CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 29,aNo. 6,a2016

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DOI: 10.3901/CJME.2016.0617.074, available online at www.springerlink.com; www.cjmenet.com

Gas Film Disturbance Characteristics Analysis of High-Speed and High-Pressure Dry Gas Seal

CHEN Yuan, JIANG Jinbo, and PENG Xudong*

Engineering Research Center of Process Equipment and Its Re-manufacturing of Ministry of Education, Zhejiang University of Technology, Hangzhou 310000, China

Received December 9, 2015; revised May 9, 2016; accepted June 17, 2016

Abstract: The dry gas seal(DGS) has been widely used in high parameters centrifugal compressor, but the intense vibrations of shafting,

especially in high-speed condition, usually result in DGS’s failure. So the DGS’s ability of resisting outside interference has become a

determining factor of the further development of centrifugal compressor. However, the systematic researches of which about gas film

disturbance characteristics of high parameters DGS are very little. In order to study gas film disturbance characteristics of high-speed

and high-pressure spiral groove dry gas seal(S-DGS) with a flexibly mounted stator, rotor axial runout and misalignment are taken into

consideration, and the finite difference method and analytical method are used to analyze the influence of gas film thickness disturbance

on sealing performance parameters, what’s more, the effects of many key factors on gas film thickness disturbance are systematically

investigated. The results show that, when sealed pressure is 10.1MPa and seal face average linear velocity is 107.3 m/s, gas film

thickness disturbance has a significant effect on leakage rate, but has relatively litter effect on open force; Excessively large excitation

amplitude or excessively high excitation frequency can lead to severe gas film thickness disturbance; And it is beneficial to assure a

smaller gas film thickness disturbance when the stator material density is between 3.1 g/cm3 to 8.4 g/cm3; Ensuring sealing

performance while minimizing support axial stiffness and support axial damping can help to improve dynamic tracking property of dry

gas seal. The proposed research provides the instruction to optimize dynamic tracking property of the DGS.

Keywords: high-speed and high-pressure; dry gas seal; gas film thickness disturbance; dynamic tracking property

1 Introduction

With the rapid development of modern industry, mechanical seals are more and more widely used at the operational conditions of high-speed, high-pressure and high-temperature[1]. At the same time, because of the superior sealing performances, such as zero wear, low power consumption, long life, and high stability of DGS[2], other forms of seals have been gradually replaced by DGS which is becoming the mainstream of high parameters centrifugal compressor which used in petrochemical, metallurgical and other industries. But the DGS’s balance and stability are often damaged during its running process because of the influence of many factors such as external excitation and installation deviation. So in order to ensure the DGS’s running stability and reliability at the extreme conditions, it is necessary for us to take a good knowledge of its dynamics rules.

Because of the nonlinear nature of the gas fluid film,

* Corresponding author. E-mail: xdpeng@zjut.edu.cn Supported by National Natural Science Foundation of China(Grant No.

51575490), National Key Basic Research Program of China(973 Program, Grant No. 2014CB046404), and Natural Science Key Foundation of Zhejiang Province, China (Grant No. LZ15E050002) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2016

some scholars research the DGS dynamic behavior by using direct numerical simulation method[3], this method can obtain the most reliable calculation results, but its computations are generally very time consuming and it is not conducive to an extensive parameter study[4]. Several linear analytical methods are usually used in the research of DGS dynamics, such as perturbation method, step jump method, and direct numerical frequency response method. Now the perturbation method is widely used in dynamics research by scholars like MALANOSKI, et al[5], PENG, et al[6], ZIRKELBACK, et al[7] and so on[8–10], and it is proved that the perturbation method can effectively predict the DGS dynamic behavior and obtain reliable calculation results. In this paper, the gas film dynamic characteristic coefficients, which used to solve motion equations, are calculated by the perturbation method.

A lot of research work of which about DGS dynamic tracking have been done. GREEN, et al[11], detailed the transient responses of flexibly mounted stator by solving the gas film lubrication and dynamics equations simultaneously. LIU, et al[12], proposed the occurring mechanism of angular wobble self-excited vibration of DGS, and then XU, et al [13] , gave a mechanical analysis to explain the phenomenon of DGS’s angular wobble self-excited vibration. LEE, et al [14], considered the

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influence of external shock interferences when the possibility of face contact, the detectable central clearance between stator and rotor, and the leakage under unsteady operation were studied. There are many others related research achievements[15–22], but little of them at extreme operating conditions. Considering the need of development of DGS, this paper first research the influence of gas film thickness disturbance on sealing performance, what’s more, the effects of some key factors on gas film thickness disturbance are systematically investigated at the operational conditions of high-speed and high-pressure, and it would provide a theoretical guide for understanding DGS’s working rules and advancing DGS’s design method.

2 Models

2.1 Physical models

Fig. 1 shows a schematic cross section of a S-DGS with a flexibly mounted stator, and this model refers to Ref. [8] which published in 2002, it considers axial vibration of shaft and installation deviation of rotor. when the shaft spins, the installation deviation of rotor will provide an angular excitation motion to stator.

Fig. 1. Schematic cross section of a S-DGS

The spiral groove geometry is shown in Fig. 2(a), when

the rotor revolves at a high speed, there is a large hydrodynamic pressure will be generated by the spiral grooves, which will help to separate the two seal faces and maintain a steady gas film thickness. But in most of the practical seals, the gas film thickness has a disturbance because of the rotor’s axial runout and misalignment.

Fig. 2. Structure of the flexibly mounted ring

Fig. 3(a) shows the relative position between stator and rotor, Fig. 3(b) shows the model of seal face kinematics, and in the seal dynamic tracking property analysis, the gas film, which have a certain stiffness and damping properties, is usually treated as a spring and damper system, and the stator can be thought of as being supported by it[3]. Stator tracks rotor’s excitation motions under the function of spring, o-ring and gas film, and a good dynamic tracking property can help the seal to continuously operate with no face contact or no excessive leakage.

Fig. 3. Model of gas film thickness disturbance analysis

2.2 Mathematical models

Assuming the ideal gas and isothermal conditions, and without considering the impact of seal faces deformation in high-pressure. The transient-state compressible Reynolds equation, which governing the pressure distribution of the gas film, expressed in the polar coordinates is given by

( ) ( )

3 3

2

1 1

12

12

,2

ph p ph pr

r r r r

ph ph

t

æ ö æ ö¶ ¶ ¶ ¶÷ ÷ç ç÷ ÷+ =ç ç÷ ÷ç ç÷ ÷ç ç¶ ¶ ¶ ¶è ø è ø

¶ ¶+

¶ ¶ (1)

where p is transient-state pressure distribution, h is transient-state gas film thickness distribution in seal face, t is time; the steady-state compressible Reynolds equation expressed in the polar coordinates is given by

( )3 30 00 0 0 0 0 0

2

1 1,

12 12 2

p hp h p p h pr

r r r r

æ ö æ ö ¶¶ ¶¶ ¶÷ ÷ç ç÷ ÷ç ç+ =÷ ÷ç ç÷ ÷÷ ÷ç ç¶ ¶ ¶ ¶ ¶è ø è ø (2)

CHEN Yuan, et al: Gas Film Disturbance Characteristics Analysis of High-Speed and High-Pressure Dry Gas Seal

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the perturbation compressible Reynolds equations are obtained based on Eqs. (1) and (2) by using perturbation method(the theoretical derivation refers to Ref. [8]). To introduce dimensionless variable as follows:

( )

( )

b 0 00 0

b

b

2

2b

; , ; ; ;

, ; , ; ( 1,2);

6; ; sin ; cos .

kjkj

i i

kj ikj i

i i

i

i i

p h p hP k z j r i P H

p p h

p hP k j r i i

p r

rrR X R Y R

r p h

= = = = =

= = = = =

= = = =

Where p0 is steady-state pressure distribution; pz, pα, pβ are, respectively, perturbation pressure distribution about z axis, x axis and y axis; pi is pressure at inner radius of seal ring; hb is equilibrium film thickness in non-groove; h0 is equilibrium film thickness distribution in seal face; μ is the gas’s dynamic viscosity; Ω is angular velocity of shaft; ω1 and ω2 are, respectively, axial and angular excitation motion angular frequency; ri is stator inner radius. The expressions of dimensionless steady-state compressible Reynolds equation and dimensionless perturbation compressible Reynolds equations are shown in Eqs. (3) and (4a)–(4f):

( )0 03 30 0

0 0 0 02

1 1;

P HP PRP H P H

R R R R

æ ö æ ö ¶¶ ¶¶ ¶÷ ÷ç ç+ =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø¶ ¶ ¶ ¶ ¶

(3)

( )

( ) ( )

( )( )

( ) ( )

202 30

1 0 0 0

20 0 02 30

0 02

031 0 0 0

0 0302

1 32

2

1 3,

2

12

1;

zrzi

zr zr

zizr

zi zi

P PPP H RH RH

R R R R

P P P H PPH H

R

P PP H P RH

R R R

P P P HH

R

é ù¶¶¶ ê ú+ + +ê ú¶ ¶ ¶ê úë ûé ù¶ ¶ +¶¶ ê ú+ =ê ú¶ ¶ ¶ ¶ê úë û

é ù¶¶ ê ú- + + +ê ú¶ ¶ë ûé ù¶ ¶¶

ìïïïïïïï

ê ú =ê ú¶ ¶ ¶

ïïïïïïí

ë û

ïïï

î

ïïïïïïïïïïïï (4a)

( )

( ) ( )

( )( )

( ) ( )

202 30

2 0 0 0

20 0 02 30

0 02

032 0 0 0

0 αi 0302

1 32

2

1 3,

2

1

;

2

1

αrαi

αr αr

αiαr

αi

P PPP H H RX RH

R R R R

P P P H P XPH X H

R

P PP H P X RH

R R R

P P P HH

R

é ù¶¶¶ ê ú+ + +ê ú¶ ¶ ¶ê úë ûé ù¶ ¶ +¶¶ ê ú+ =ê ú¶ ¶ ¶ ¶ê úë û

é ù¶¶ ê ú- + + +ê ú¶ ¶ë ûé ù¶ ¶¶ ê ú =ê ú¶ ¶ ¶ë

ìïïïïïïïïï

û

ïïïïíïïïïïïïïïïïïïïïî

(4b)

( )

( ) ( )

( ) ( )

( ) ( )

202 30

2 0 0 0

20 0 02 30

0 02

032 0 0 0

0 0302

1 32

2

1 3,

2

12

1.

βrβi

βr βr

βiβr

βi βi

P PPP H H RY RH

R R R R

P P P H P YPH Y H

R

P PP H P Y RH

R R R

P P P HH

R

é ù¶¶¶ ê ú+ + +ê ú¶ ¶ ¶ê úë ûé ù¶ ¶ -¶¶ ê ú+ =ê ú¶ ¶ ¶ ¶ê úë û

é ù¶¶ ê ú- - + +ê ú¶ ¶ê úë ûé ù¶ ¶¶ ê ú =ê ú¶

ìïïï

¶ ¶

ïïïïïï

í

ê úë û

ïïïïïïïïïïïïïïïïïïïïïïïïïî

(4c) The pressure boundary conditions are specified at the

outer and inner radius:

( )( )

( )

0

0 o o

o

1 1 ;

;

0 1 ; , , ; , .

i i

kj i

P R

P p p R r r

P R or R r r k z j r i

= =ìïïïï = =

= =

ïíïïïïïî = = =

And the pressure periodic boundary conditions are specified as

( ) ( )( ) ( )

0 02 ;

2 ; , , ; , .

kj kj

P P

P P k z j r i

+ =

+ = =

ìïïïí=ïïïî

Combining and solving Eqs. (3) and (4a)–(4c) to obtain

dimensionless perturbation gas film pressure distribution, and according to Eq. (5) to obtain dimensionless gas film dynamic coefficients:

1 2 2

1 2 2

, dzz zx zy zr αr βr

xz xx xy zr αr βr

Ayz yx yy zr αr βr

βizi αi

zz zx zyβizi αi

xz xx xy

yz yx yy

K K K P P P

K K K XP XP XP A

K K K YP YP YP

PP P

C C CXPXP XP

C C C

C C CYP

æ ö÷ç ÷ç ÷ç ÷=- ç ÷ç ÷ç ÷÷ç ÷- - -çè ø

æ ö÷ç ÷ç ÷ç ÷=-ç ÷ç ÷ç

æ ö÷ç ÷ç ÷

÷÷ç ÷çè ø

ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

-

òò

1 2 2

d .A

βizi αi

A

YPYP

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç -- ÷ç ÷ç ÷ç ÷÷çè ø

òò

(5)

And the dimensional gas film dynamic coefficients are

2 2

b b

3 3

b b

3 3

b b

4 4

b b

,

,

( , , ),

,

,

zz i i zz i izz zz

jz i i jz i ijz jz

zj i i zj i izj zj

jg i i jg i ijg jg

K p r C p rk c

h h

K p r C p rk c

h hj g x y

K p r C p rk c

h h

K p r C p rk c

h h

üïï= = ïïïïïïïï= = ïïïï =ýïïï= = ïïïïïïïï= = ïïïþ

(6)

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where kjg and cjg(j, g=x, y) are gas film dynamic stiffness coefficients and damping coefficients respectively. Under the influence of support spring and secondary seal and gas film, stator will track the rotor’s motions which consist of the axial pulsation and angular wobbling. The equations of motion for both the axial mode and angular modes are expressed as follows:

s s r r( ) ) ( ,zz zz zz zzmz c c z k k z c z k z+ + + + = + (7)

r r r r

( ) ( )

,

x xx sx xy xx sx xy

xx xy xx xy

I c c c k k k

c c k k

+ + + + + + =

+ + +

(8)

r r r r

( ) ( )

.

y yx yy sy yy sy yx

yx yy yx yy

I c c c k k k

c c k k

+ + + + + + =

+ + +

(9)

In Eqs. (7)–(9), z and α, β are, respectively, stator’s axial response motion and angular response motion; zr and αr, βr are, respectively, rotor’s axial excitation motion and angular excitation motion; m is the mass of the stator; Ix and Iy are, respectively, the stator’s transverse moment of inertia about x axis and y axis; ks is the axial stiffness of the spring, cs is the axial damping of the secondary seal; ksx and ksy are, respectively, the angular stiffness of spring about x axis and y axis; csx and csy are, respectively, the angular damping of secondary seal about x axis and y axis.

( ) ( )2 2 2o i

2 2s s s s1 s s s s2

1 3 2 3 , d ,

0.5 , 0.5 ,

x y

x y x y

m B l r r I I r m

k k k r c c c r

= + - = =

= = = =

ò

rs1 and rs2 are, respectively, the radial position of spring and secondary seal; and rs1=0.5(ro+rb), rs2=rb, ro is stator outer radius, rb is stator balance radius. The following excitation motions are assumed for the rotor:

r rz 1 r r 2 r r 2sin , cos , si n .z A t A t A t = = = (10)

The initial conditions for the stator are

( ) ( ) ( ) ( ) ( ) ( )r r 20 0 0 0 0, 0 , 0 . z z A A = = = = = =

It is easy and inerrable to solve Eq. (7) by using analytical method as the Eq. (7) is the linear homogeneous second-order differential equation with constant coefficients, but Eqs. (8) and (9) are coupled equations which solved much easier by using finite difference method. The seal face gas film thickness disturbance distribution Δh at any time will be obtained by Eqs. (11) and (12) after the motion equations have been solved:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

r

r

r

,

,

;

z t z t z t

t t t

t t t

D = -

D = -

D = -

(11)

( ) ( ) ( ) ( ), , sin cos .h r t z t t r t r =D +D -D (12)

The transient-state gas film thickness distribution h in

seal face and pressure distribution p are expressed as follows:

( ) ( ) ( )0, , , , , ,h r t h r h r t = + (13)

( ) ( ) ( )0, , , , , ,r t p rp p r t = + (14)

p can be calculated by substituting the h into the expression of Eq. (1). Steady-state open force F0 and transient-state open force F are calculated from the obtained p0 or p by

o

i

o

i

2

0 00

2

00

d d ,

d d ,

r

r

r

r

F rp r

F F F rp r

=

= +D =

ò ò

ò ò (15)

where ΔF is a variation of F caused by gas film thickness disturbance. Steady-state volume leakage rate Q0 and transient-state volume leakage rate Q at inner radius pressure pi are calculated by

320 0 0

00

i

32

00

i

d ,12

d ,12

rh p pQ

p r

rh p pQ Q Q

p r

¶=

¶

¶= +D =

¶

ò

ò (16)

where ΔQ is a variation of Q caused by gas film thickness disturbance. The disturbance rates η of gas film thickness distribution h, open force F and volume leakage rate Q at any given time are defined by the expressions as following:

maxh

b

F0

Q0

( , ) ,

,

.

h r

h

F

F

Q

Q

D=

D=

D=

(17)

3 Results and Discussion

The basic calculation parameters are given in Table 1,

and the choice of stator’s seal face geometry and structure parameters refers to Ref. [23]. Generally, it prescribes that the sealing medium pressure of high pressure DGS is between 3MPa and 15MPa, and the mean linear velocity of seal face of high speed DGS is between 25 m/s and 100 m/s. So this paper assigns 10.1 MPa to po and assigns 700 to Λ which is the equal of assigning 107.3 m/s to the mean linear velocity of seal face that is very proper to represent an operational conditions of high-speed and high-pressure. And the selection of other basic calculation parameters meets the engineering practice. The parameters will remain

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unchanged during the following analysis unless otherwise stated.

Table 1. Parameters for dynamic analysis of the S-DGS

Parameter Value

Stator inner radius ri/mm 58.42 Stator outer radius ro/mm 77.78 Number of spiral groove Ng 12 Groove-to-land ratio κ=wl/wg 1 Groove-to-dam ratio δ=(rorg)/(rgri) 1 Spiral angle φ/(°) 15 Spiral groove depth hg/μm 5 Equilibrium film thickness in non-groove hb/μm 3 Pressure at inner radius pi/MPa 0.101 Pressure at outer radius po/MPa 10.1 Dynamic viscosity μ/(Pa·s) 19.73 Balance ratio B=(ro

2rb2)/(ro

2ri2) 0.6

Stator length l/mm 50 Stator density ρ/(kg·m–3) 3.1´103 Compressibility number Λ 700 Dimensionless axial excitation frequency Г1 1 Dimensionless angular excitation frequency Г2 1 Support axial stiffness ks/(MN·m–1) 10 Support axial damping cs/(kN·s·m–1) 1 Axial excitation amplitude Arz/μm 10 Angular excitation amplitude Ar/rad 1´10-4

3.1 Gas film thickness and pressure disturbance

transient distribution Fig. 4 and Fig. 5 present, respectively, the gas film

thickness disturbance distributions and gas film pressure disturbance distributions at four different moments during the cycle n+1, where cycle time T is period of rotation of shaft. From the Fig. 4, the positive values represent increment of gas film thickness and the negative values represent decrement of gas film thickness, and there is a maximum gas film disturbance |Δh(r,θ)|max on seal face at every moment. Fig. 4 shows that |Δh(r,θ)|max changes with time, and too much increment or decrement of gas film thickness all will lead to too much |Δh(r,θ)|max, this is not benefit to stable operation of S-DGS, and serious it will lead to ultimately premature seal failure because of excessive leakage or face wear, so the |Δh(r,θ)|max is an important parameter to represent whether S-DGS will work effectively.

The presence of gas film thickness disturbance will cause the disturbance of gas film pressure. A comparative analysis of Fig. 4 and Fig. 5 indicates that the more decrease of film thickness will lead to the more increase of film pressure, in other words, the more increase of film thickness will also lead to the more decrease of film pressure, and the Fig. 5 shows that gas film thickness disturbance have a bigger impact on film pressure disturbance in non-groove.

3.2 Influence of gas film thickness disturbance

on sealing performance Fig. 6 shows that the disturbance rates ηh, ηF and ηQ

change with time, the ηh represents the biggest degree of

gas film thickness change in the seal face. From the Fig. 6, it can be deduced that the ΔF and the opposite number of ΔQ have the same change rule with gas film thickness disturbance. When the maximum of ηh is 3.8%, the maximum of ηF and ηQ are, respectively, 2.2% and 19.7%, so the gas film thickness disturbance have a significant effect on leakage rate, but have relatively litter effect on open force.

Fig. 4. Gas film thickness disturbance distribution

Δh(r,θ) ( n is integer, n≥1, Arz=50 μm)/μm

Fig. 5. Gas film pressure disturbance distribution Δp(r,θ) ( n is integer, n≥1, Arz=50μm)/MPa

3.3 Influence of operation parameters on gas film thickness disturbance

3.3.1 Axial or angular excitation amplitudes

on |Δh(r,θ)|max

Fig. 7 shows that the |Δh(r,θ)|max changes with time under the different axial excitation amplitudes Arz. From the Fig. 7 it can be seen that the |Δh(r,θ)|max first changes

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sharply in a very short time and then the change of |Δh(r,θ)|max gradually becomes slowly, at last, the |Δh(r,θ)|max presents a cyclical change rule. We defined the cyclical change period of |Δh(r,θ)|max as stable phase and defined the period before the stable phase as adaptive phase. The reason of presenting the adaptive phase is that the stator will not respond quickly under the function of inertial force when it suddenly suffers the axial excitation motion of rotor, and then gradually adapts itself to the excitation. In the stable phase, the difference between the peak and the trough of |Δh(r,θ)|max increases linearly with the Arz, and the troughs of |Δh(r,θ)|max are equal under the different Arz.

Fig. 6. η changes with time (Arz=50μm)

Fig. 7. |Δh(r,θ)|max changes with time under the different Arz

Fig. 8 shows that the |Δh(r,θ)|max changes with time

under the different angular excitation amplitudes Ar. It is evident from Fig. 8 that the |Δh(r,θ)|max also experiences the adaptive phase and the stable phase, in the stable phase, the difference between the peak and the trough of |Δh(r,θ)|max remains unchanged, and the troughs of |Δh(r,θ)|max increase linearly with the Ar.

A comparative analysis of Fig. 7 and Fig. 8 indicates that, when other parameters remain the same, the troughs of |Δh(r,θ)|max in stable phase are all about the Ar and the difference between the peak and the trough is all about the Arz. From the two figures we can conclude that the increase of Arz or Ar aggravates the gas film thickness disturbance and deteriorates the DGS’s stability. It is hard for the DGS designers to control Arz, but easy to control Ar. So in order

to increase dynamic stability of DGS, designers can try to decrease installation deviation of rotor from the source, it demands that designers should not only ensure a strict geometrical tolerance range for rotor’s seal face but also for the rotor's back.

Fig. 8. |Δh(r,θ)|max changes with time under the different Ar

3.3.2 Dimensionless axial excitation frequency

on |Δh(r,θ)|max Fig. 9 illustrates the influence of dimensionless axial

excitation frequency Г1 on |Δh(r, θ)|max. In this article, the angular excitation angular frequency ω2 always equals angular velocity Ω of shaft (dimensionless angular excitation frequency Г2=1) because the angular excitation is only caused by rotor misalignment. As shown in Fig. 9, the peak of |Δh(r,θ)|max increases with the Г1 in the adaptive phase, this is because the higher excitation frequency will lead to greater excitation displacement in a certain time, meanwhile, stator can’t track the excitation motion quickly under the function of inertial force in a very short time. Moreover, the peak of |Δh(r,θ)|max in adaptive phase will be greater than the peak of |Δh(r,θ)|max in stable phase when the Г1 goes up to some extent, therefore, by this time the S-DGS will be more likely to happen failure of face wear or crack when it suddenly suffers the axial excitation.

Fig. 9. |Δh(r,θ)|max changes with time under the different Г1

In the stable phase, the peak of |Δh(r,θ)|max increases and

then decreases with the increase of Г1, this is because the increase of Г1 can lead to the increase of axial gas film

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stiffness[8] which is good for DGS’s stable operation, but on the other hand, the increase of Г1 also can intensify the gas film disturbance if the axial gas film stiffness remain unchanged. When the Г1 increases from 1 to 2, the axial gas film stiffness plays a more important role in gas film thickness disturbance than excitation frequency, but when the Г1 increases from 2 to 5 and to 10, the excitation frequency plays a leading role instead. It is concluded that when Г1 is more than 2, the Г1 is higher, the gas film thickness disturbance will become stronger, and the stability of the DGS will be worse. 3.4 Influence of structure parameters on gas film thickness disturbance 3.4.1 Stator material density on |Δh(r,θ)|max

Fig. 10 shows that |Δh(r,θ)|max changes with time under four different stator densities ρ. It can be observed from Fig. 10 that the peak of |Δh(r,θ)|max increases with ρ in the adaptive phase, this is because the inertial force increases with ρ, and the bigger inertial force will make the stator harder respond quickly to a sudden axial excitation motion of rotor. In the stable phase, the peak of |Δh(r,θ)|max decreases with the increase of ρ, it is indicated that the inertial force is benefit to decrease the gas film thickness disturbance and improve the dynamic tracking property of stator. What’s more, it is obvious from Fig. 10 that the peak of |Δh(r,θ)|max in the adaptive phase will be bigger than which in the stable phase when ρ is big enough, thus, the seal face will be easy to be worn or be crashed when the axial excitation motion just happened. However, to choose a density of stator material which is between 3.1 g/cm3 to 8.4 g/cm3 can guarantee a smaller gas film thickness disturbance.

Fig. 10. |Δh(r,θ)|max changes with time under the different ρ

3.4.2 Support axial stiffness or damping on |Δh(r,θ)|max

Figs. 11 and 12 show that, respectively, the |Δh(r,θ)|max changes with time under the different support axial stiffness ks and damping cs. From the Figs. 11 and 12 we can see that the peak of |Δh(r,θ)|max increases with ks or cs, this means that the ks or cs is bigger, the dynamic tracking property of stator will be worse, and the probability of DGS failure also

will be greater.

Fig. 11. |Δh(r,θ)|max changes with time under the different ks

Fig. 12. |Δh(r,θ)|max changes with time under the different cs

When the engineers want to design a DGS with good

dynamic stability, they can choose to use the springs with smaller stiffness, but they have to design a bigger spring compression to offset the close force. On the other way,

designers can also use a kind of auxiliary seals material, which not only can adapt to the conditions of high-pressure and high-speed but also have a low damping, to make sure DGS run stable.

4 Conclusions

(1) The gas film thickness disturbance has a significant effect on leakage rate, but has relatively litter effect on open force.

(2) The dynamic tracking property of stator will be bad and the gas film thickness disturbance will be drastic when the excitation amplitude is very big or the excitation frequency is very high, and at this very moment the DGS is more likely to fail because of excessive leakage or face wear.

(3) No matter the ρ is too big or too small, it will result in increase of the peak of gas film disturbance, especially when the ρ is very big, the stator is easier to be crashed because it is hard to respond quickly to the sudden axial excitation motion, so it is better to choose a kind of stator material which the ρ is between 3.1 g/cm3 to 8.4 g/cm3;

CHINESE JOURNAL OF MECHANICAL ENGINEERING

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Moreover, the peak of gas film thickness disturbance almost increases linearly with ks or cs , thus, in order to ensure a good operation stability for DGS we should try to decrease ks and cs while guaranteeing sealing performance.

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Biographical notes CHEN Yuan, born in 1990, is currently a PhD candidate at Zhejiang University of Technology, China. His research interest is the modern fluid sealing technology. E-mail: chenyuan_1221@163.com JIANG Jinbo, born in 1989, is currently a PhD candidate at Zhejiang University of Technology, China. His research interest is the modern fluid sealing technology.

E-mail:jinbo_110@163.com PENG Xudong, born in 1964, is currently a professor at Zhejiang University of Technology, China. His research interests include the modern fluid sealing technology and tribological design of mechanical equipment. Tel: +86-571-88320212; E-mail:xdpeng@zjut.edu.cn