AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH © 2010, Science Huβ, http://www.scihub.org/AJSIR

ISSN: 2153-649X doi:10.5251/ajsir.2011.2.58.68

Advanced seal design for rotating machinery E. Saber and Khaled M. Abdou

Mechanical Engineering Department, College of Engineering, Arab Academy for Science and Technology, Alexandria – Egypt

ABSTRACT

Labyrinth and more recently brush seals have been used extensively in high speed rotating machinery. Development is still in progress to reduce the leakage rate, and the frictional resistance of these seals beside the awareness of their stability characteristics. A seal very similar to the brush seal geometry is proposed here. It is composed of fine strips replacing the wires of the ordinary brush seal. A simple theoretical model for this strip seal is put forward here based on the assumption that the flow through it is steady laminar and incompressible. A Reynolds like equation is devised, governing the pressure inside the seal. Calculations of the leakage rate and the friction resistance are carried out for the various parameters considered and the results are presented and discussed in detail along with comparisons with previous work. It is shown in this work that the strip angle has a major effect on the performance of the seal as well as on the friction resistance. It is found that the larger the angle the lesser the leakage rate and the larger the friction force. The optimum value for minimum leakage is 90o but for mechanical stability and durability reasons it should be smaller. It is also found that the wider the strips the lesser the leakage rate for all values of the strip angle. The opposite holds for the friction resistance. The spacing between the strips has a minor effect on the leakage rate for small values of the strip angle but for larger values of the angle the increase in leakage becomes considerable when increasing the spacing for angles of order 70o. Comparisons with previous works on different types of seals showed that the present seal model can satisfactorily be implemented in practice. Keywords: Non-Contact seals, Labyrinth seals, Brush seals, Rotating machinery.

INTRODUCTION

Seal technology plays an important role in controlling end leakage flow. Although the labyrinth seal technology is very well developed, leakages are still high. The need for a sealing technology with good leakage sealing performance and compact size has risen. Brush seals have fulfilled such conditions. The brush seals are designed to contact the rotor. They are several times better leakage performance, weight reduction and space requirements than labyrinth seals. Compared to finned labyrinth seals, brush seals can maintain a much higher pressure difference. The outcome is a much more stable and balanced sealing system that is unlikely to let leakages take place. Mullen et al [1] modeled the two-dimensional flow field in a linear brush seal configuration and determined the flow field numerically using the finite element method. They studied the temporal disturbance of the upstream flow field. They studied also the effect of tangential flow of the upstream boundary. From their results the calculated flow rates were in agreement with previous

experimental results. Braun et al [2] carried out flow visualization in simulated brush seals using a shining of a planar laser sheet of light through the region of interest in the brush thus making visible the magnesium oxide seed particles that trace the fluid flow. Their flow visualization technique allowed definition of both local flow structure inside the voids of the porous matrix and qualitative and quantitative flow description at the entry section and at the exit of porous matrix. They investigated also the pressure drop between single and two sequentially positioned brush seals. They introduced a bulk flow model and compared their results against published engine brush seal data. Schlumberger et al [3] carried out experimental investigations on the effects of eccentricity on brush seal leakage for air at ambient temperature and saturated steam, both at non-rotational and low rotational speeds. They compared their results with an annular seal for air and standard labyrinth seal for steam. They determined that large eccentricities did not damage the brush seal; however, the rotor surface was worn, indicating that a

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

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harder surface was required. Shaft rotation did not affect brush seal performance in air, but it significantly increased leakage in steam. The brush seal had lower leakage rates than those predicted for comparable annular and labyrinth seals. Hendricks et al [4] discussed design similitude of brush seal with a bulk flow model based on flows in porous media, and rub surface coating. Their flow calculations were the sum of leakage flow through the bristles, the clearance at the interface between the surface and bristles, and the clearance between the surface and backing plate. Their results of flow calculations were in agreement with the previous experimental results. Carlile et al [5] investigated the leakage performance of a brush seal with gaseous working fluids; air, helium, and carbon dioxide for several bristle /rotor interference, by using a tapered rotor at static and low rotor speed conditions. They concluded that the brush seal reduced the leakage in comparison to the annular seal, up to 9.5 times. Addition of a lubricant reduced the leakage by 2.5 times when compared to a non-lubricant brush seal. Dowler et al [6] presented a model of bristle bed consisted of a hexagonal circular cylinder arrangement in order to study the curvature effects on the leakage flow through the brush seal. They used a single parameter, effective brush thickness, to correlate the leakage flow through the brush seal. The effective brush thickness was defined as a constant between the journal diameter and the mean diameter of the brush. They compared their results against available experimental data for the same model from different sources. They demonstrated that the model effectively accounts for the effect of the seal curvature and the signal parameter was indicator of brush seal leakage effectiveness. Chupp and Dowler [7] carried out experimental investigations on a single stage and multi stage brush seals, and one reference labyrinth seal to determine the performance characteristics of brush seals for application in limited-life gas turbine engines. They designed and built an elevated temperature, rotating test rig to test labyrinth and brush seals in simulated subsonic and supersonic engine conditions. The results from their tests demonstrated that brush seals leaked less compared to labyrinth seals, especially for subsonic, limited-life engine applications. Thus resulting in significant engine performance improvements. Bayley and Long [8] carried out experimental and theoretical studies of the flow and pressure distributions in a brush seal. Their model assumed laminar and compressible flow through porous media. Their measurements of the mass flow rate together with radial and axial

distributions of pressure were taken on a non-rotating experimental rig. They used the experimental data to estimate the seal porosity. They assumed the overall porosity constant with a variation in the bristle matrix. Comparison between these two cases showed some difference in the distribution of pressure in the bristle matrix. Their comparisons between measured and predicted flows through the seal was generally good considering the overall simplicity of their theoretical model. Handericks et al [9] carried out experiments on dual brush and labyrinth compressor discharge seals in a T-700 engine test.. The test condition included compressor discharge pressure up to 1 MPa, temperatures 407C, operating speeds up to 43000 rpm, and surface speeds up to 160 m/s with dry ambient air as the working fluid. The fuel consumption was less for the dual brush than for the labyrinth seal. They concluded that the brush seal leaked less than the labyrinth seal. Chew et al [10] developed a model of brush seal in which the bristle pack was treated as an axisymmetric, anisotropic porous medium with nonlinear porous resistance coefficients. The equations were solved using conventional computational fluid dynamics techniques to give prediction of flow rate, pressure distribution, velocity field, and bending forces on the bristles. The bristle aerodynamic forces were used in a separate calculation to estimate bristle bending and reaction forces on the shaft and backing plate. Their prediction for the brush seal were in agreement with experimental observations. From their results a nonlinear porous resistance law gave better agreement with measurement than the linear Darcian treatment. Sharatchandra and Rhode [11] presented a numerical study of the effects of rotor induced swirl velocity on the performance of brush seals. They used the Navier-Stokes equations in a two-dimensional analysis to obtain the velocity and pressure fields for an idealized brush seal configuration where the inclination of the bristles was neglected. Their computations were in good agreement with measurements for the similar flow across tube banks. Sharatchandra and Rhode [12] investigated the aerodynamic bristle force distributions in brush seals used in aircraft gas turbine engines. These forces were responsible for the onset of bristle tip lift-off from the rotor surface which significantly affected brush seal performance. In order to achieve that, they used Navier-Stoke flow simulation in a streamwise periodic module of bristles corresponding to the staggered square configuration. They studied also the effect of bristle spacing and bristle inclination angle. They found that the lifting

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60

bristle force increased with reduced intra-bristle spacing and increased inclination angle. The primary effects of decreasing the bristle inclination angle were a decrease in the tangential force component and a slight increase in the axial force component. Short et al [13] carried out experiments on a single stage brush seals in order to increase pressure capability per stage, reduce bristle tip contact pressure and insuring stable leakage performance of brush seals. They developed a design for the seal with thicker bristle diameter, wider bristle pack and wider clearance between back plate and bristles pack. Hendricks et al [14] studied and tested a set of inner diameter /outer diameter bidirectional brush seals to reduce the leakage losses in case of the leakage flow changed directions. From their data, they indicated that the brush seals were capable of sealing positive or negative pressure drops with respect to the axial direction and the performance of the machine increased from 3% to 12%. Chew and hogg [15] developed a model for predicting leakage flows through the bristle pack of a brush seal. They treated the bristle pack as a porous medium. Their predictions from a one-dimensional model were in agreement with a wide range of experimental data available from literature, for seals with a bristle pack to rotor interference fits. They also predicted a reasonable estimation of the bristle pack thickness, and concluded that the leakage flow increased proportional to the area available between the backing ring and shaft. O’Neill et al [16] carried out experiments on different designs of brush seal, multiple seals, and a single brush seal in order to study the effect of instability on sealing performance when multiple seals were operated in series at high pressure loading. Demiroglu et al [17] presented a 2-D laminar flow analysis, using the finite element analysis. Their model predicted flow rate versus pressure drop for a given flow region. The brush seal flow domain was divided into representative smaller cells in which simulation takes place. The cells were chosen such that together they can represent the whole seal. A cell is placed in the beginning, a second one in the middle and four cells at each corner of a square. Then, other cells were added in order to extrapolate the results for a full seal. In parallel to the numerical work, they developed a simple analytical model. This simplified solution of the 2-D laminar Navier-Stokes equation predicted pressure drop across two bristles. Their results from the numerical work were also compared to their analytical solution. Their results showed that the presented model was capable of analyzing bristles

configurations with realistically small inter-bristle gaps and in a staggered fashion. The model showed reasonable accuracy when compared with available experimental data. It could be used to evaluate various seal designs before an expensive testing process. The model predictions confirmed the trend of linear pressure drop across the seal thickness. Turner et al [18] presented experimental investigation and a CFD mathematical modeling of clearance brush seals with the rotor to measure the distribution of axial and radial pressure and their leakage characteristics. They used 0.27mm and 0.75 mm clearances, with pressure ratios up to 4, with the seal flow exhausting to atmospheric conditions. Their model treated the bristle pack as an axisymmetric, anisotropic porous region. They calculated from the CFD analysis the aerodynamic forces on the bristles, the bristle movements, stresses, and bristle and rotor loads. The results showed that the flow through a brush seal was very dependent on the position of the bristles. If a brush seal was built with a clearance, the aerodynamic forces will tend to move the bristles towards the rotor. The movement was very dependent on the balance between the normal aerodynamic forces and the friction levels in the brush pack. Their studies confirmed that the use of a non-Darican approach gave better agreement to experimental results. Laos et al [19] described a recently developed hybrid brush pocket damper seal that combines high damping with low leakage. The main objective of their experiments was to measure the damping and leakage of brush hybrid seals and six-bladed labyrinth seals of the same working dimension for comparison. Their test rig was connected to an electromagnetic shaker. The shaker contains an impedance head that outputs acceleration and force signals in time to excite the seal journal. The flow rate was controlled using a pneumatic control valve in the pressure range from 1 to 9.2 bars. They concluded that the brush hybrid pocket damper seal leaked less than the labyrinth seal while producing two to three times more damping than the original pocket seal. The aim of the present work is to design a new type of non-contact seal introduced as strip seal. The seal is very similar to the brush seal geometry.

MATHEMATICAL MODEL Fig.1 shows the strip seal geometry and coordinate system. In Cartesian coordinate ,z and y ,x ′′′ the Navier-Stokes and the continuity equations for steady, laminar, incompressible and iso-viscous flow are,

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

61

)1(zu

yu

xu

xp

zuw

yuv

xuu

2

2

2

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

′∂

′∂+

′∂

′∂+

′∂

′∂μ+

′∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂′+

′∂′∂′+

′∂′∂′ρ

)2(zv

yv

xv

yp

zvw

yvv

xvu

2

2

2

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

′∂

′∂+

′∂

′∂+

′∂

′∂μ+

′∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂′+

′∂′∂′+

′∂′∂′ρ

)3(zw

yw

xw

zp

zww

ywv

xwu

2

2

2

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

′∂

′∂+

′∂

′∂+

′∂

′∂μ+

′∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂′+

′∂′∂′+

′∂′∂′ρ

)4(0zw

yv

xu

=′∂′∂

+′∂′∂

+′∂′∂

Where u ′ , v′ and w ′are the velocities in the x ′ , y′ and z′ directions respectively.

(a) The strip seal configuration and

coordinate system.

(b) The strips and rotor.

Fig.1: The strip seal geometry and coordinate system

The strip array of strip seal is inclined to the rotor by an angle α . To simplify the equations of motion the Cartesian coordinate can be transformed to an oblique type of coordinates. Fig.2 shows the oblique system of coordinates. The Cartesian coordinates x ′ , y′ and z may be written in terms of the oblique coordinates x, y, z as follows:

)5(cosyxx,sinyy,zz α+=′α=′=′ The velocity components u′ , v′and w′ in terms of u, v and w may be written as.

)6(ww,sinvv,cosvuu ′=α=′α+=′

Fig. 2 System of oblique coordinates Substituting from the relations (5) and (6) into Eqns.1 to 4, considering the order of magnitudes, ( ),1O=θ

( ),1Oz = ( )1Oy = , ( )1Ou = , ⎟⎠⎞

⎜⎝⎛=

RbOv

and ( )1ORLOw =⎟⎠⎞

⎜⎝⎛= , and neglecting higher order

terms we get the following dimensionless equations:

)7(psinyu 22

2

θ∂∂

α=∂

∂

)8(yp0∂∂

=

)9(zpsin

LR

yw 22

2

∂∂

α⎟⎠

⎞⎜⎝

⎛=∂

∂

Where,

, uww ,

uvv ,

uuu

,Lzz ,

byy ,

Rx

000===

===θ

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

62

and pRu

bpo

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

μ=

The boundary conditions of the velocity component are,

0y = , 0wu ==

hbhy == , k

uRuo=

ω= and 0w =

Integrating Eqns.7 and 8 twice with respect to y and using the velocity boundary conditions we have,

[ ] )10(khyhyypsin

21u 22 +−

θ∂∂

α=

[ ] )11(yhyzpsin

LR

21w 22 −

∂∂

α⎟⎠⎞

⎜⎝⎛=

Substituting expressions (10), and (11) into the continuity Eqn.8 and taking the spatial average by integrating it w. r. t. y across the film we have,

)12(th

sink12h

sink6

zph

zLRph

22

32

3

∂∂

α+

θ∂∂

α=

⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛θ∂∂

θ∂∂

And in the absence of any squeeze velocity (steady

state) we can but 0th=

∂∂ .

The following Reynolds-like equation governing the dimensionless pressure distribution inside the seal is obtained,

)13(θh

αsin6

zph

zLR

θph

θ

2

*3

2*3

∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

⎟⎠

⎞⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

where, pb

u

kp

p2

2o*⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

ω

μ==

Eqn.13 is subject to the boundary conditions: ( )( ) (14)

)z,2(p)z,0(p

p1,p

p0,p

**

*e

*

*i

*

⎪⎪⎭

⎪⎪⎬

⎫

π=

=θ

=θ

Where *ip and *

ep are the dimensionless pressure at inlet and exit respectively. Because the shaft may be eccentric with the seal ring (casing surface), the gap that exists between the shaft and casing surfaces is not uniform along the circumference of the casing. When the width of the gap is measured along the casing radius, it is referred to as the gap width h′ .

Figs. 3 and 4 show the seal gap and the strip shape along with the systems of coordinates.

Fig.3 The seal gap and coordinate system.

Fig.4 Strip shape and coordinate system

To order ( Re ), the approximate expression for the gap width in aligned rigid shaft is,

)15(Rxcosebhg ⎟⎠⎞

⎜⎝⎛ ′

+=′

Using the transformation:

)16(tanh

-x xand sinhh ggg α

′′=α=′

Substituting from (15) into (16)

)17(cosRbhcosc1sinh gg α⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+θε+=α

bh

h gg = ,

bc c = ,

ce

=ε are the dimensionless gap

width, the dimensionless clearance and the eccentricity ratio respectively. The variation of the strip width may be written in dimensionless form as,

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

63

( ) )18(sinsin

c1H θα

−= φ

where φ is defined as strip shape factor. A small value of φ leads to a thick strip while large one gives a thin strip. The dimensionless film thickness may then be written as,

( ){ } )19(sinc1sin

1

cosRbhcosc1

sin1h g

θ−α

−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+θε+

α=

φ

The leakage flow rate through the seal ring takes place from the high pressure side to the low pressure side in the axial direction.

∫ ∫θ

θ=

0

h

0

wRdydQ ∫ ∫π

θ=2

0

h

0

o dyd wRbu

)20(dzphsin

LR

121

kLbuQQ

*2

0

322

o

*

θ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

α⎟⎠⎞

⎜⎝⎛−=

=

∫π

The friction force on the shaft is obtained by integrating the shear stress τ over the shaft surface,

dzd R FL

0

2

0

f θτ= ∫ ∫π

)21(zddθh1

xp

2hαsin

FRLu

bF

1

0

2π

0

*2

fo

f

∫ ∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡+

∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

μ=

where fF is the dimensionless friction force. RESULTS AND DISCUSSION Fig.5 shows the dimensionless leakage flow rate as a function of the dimensionless pressure drop for different values of strip angle. The results show that the relationship between the dimensionless leakage flow rate and the dimensionless pressure drop is linear irrespective of the value of the strip angle. It is clear that the larger the strip angle the lesser the leakage rate. The optimum value is the 090 angle but the strip angle should always be less than 090 for mechanical stability reasons. On other hand Fig.6

shows the variation of the dimensionless friction force with the dimensionless pressure drop for the same strip angles selected previously. It is seen that for small pressure drop values the rate at which the friction force increases with the increase of pressure drop is very much larger than that for larger dimensionless pressure drop values, say larger than approximately 1.5. This behavior is very much pronounced for larger values of the strip angle. The effect of the strip angle on the leakage rate can be shown more clearly from Fig.7. A tentatively selected value of dimensionless pressure drop say 1p* =Δ is used to produce the relationship for different values of the dimensionless spacing parameter sg /tt . It is obvious that the spacing parameter has in general minor effect on the leakage rate for small values of the strip angle but for larger values say for o70 angle double sg /tt leads to an increase in the leakage rate by approximately 20 percent. A reproduction of this figure is in Fig.8 which the effect of spacing parameter on the performance.

0 1 2 3 4 5DIMENSIONLESS PRESSURE DROP

0

2000

4000

6000

8000

10000D

IMEN

SIO

NLE

SS L

EAKA

GE

RAT

E α=30ο

α=50ο

α=70ο

Fig.5 The variation of the dimensionless leakage rate versus the dimensionless pressure drop for different values of the strip angle.

0 1 2 3 4 5DIMENSIONLESS PRESSURE DROP

0

50

100

150

200

250

300

350

400

450

500

550

600

650

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E

α=30ο

α=50ο

α=70ο

Fig.6 The variation of the dimensionless friction force versus the dimensionless pressure drop for different values of the strip angle.

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

64

20 30 40 50 60 70 80 90 STRIP ANGLE

4000

6000

8000

10000

12000

14000

16000

18000

DIM

ENSI

ON

LESS

LEA

KAG

E R

ATE tg / ts = 1

tg / ts = 1.5tg / ts = 2

Fig.7 The variation of the dimensionless leakage rate

versus the strip angle for different values of the spacing parameter sg /tt .

0.5 1 1.5 2 2.5

4000

6000

8000

10000

12000

0.5 1 1.5 2 2.5

4000

6000

8000

10000

12000

0.5 1 1.5 2 2.5

Spacing parameter(tg/ts)

4000

6000

8000

10000

12000

dim

ensi

onle

ss le

akag

e ra

te

α=70οα=50οα=30ο

Fig.8 The variation of the dimensionless leakage rate versus the spacing parameter sg /tt for different values of the strip angle.

The effect of the spacing parameter on the dimensionless friction force is given in Fig.9. Increasing the value of the spacing parameter sg /tt leads to a considerable decrease in the friction force and this is very much pronounced at smaller values of the strip angle α .

20 30 40 50 60 70 80 90STRIP ANGLE

50

100

150

200

250

300

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E tg / ts = 1tg / ts = 1.5tg / ts = 2

Fig.9 The variation of the dimensionless friction force versus the strip angle for different values of the spacing parameter sg /tt .

Fig.10 shows the variation of the dimensionless leakage rate with the strip angle for different values of strip thickness to shaft radius ratio. It seen that this ratio ( /Rt s ) has a minor effect on the performance irrespective of the value of strip angleα . The opposite holds in Fig.11 for the friction force where the ratio /Rt s has a surprisingly considerable effect on it. An increase of the ratio /Rts from say 0.002 to 0.005 reduces the dimensionless frictional force approximately four time, over the whole range of strip angle α .

20 30 40 50 60 70 80 90STRIP ANGLE

4000

6000

8000

10000

12000

14000

16000

DIM

ENSI

ON

LESS

AXI

AL L

EAKA

GE ts / R = 0.002

ts / R = 0.003ts / R = 0.004ts / R = 0.005

Fig.10 The variation of the dimensionless leakage rate

versus the strip angle for different values of dimensionless strip thickness.

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65

20 30 40 50 60 70 80 90STRIP ANGLE

0

50

100

150

200

250

300D

IMEN

SIO

NLE

SS F

RIC

TIO

N F

OR

CE

ts / R = 0.002ts / R = 0.003ts / R = 0.004ts / R = 0.005

Fig.11 The variation of the dimensionless friction force versus the strip angle for different values of dimensionless strip thickness.

The clearance between the strip and the shaft is seen from Fig.12 to have a little effect on the leakage rate. Although the value of the dimensionless clearance c/b does not affect the leakage it does affect the friction force to a great extent as can be seen in Fig.13. Increasing the value of c/b from 0.05 to 0.15 results in a decrease from about 50 % at 090=α to less than 10 % of the original value at α = o20 . Fig.14 shows the variation of the dimensionless leakage rate with the strip angle for various strip width to shaft diameter ratio. It is clear that the wider strip the lesser the leakage rate for all the values ofα . The opposite effect is true for the dimensionless friction force for all values of α as seen as in Fig.15. Reproductions of Fig.14 and 15 are given in Fig.16 and 13 respectively. It is quite clear that for relatively wide strip the leakage rate is very for all values ofα . The relationship between the friction force and the strip width is found to linear for all values of α .

20 30 40 50 60 70 80 90STRIP ANGLE

4000

6000

8000

10000

12000

14000

16000

DIM

ENSI

ON

LESS

LEA

KAG

E R

ATE

C / b = 0.05C / b = 0.1C / b = 0.15

Fig.12 The variation of the dimensionless leakage rate versus the strip angle for different values of the dimensionless clearance.

20 30 40 50 60 70 80 90STRIP ANGLE

50

100

150

200

250

300

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E C / b = 0.05C / b = 0.1C / b = 0.15

Fig.13 The variation of the dimensionless friction force versus the strip angle for different values of the dimensionless clearance.

20 30 40 50 60 70 80 90STRIP ANGLE

0

2000

4000

6000

8000

10000

12000

14000

16000

DIM

ENSI

ON

LESS

LEA

KAG

E R

ATE L / D = 0.02

L / D = 0.03L / D = 0.04L / D = 0.05

Fig.14 The variation of the dimensionless leakage rate versus the strip angle for different dimensionless strip width.

20 30 40 50 60 70 80 90STRIP ANGLE

50

100

150

200

250

300

350

400

450

500

550

600

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E

L / D = 0.02L / D = 0.03L / D = 0.04L / D = 0.05

Fig.15 The variation of the dimensionless friction force versus the strip angle for different dimensionless strip width.

Am. J. Sci. Ind. Res., 2011, 2(1): 58-68

66

0.02 0.04 0.06 0.08 0.10L / D

0

2000

4000

6000

8000

10000

12000D

IMEN

SIO

NLE

SS L

EAKA

GE

RAT

E α=30ο

α=50ο

α=70ο

Fig.16 The variation of the dimensionless leakage rate versus the dimensionless strip width for different values of the strip angle.

0.02 0.04 0.06 0.08 0.10L / D

0

100

200

300

400

500

600

700

800

900

1000

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E α=300

α=500

α=700

Fig.17 The variation of the dimensionless friction force versus the dimensionless strip width for different values of the strip angle.

For all the results considered so far the shaft and strips pack inner diameters are concentric. Fig.18 shows the variation of the leakage rate with eccentricity ratio for various strip angles. It can easily be argued that the eccentricity ratio has no sensible effect on the leakage rate for all values ofα , but in Fig.19 has a noticeable effect on the friction for eccentricity ratio greater than approximately 0.4 for the values of α considered here.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ECCENTRICITY RATIO ( e / c )

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

DIM

ENSI

ON

LESS

LEA

KAG

E R

ATE

α=30οα=50ο

α=70ο

Fig.18 The variation of the dimensionless leakage rate versus the eccentricity ratio for different values of the strip angle.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ECCENTRICITY RATIO ( e / c )

30

40

50

60

70

80

90

100

DIM

ENSI

ON

LESS

FR

ICTI

ON

FO

RC

E α=30ο

α=50ο

α=70ο

Fig.19 The variation of the dimensionless friction force versus the eccentricity ratio for different values of the strip angle.

CONCLUSIONS In this work the following conclusions are drawn:

1- The strip angle has a major effect on the performance of the seal as well as on the friction force. Since the larger the strip angle the lesser the leakage rate and the larger the friction force. The optimum value for minimum leakage rate is 090 but for mechanical stability and durability reasons it should be less than that.

2- The spacing between the strips has a minor effect on the leakage rate for small values of the strip angle but for larger values say for

070 angle the increase in leakage rate

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67

reaches approximately 20 percent when the spacing is doubled.

3- The strip thickness has a minor effect on the performance irrespective of the value of the strip angle.

4- The clearance between the strip and the shaft has a little effect on the leakage rate but increasing the clearance results in a considerable decrease in the friction force which approximately reaches 50 percent for

090 strip angle. 5- The wider the strip the lesser the leakage

rate for all values of strip angle opposite holds for the friction force.

NOMENCLATURE b the seal gap length, m. c radial clearance, m. c dimensionless clearance, c =c/b e eccentricity, m.

fF friction force, N.

fF dimensionless friction force, ( )RL/bμu/FF off =

gh ′ gap width, m.

gh oblique gap width, /sinαhh gg ′= , m.

gh dimensionless oblique gap width, gh = gh /b. h film thickness, m. h dimensionless film thickness, h =h/b k Velocity ratio, oωR/uk = . L strip width, m. p pressure, Pa. p dimensionless pressure, ( )2

o b/Rupp μ=

*p Characteristic dimensionless pressure, *p = p / k .

R shaft radius, m. eR Reynolds number, eR = b/μρuo .

Q Leakage flow rate through the seal, s/m3 . *Q Dimensionless leakage flow rate through the

seal, kLbQ/uQ o* = .

gt The gap between the strips, m.

st Strip thickness, m.

u ′ Velocity component in x'-direction, m/s. u Velocity component in x-direction, m/s. u Dimensionless flow velocity component in x-

direction, ou/uu = .

ou Reference velocity, m/s

v′ Velocity component in y ′ -direction, m/s. v Velocity component in y -direction, m/s.

v Dimensionless velocity component in y-direction, ov/uv =

w ′ Velocity component in z′ -direction, m/s. w Velocity component in z -direction, m/s. w Dimensionless velocity component in z-

direction, 0w/uw = . )z,y,x( ′′′ Cartesian co-ordinate system.

(x ,y ,z) Oblique co-ordinate system. ( z,yθ, ) dimensionless oblique co-ordinate system,

z/Lz y/b,y x/R,θ === Greek Symbols α Strip angle. ε Eccentricity ratio, c/e=ε . φ Strip shape factor. μ viscosity, Pa s. ν Kinematic viscosity /sm2 θ Angle coordinate in circumferential, Rx=θ ρ Density, kg/ 3m . τ

Shear stress over the shaft surface, N/ 2m . ω Angular velocity, rad/s. REFERENCES [1] R. Mullen, M. Braun, and R. Hendricks “Numerical

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[4] R.C. Hendricks, S. Schlumberger, M.J. Braun, F. Choy and R.L. “A Bulk Flow Model Of A Brush Seal System” ASME, Paper No. 91-GT- 325, 1991.

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