Labyrinth Seals Diameter and Length Effect Study on Nonlinear Dynamics

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1 Available online at SieneDiret Proedia Engineering 14 APISAT14, 14 Asia-Paii International Symposium on Aerospae Tehnology, APISAT14 Labyrinth Seals Diameter and Length Eet Study on Nonlinear Dynamis a Wensheng a, *,Huang Hai a,feng Guoquan a,chen Zhaobo b,r.g Kirk a AVIC Shenyang Aeroengine Researh Institute, Shenyang 111, China b Shool o ehatronis Engineering, Harbin Institute o Tehnology, Harbin 11, China Rotor Dynamis Laboratory, Virginia Teh, Blaksburg, VA 461, USA Abstrat The stability study has beome the most important part o rotor dynamis analysis. In this researh, labyrinth seals nonlinear dynamis model is built to evaluate the biuration rotor speed o rotor-seals system. It is important to know the aurate biuration rotor speed or prediting the instability o rotor-seals system. Labyrinth seals geometri dimension eets on system biuration are studied in this paper. Biuration rotor speed o rotor-seal system at our seals diameter and our seals lengths are alulated using Runge-Kutta method. The results show that rotor-seals system biuration rotor speed will inrease with seals diameter and seals length inreasing. 14 The Authors. Published by Elsevier Ltd. Peer-review under responsibility o Chinese Soiety o Aeronautis and Astronautis CSAA. Keywords: Labyrinth seals, nonlinear, dynamis model, Runge-Kutta method 1.Introdution Rotating mahines suh as ompressors, turbines, pumps, jet engines, turbohargers, et are subjet to rotor-seal system stability. The labyrinth seal has many advantages, suh as simpliity, reliability, and tolerane or high temperature and high pressure. The labyrinth seal has improved over the years. Originally, most o the work oused upon eperimental data and simpliied rotor dynami systems. Alord[1] studied a labyrinth seals with varying inlet * Corresponding author. Tel.: address: mawensheng198@gmai.om The Authors. Published by Elsevier Ltd. Peer-review under responsibility o Chinese Soiety o Aeronautis and Astronautis CSAA.

2 a Wensheng/ Proedia Engineering 14 and outlet areas. He onluded that when the inlet area eeeds the outlet area, an eitation ore is produed that auses whirl in the diretion o the rotation. Conversely, Alord onluded that no eitation ore would be produed when the outlet area eeeds the inlet area. Wyss ann et al.[,3] used a two ontrol volume approah to model the low through a labyrinth seals. The two ontrol volume approah is a simpliied approah to a inite volume method. For that reason, the modeled low through the seals was oversimpliied and ould only be used or general omparisons and trends. Kirk[4] developed a bulk ode to determine the dynami oeiients o a labyrinth seals, the ode was alled DNLAB, but reently has been renamed LabyL. The paper presents a theoretial study o a Jeott rotor-seal system. The nonlinear dynamis model o unbalane rotor-seal system is established in this paper. The ollowing results will demonstrate the inluene o seals struture on the biuration. The omparison biuration harts in dierent parameters o rotor-seal nonlinear system is analyzed in this paper..labyrinth seals nonlinear dynamis model analysis Labyrinth seals an be used to redue leakage rom high pressure region into lower pressure region. A rotor-seal system is shown in Fig.1. Then the equations o motion an be written in Eq. 1. Dynamis model: m m y uszynska seals model: F F y K m τ ω τωd τωd K m τ ω It an get Eq. 3 by Eq. 1 and Eq. m m m m y y D τ m ω τ m ω D Fig.1 Jeott rotor-seal system y y y F F y D τ m ω τ m ω D τ ω τωd τωd τ ω mg mrω y y m osωt sinωt m y mrω osωt mg mrω sinωt Deine the dimensionless parameters o oordinates:, y, wt t, d w d dt, d dt dt, y,, y. Assume m m,and Eq. 3 beome ω ω D τ m ω τ m ω D ω ω τ ω τωd τωd K e τ ω mrω osωt mg mrω sinωt d w, dt 1 3 4

3 a Wensheng/ Proedia Engineering D ω τ m τ m D ω Deine the dimensionless parameters: τ ω τ D ω τ D ω τ ω mr osωt mg mrω sinωt ω De D t m Ke K - mt w t D mg mr D1, D, K1, K, G, r 6 w w w w The papameters are into Eq D 1 D D D 1 K 1 K ρ osωt 7 K K 1 G ρ sinωt Then, Eq. 7 beome Assume [ ] T 1 K 1 D 1 K D 1 K D K 1 D 1 T,and [ ] T System state equation is obtained rom Eq K 1 D 1 K D K D K 1 D 1 Dynamis parameters solution: ρ osωt G ρ sinωt T ρ osωt G ρ sinωt - n - n 1 K K1 - e, D D1 - e, n ~ 8 1 t - b t 1 e, b < < 11 K mm 3, D mm 1 3T, m mm 3T 1 s m 1 1 s E - m σ E µ 1 1ξ σ σ B 1 6 E The dynamis parameters, rom Eq.6, Eq. 1 to Eq. 16, are s 1 m 1 s 6 E 1 p RDP m3 16 l

4 4 a Wensheng/ Proedia Engineering 14 Where / D 1 D ω σ E 1ξ σ σ B 1 6 E * σ 1, 1ξ σ 6 E *,T 1 e σ 1 m 1ξ σ 6 E π RΔP * T,ω λ D τ m τ 1 e b σ 1 1ξ σ 6 E *, π RΔP T λ / σ 1 m 1ξ σ 6 E *, π RΔP T 3 1 λ 4 K 1 K K m e τ ω σ 1ξ σ E1 m π RΔP 1 e n λ / σ 1 m 1ξ σ 6 E *, π RΔP T 3 1 λ 4 K τ D ω τ 1 e b D 1 e n σ 1 m 1ξ σ 6 E π RΔP * T,ω λ σ 1 1ξ σ 6 E *, π RΔP T τ λ 1 e b ω σ 1 m 1ξ σ 6 E π RΔP * T,ω 1 λ l l 1 s s n 17, D P 1 v 18 1m m λ n R a 1 R v 19 R a 1 z E, B - v / a v a R R - m 1 z s R / R 1 v R R d wd a, Rv m 1 m 3.Labyrinth seals nonlinear dynamis analysis The labyrinth seals geometri dimensions are shown in Table 1 Table1 Seals geometri dimensions Dimensions Value Dimensions Value Diameter, R mm Gas aial speed, v 3m/s Radial Clearane, Cr.3mm Inlet loss, ξ.1 Seals length, L Rotor damp,de N/mse 76mm Seals mass, m kg Rotor stiness, Ke 1 7 N/m

5 3.1.Seals diameter eet study on biuration a Wensheng/ Proedia Engineering 14 Fig. is the seals diameter eet on rotor-seal system biuration. Fig. a is the biuration when the seals diameter is mm, rotor-seal system is synhronous motion when rotor speed is less than 4 rad/s. Rotor-seal system is quasi-periodi motion when 4rad/s<w<61rad/s. Rotor-seal system is subsynhronous motion with two-period when 61rad/s<w<638rad/s, and it will beome quasi-periodi motion when ω is more than 638rad/s. Fig. b is the biuration when the seals diameter is 3mm, rotor-seal system is synhronous motion when rotor speed is less than 7 rad/s. Rotor-seal system is quasi-periodi motion when 7rad/s<w<616rad/s. Rotorseal system is subsynhronous motion with two-period when 616rad/s<w<66rad/s, and it will beome quasiperiodi motion when w is more than 66rad/s. Fig. is the biuration when the seals diameter is 3mm, rotor-seal system is synhronous motion when rotor speed is less than 6 rad/s. Rotor-seal system is quasi-periodi motion when 6rad/s<w<63rad/s. Rotorseal system is subsynhronous motion with two-period when 63rad/s<w<68rad/s, and it will beome quasiperiodi motion when w is more than 68rad/s. Fig. d is the biuration when the seals diameter is 4mm, rotor-seal system is synhronous motion when rotor speed is less than 617 rad/s. Rotor-seal system is quasi-periodi motion when 617rad/s<w<6rad/s. Rotorseal system is subsynhronous motion with two-period when 6rad/s<w<68rad/s, and it will beome quasiperiodi motion when w is more than 699rad/s. a Seals diametermm b Seals diameter3mm Seals diameter3mm d Seals diameter4mm Fig. Seals diameter eet study on biuration Table Nolinear motion proess under dierent seals diameter Seals diameter Nolinear motion proess Speed o quasi-periodi mm Sà APà DPà AP 638rad/s mm Sà APà DPà AP 66rad/s mm Sà APà DPà AP 68rad/s mm Sà APà DPà AP 699rad/s Synhronous motion---s; Double period motion---dp; Quasi-periodi motion---ap;

6 6 a Wensheng/ Proedia Engineering Seals length eet study on biuration Fig.3 Seals diameter eet study on quasi-periodi motion Fig.4 is the seals length eet on rotor-seal system biuration. Fig.4 a is the biuration when the seals length is 7mm, rotor-seal system is synhronous motion when rotor speed is less than 7 rad/s. Rotor-seal system is quasi-periodi motion when 7rad/s<w<6rad/s. Rotor-seal system is subsynhronous motion with two-period when 6rad/s<w<638rad/s, and it will beome quasi-periodi motion when w is more than 638rad/s. Fig.4 b is the biuration when the seals length is 1mm, rotor-seal system is synhronous motion when rotor speed is less than 9 rad/s. Rotor-seal system is quasi-periodi motion when 9rad/s<w<6rad/s. Rotor-seal system is subsynhronous motion with two-period when 6rad/s<w< 66rad/s, and it will beome quasi-periodi motion when w is more than 66rad/s. Fig.4 is the biuration when the seals length is 1mm, rotor-seal system is synhronous motion when rotor speed is less than 631 rad/s. Rotor-seal system is quasi-periodi motion when 631rad/s<w<69rad/s, and it will beome quasi-periodi motion when w is more than 69rad/s. Fig.4 d is the biuration when the seals length is mm, rotor-seal system is synhronous motion when rotor speed is less than 749 rad/s, and it will beome quasi-periodi motion when w is more than 6749rad/s. From Table 3 and Fig., it shows that quasi-periodi rotor speed will inrease with the seals length in reasing. Rotor-seal system has not irt quasi-periodi motion when seals diameter is more than.7, and it only has synhronous motion and seond quasi-periodi motion when seals length is more than mm. a Seals length7mm b Seals length1mm Seals length1mm d Seals lengthmm Fig.4 Seals length eet study on biuration

7 a Wensheng/ Proedia Engineering 14 7 Table3 Nolinear motion proess under dierent seals length Speed o quasiperiodi Seals length Nolinear motion proess 7mm Sà APà DPà AP 638rad/s 1mm Sà APà DPà AP 66rad/s 1mm S à DPà AP 69rad/s mm S à AP 749rad/s Synhronous motion---s; Double period motion---dp; Quasi-periodi motion---ap; Fig. Seals diameter eet study on quasi-periodi motion 4.Conlusions The paper presents the impat o labyrinth nolinear dynamis motion proess by mathematial methods. The study results and disussion give an enhaned ause-eet understanding o pressure eet. It was ound that the dierene o seals diameter and seals length are important in determining hanges o rotor-seal system biuration. The rotor speed o quasi-periodi will inrease with the seals diameter and seals length inreasing. Rotor-seal system is more stability when quasi-periodi rotor speed is greater. The labyrinth seals non-linear dynamis study results have shown that the seals diameter and seals length are beneiial to system stability. Aknowledgements This work was supported by the key projet o National Nature Siene Foundation o China No.1634, the projet o National Nature Siene Foundation o China No.1874, No.187, No.949 and No Reerenes [1]Alord, J. S 196. Proteting turbomahinery rom sel-eited rotor whirl. Journal o Engineering or Power, v. 87, []Kirk, R.G Evaluation o aerodynami instability mehanisms or entriugal ompressors part II: advaned analysis. Journal o Vibration, Aoustis, Stress, and Reliability in Design, v. 11, 7-1. [3]R.G. Kirk and Z.Guo. 9. Inluene o leak path rition on labyrinth seals inlet swirl. Tribology Soiety o Tribologists and Lubriation Engineers4, v., [4]Wyssmann, H.R., Pham T.C., Jenny R.J Predition o stiness and damping oeiients or entriugal ompressor labyrinth seals. Journal o Engineering or Gas turbines and Power, v.16, []Iwatsubo, T. Evaluation o instability ore o labyrinth seals in turbines or ompressors. Rotordynami Instability Problems in Highperormane Turbomahinery. NASA CP No. 198, proeedings o workshop held at Teas A University. [6]Childs, D. W. and Sharrer, J. K. An Iwatsubo-based Solution or Labyrinth Seals: Comparison to Eperimental Results, Journal o Engineering or Gas Turbines and Power1986, Vol. 18, pp