($x$, $y$), $O$ is the stator center, ${O}^{\text{'}}$ is the rotor center. $t$, $r$ denote the tangential and radial direction, respectively. The whirling orbit of the rotor can be approximately assumed as a cycle with a radius $e$ [29]. The rotational and whirling speed are $\omega $, $\mathrm{\Omega}$, respectively. The seal reaction force can be divided into two components, ${F}_{x}$, ${F}_{y}$ in two orthogonal directions.
Fig. 1. Rotordynamic model for annular gas seals
According the firstorder perturbation of the bulkflow governing equations by Childs [1, 30], the seal reaction force is significantly more linear function of displacement than bearings. The linearized forcedisplacement model for seals can be stated as:
Under small perturbations about a centered position, the seal reaction force for a gas seal can be expressed by Eq. (2):
Generally, the direct stiffness coefficient $K$ influences the system stiffness and critical speed of the rotor system. The tangential force characterized by the crosscoupled stiffness coefficient $k$ is a major contributing factor to the nonsynchronous whirl. The direct damping coefficient $C$ reflects the capacity to suppress the unstable whirling motion. An increase of $C$ is benefit for the system stability. The crosscoupled damping coefficient $c$ is insignificant to the rotor seal system.
The moving reference frame (MRF) [31] is employed to identify the dynamic characteristics of the annular seal. As shown in Fig. 2, when the rotor motion is observed from a stationary frame of reference, the relative position of rotor and stator seems to be changing all the time. Therefore, the moving grid and transient analysis are necessary. From a moving frame of reference with $\mathrm{\Omega}$, the rotor becomes to be a relatively stationary position. In this case, the analysis can be simplified as a steady state problem. Compared with the transient analysis, MRF method could save more computing resources with finer mesh grids. Particularly, this method is suitable for the analysis of the labyrinth seal [31].
Fig. 2. Schematic diagram of the quasisteady state model
The orbit equation of the whirling rotor in the ($x$, $y$) coordinate system is defined by Eq. (3):
In the rotating coordinate system, the radial component ${F}_{r}$ and the tangential component ${F}_{t}$ of the seal reaction force acting on the rotor can be expressed as Eq. (4):
Eq. (4) shows that the radial force ${F}_{r}$ and tangential force ${F}_{t}$ are composed of stiffness and damping terms. When the whirling frequency equals to 0 Hz, only the stiffness term plays a role. The crosscoupled stiffness coefficient $k$ can be expressed as Eq. (5):
3. Numerical method
3.1. Geometrical parameters
Fig. 3 shows the cross section of the test labyrinth seal. Table 1 gives its detailed dimensions. The test seal includes 10 blades and 9 cavities (#1~#9). In order to study the effect of the blade number on the dynamic characteristics, the labyrinth seal with 5 and 15 blades are also employed in the numerical simulation.
Fig. 3. Twodimensional geometry of the test labyrinth seal
Table 1. Seal dimensions
Parameter

Value

Rotor diameter $d$ /mm

60

Radial clearance ${C}_{r}$ / mm

0.2

Cavity width ${w}_{1}$ / mm

3.8

Cavity depth $h$ / mm

3.5

Blade thickness $t$ / mm

0.25

Blade space width ${w}_{2}$_{}/ mm

2.3

Number of blades

5, 10, 15

3.2. Numerical model
A commercial CFD software [28], which solves the compressible Reynoldsaveraged NavierStokes equations, is employed in this paper. The standard $k$$\epsilon $ is used as the turbulence model with the turbulence intensity of 5 %. The scalable wall function method combines the physical quantity of the wall and the highlevel turbulent zone. And the value of ${y}^{+}$ is controlled within 30200. In order to investigate the effect of mesh densities for an accurate prediction of the rotordynamic coefficients, coarse and fine mesh grids were tested for the labyrinth seal. The fine mesh placed more nodes in radial and circumferential direction. For the labyrinth seal with 10 blades, the size of the coarse mesh and fine mesh are 2.4×10^{6} nodes and 5.6×10^{6} nodes, respectively. The difference of rotordynamic coefficients predicted by the coarse and fine mesh is under 1.68 %. Therefore, for the studied labyrinth seal in this paper, the mesh with nodes 2.4×10^{6} is accurate enough for the investigation. The grid distribution for the simulation(coarse) is shown in Fig. 4.
Fig. 4. Grid distribution
3.3. Boundary conditions
Table 2 lists the detailed parameters for CFD calculation. The fluid (air) is assumed to be ideal gas. The inlet and outlet boundary were placed at the upstream of the inlet extension and the downstream of the outlet extension, respectively. The walls of rotor and stator are defined to be adiabatic, smooth and no slip. The total pressure and temperature are defined at the inlet boundary, while the average static pressure is specified at the outlet. The inlet preswirl ratio is defined as Eq. (6):
where, ${V}_{\theta}$ is the circumferential velocity at the seal inlet.
The desired convergent target of each numerical simulation is that the RMS residuals of the momentum and mass equations, energy equations, and turbulence equations are below 10^{6}.
Table 2. Calculation conditions
Simulation condition

Value

Fluid

Air (ideal gas)

Turbulence model

Standard $k$$\epsilon $

Wall properties

Adiabatic, smooth surface

Temperature $T$ /K

287

Supply pressure ${P}_{in}$_{}/ bar

6.9

Discharge pressure ${P}_{out}$_{}/ bar

1

Inlet preswirl ratio $\lambda $

–0.3, –0.15, 0, 0.15, 0.3

Rotational speed $\omega $ /RPM

15000

Whirling frequency $\mathrm{\Omega}$ /Hz

0, 50, 100, 150

Whirling orbit

Circle orbit

Whirling radius $e$ /mm

0.01

3.4. Validation
In order to validate the reliability of the CFD simulation, this paper measured the static seal force acting on the rotor with various eccentric ratios, and compared the numerical results with experiments. Fig. 5 shows the experiment facility.
Fig. 5. Gas facility, experiment part and testing system
Fig. 6 shows detailed experiment device of the rotorseal system. Two hydrodynamic bearings are used to support the rotor. The rotor diameter of the test section is 60 mm. The stator is supported by eight springs in vertical and horizontal directions. The labyrinth seal contains 10 blades, and the detailed dimension is shown in Fig. 3 and Table 1. Four inlet ports are arranged symmetrically around the middle plane of the cylindrical stator surface. Highpressure gas enters the seal from four centered inlet ports and discharges into the atmosphere. The inlet pressure is ~6.9 bar, and the rotational speed is zero. In the experiment, the zero calibration for all the test instruments is guaranteed to avoid the errors caused by their deviation.
Fig. 6. Experimental device of the rotorseal system: 1. Rotor, 2. #1 Journal bearing, 3. Turbine oil, 4. Stator, 5. Elastic support, 6. Inlet pipe, 7. #2 Journal bearing, 8. Displacement sensor, 9. Dial indicator, 10. Pressure sensors, 11. Balance disk, 12. Tachometric transducer, 13. Inlet port, 14. Blades, 15. Elastic metal insert
To improve the identification accuracy, the seal housing is supported by eight elastic springs in both horizontal and vertical directions. The dynamic response of the stator can be amplified by the supporting springs. By multiplying the stator displacement with the spring stiffness, the seal force acting on the rotor can be obtained. The stator displacement is measured by averaging 10 tests under the same working condition. Table 3 gives the comparison of the static experimental force and the results obtained by the CFD method. Because of the machining accuracy of the rotor and seal, a weak seal force can be found even the eccentric ratio $\epsilon $ is zero, which is small enough to be neglected. The difference between experiment and CFD simulation is less than 2.9 %, which verifies the accuracy of the current CFD simulation.
Table 3. Comparison of the seal reaction force
Eccentric ratio $\epsilon $

EXP / N

CFD / N

Error

0

0.01

0

1.0 %

0.05

0.42

0.41

2.4 %

0.10

0.79

0.78

1.6 %

0.15

1.12

1.16

2.9 %

0.20

1.65

1.63

1.9 %

4. Results and discussion
4.1. Static characteristics
The crosscoupled stiffness coefficient $k$ is one of the main criteria to evaluate the system stability. For a steady flow, the $k$ is the function of the tangential seal reaction force, as shown in Eq. (4). Fig. 7 shows the tangential seal reaction force ${F}_{t}$ vs. preswirl ratios for the labyrinth seal with different blade numbers. With the increasing preswirl ratio, the tangential seal reaction force of the labyrinth seal with 5 blades exhibits an opposite trend compared with that of the labyrinth seal with 15 blades. For the labyrinth seal with 5 blades, the tangential seal reaction force decreases with the increasing preswirl ratio, and corresponding crosscoupled stiffness coefficient $k$ becomes to be negative. From the stability viewpoint, the increasing preswirl ratio tends to enhance the stability. This result shows a contrary trend to the conventional conclusion. The tangential seal reaction force of the labyrinth seal with 15 blades increases with the rising preswirl ratio. The crosscoupled stiffness coefficient changes in signs from negative to positive, which processes the decreasing stability. One notable phenomenon for the labyrinth seal with 10 blades is that it shows little dependency on preswirl ratios and keeps an almost constant value. Above results show the tangential seal reaction force is greatly correlated with the blade number. The seal reaction force from each seal cavity is extracted to clarify the phenomenon in following section.
Figs. 810 show the tangential seal reaction forces inside each cavity for the labyrinth seal with 5, 10, 15 blades, respectively. For #1 cavity, the increasing preswirl decreases the tangential seal reaction force. However, the effect of preswirl on the seal reaction force inside #2~#14 cavity shows an opposite trend, and the increasing preswirl ratio tends to produce a larger seal reaction force.
Fig. 7. Tangential seal reaction forces ${F}_{t}$ vs. preswirl ratios
Fig. 8. Tangential seal reaction forces ${F}_{t}$ vs. seal cavities (5 blades)
Fig. 9. Tangential seal reaction forces ${F}_{t}$ vs. seal cavities (10 blades)
Fig. 10. Tangential seal reaction forces ${F}_{t}$ vs. seal cavities (15 blades)
For the labyrinth seal with 5, 10, 15 blades, the preswirl mainly influences the fluid flow in #1 cavity, and it has a relatively minor effect on the flow in other seal cavities. Hence, when the number of seal cavities is relatively small (such as 5 blades in this paper), the changing trend of the tangential seal reaction force acting on the whole rotor largely depends on the force variation inside #1 cavity.
When the cavity number becomes relatively large (such as 15 blades in this paper), the tangential force inside #2~#14 cavities dominates the whole seal. The labyrinth seal with 10 blades shows little dependency on the preswirl ratio. This can be attributed to the counteracting effect of the flow forces between #1 cavity and other cavities.
Fig. 11 shows the distribution of highpressure spots for the labyrinth seal with various blade numbers ($\lambda =$0). $\theta $ is the circumferential angle of high pressure spots in spatial polar coordinate system. All highpressure spots inside #1 cavity locate at the first quadrant (divergent gap), while $\theta $ in other cavities becomes more than 150°. Along the leakage direction, the highpressure spot for each cavity continuously moves toward the rotational direction.
Fig. 11. High pressure spots vs. seal cavities without inlet preswirl
Fig. 12. Effects of high pressure spots on the tangential seal reaction force
Fig. 12 depicts the direction and magnitude of the tangential seal reaction force due to the changing position of highpressure spots affects. As the highpressure spot locates in the first and second quadrants, the tangential seal reaction force shows a negative sign, which is consistent with the results for the labyrinth seal with 5 and 10 blades shown in Fig. 7. The shifting of high pressure spots inside cavities (except for the #1 cavity) from the second quadrant to the third quadrant results in a sign change of the tangential seal reaction force from negative to positive, which agrees with the variation of tangential seal reaction forces in Fig. 10. The main reason for the variation of the seal reaction force is that the circumferential location of the highpressure spot for each seal cavity varies along the leakage direction.
Taking the labyrinth seal with 15 blades as an example, the variation of highpressure spots for #1, #3, #7, #13 cavity is shown in Fig. 13. The pressure high spot shifts toward the inlet preswirl direction. Additionally, the inlet preswirl has a more dramatic effect on the seal cavity close to the inlet. Combining with Fig. 11, it’s obvious that the highpressure spots of #1 cavity locate at the first and fourth quadrants. While the highpressure spots of #2~14 cavities locate at the third and fourth quadrants.
Fig. 13. Position of highpressure spots vs. preswirl ratios
Fig. 14 presents the effect of various highpressure spots on the tangential seal reaction force. Fig. 14(a) depicts the transition of the highpressure spot in different quadrants. Fig. 14(b) shows the projection of the seal reaction force in the tangential direction ($t$ axis). According to the distribution of highpressure spots, the cavities, for which the high pressure spot locates at the first and fourth quadrant, are classified as Type A Cavity (e.g. # 1 cavity). And the other cavities, for which the highpressure spot locates at the second and third quadrant, are classified as Type B Cavity (e.g. #2#14 cavities). When the preswirl ratio is positive ($\lambda >$ 0), the tangential seal reaction force in the Type A cavities shifts to be negative, while that in the Type B cavities converts to be positive. When the inlet preswirl ratio is negative ($\lambda <$ 0), the tangential seal reaction force inside both Type A and B cavities show the opposite result. This phenomenon is consistent with the variation trend of tangential seal reaction forces shown in Fig. 10.
Fig. 14. Effects of high pressure spot shifts caused by preswirls on the tangential seal reaction force
a) Hight pressure spot position
b) Projection of seal reaction force in $t$ axis
4.2. Dynamic characteristics
The effective damping coefficient ${C}_{eff}$ is a main parameter to evaluate the system stability [32, 33], which is defined as follows:
where ${F}_{t}$, $e$, $C$, $k$, denote the tangential seal reaction force, radius of whirling orbit, direct damping coefficient, crosscoupled stiffness coefficient, respectively. The direct damping coefficient $C$ reveals the capacity for reducing the rotor whirling motion. The crosscoupled stiffness coefficient $k$ shows the destabilization effect. As a result, the effective damping coefficient ${C}_{eff}$ could be employed to depict the resultant effect.
Fig. 15 presents the tangential seal reaction force inside seal cavities (15 blades) with whirling frequencies 0 Hz, 50 Hz, 100 Hz and 150 Hz. With the increasing whirling frequency, the seal reaction force in the tangential direction for each cavity becomes toward negative gradually. This is mainly because the increasing direct damping effect can suppress the whirling motion.
Fig. 15. Distribution of the tangential seal reaction force inside seal cavities with different whirling frequencies (15 blades)
Fig. 16 shows the distribution of tangential seal reaction forces inside the cavities with a whirling frequency 50 Hz. It’s obvious that the responses of the seal reaction force inside the cavities working in dynamic conditions are similar with that working in the static condition.
Fig. 17 shows the effective damping coefficient ${C}_{eff}$ versus inlet preswirl ratios with a whirling frequency 50 Hz. The labyrinth seal with different blade numbers differs greatly in the response to preswirl ratios. The increasing preswirl ratios enhances the stability of the labyrinth seal with 5 blades, while the labyrinth seal with 15 blades shows the opposite trend. The labyrinth seal with 10 blades shows the lowest preswirl dependency.
Fig. 16. The tangential seal reaction force in seal cavities under different preswirl ratios (50 Hz)
Fig. 17. Effective damping coefficients for different blades numbers vs. preswirl ratios (50 Hz)
Under static and dynamic conditions, the stability of the labyrinth seal with 5 blades increases with the rising preswirl ratio, which is contrary to the traditional viewpoint. The main reason is that the circumferential position of the high pressure spot inside the #1 cavity is different from the others, which results in the different effect of the preswirl on the tangential force. When the blade number is small (e.g. 5 blades), the effect of preswirl on #1 cavity plays a dominant role, and the high pressure spot inside #1 cavity appears in the divergence gap. Meanwhile, the positive preswirl is conducive to improving the overall stability of the seal system. When the blade number is large, the effect of preswirl on #1 cavity is reduced by the other cavities, and the high pressure spot mainly appears in the convergence gap. The overall tangential seal reaction force is dominated by the downstream cavities, and the negative preswirl is beneficial to improve the system stability.
5. Conclusions
In this paper, the effect of inlet preswirls on the static and dynamic stability of the labyrinth seal with different blade numbers is studied based on experimental test and numerical simulation. Following conclusions are summarized.
The variation of the highpressure spot for each seal cavity contributes to the change of tangential seal reaction forces in magnitude and direction. The inlet preswirl has a dramatic effect on the circumferential position of the highpressure spot. For the labyrinth seal with different blade numbers, the difference of highpressure spot position between #1 cavity and the others results in an opposite effect of the inlet preswirl ratio on the system stability.
For the labyrinth seal with fewer blades (e.g. 5 blades in this paper), a positive inlet preswirl could improve the system stability. For the labyrinth seal with 10 blades, it shows little dependency on preswirl ratios. For the labyrinth seal with more blades (e.g. 15 blades in this paper), a negative inlet preswirl will increase the system stability.
Acknowledgements
The authors are grateful for the Grants from the National Natural Science Foundation of China (51875361).
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