rectengular gas bearing

7/30/2019 Rectengular Gas Bearing

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P e r f o r m a n c e o f an e x t e r n a l l yp r e s s u r i z e d r e c t a n g u l a r g a sb e a r i n g u n d e r c o n s t a n te f f e c t i v e r e c e s s p r e s s u r eSadek Z. KassabThe present study predicts the performance of an external lypressurized rectangular recessed gas bearing under dif ferentoperat ing condit ions. The analysis m akes use of simp li fyingassumptions and an empirical formula which relates the recesspressure to the bearing supply pressure. A comparison betweenthe theoretical results obtaine d and av ai lable expe rime ntal datashows qual i ta t ive agreement . There are some deviat ions inpressure distribution and load-carrying capacity especially atgreater f i lm thickness and supply pressures. The reasons for thesedeviat ions are discussed.Keywords: gas bearing performance, recess pressure, bearing sup plyp r e s s u r e

Introduct ionB e c a u s e o f t h e i n d u st r ia l i m p o r t a n c e a n d w i d e s c o p eo f a p p l i c a ti o n s o f t h e e x t e rn a l l y p r e s s u r i z e d r e c t a n g u l a rg a s b e a r i n g , a t h e o re t i c a l a n d e x p e r i m e n t a l r e s e a r c hp ro g ra m m e c o n c e rn i n g t h is t y p e o f b e a r i n g s t a r t e d a tt h e M e c h a n i c a l En g i n e e r in g D e p a r t m e n t , A l e x a n d r i aUniv ers i ty , in the ear ly 1970s 1 . A t th e beg inn ing o fth i s p roga mm e bo th theo re t i ca l inves t iga t ions 2,3 ande x p e r i m e n t a l w o r k 4,5 w e re p e r fo r m e d u n d e r c o n s t a n tin le t recess p ressu re in o rder to fac i l i t a te compar i sons .Ev e n t u a l l y d e s i g n e r s r e a l i z e d t h a t t h e k n o w n p re s s u rei s the supp ly p ressu re , no t the in le t p ressu re tot h e b e a r i n g r e c e s s . F o r t h i s r e a s o n , e x p e r i m e n t a li n v e s t i g a t i o n s w e re s t a r t e d i n t h e s e c o n d s t a g e o f t h ep ro g ra m m e u n d e r a c o n s t a n t s u p p l y p r e s s u re 6 -1 1.U n fo r t u n a t e l y , i t w a s n o t e a s y t o c h a n g e f ro m t h e o re t i -c a l a n a l y s i s b a s e d o n i n l e t p r e s s u re t o t h e b e a r i n gre c e s s t o a s t u d y o f t h e fu n c t i o n o f t h e s u p p l yp re s s u re t o t h e b e a r i n g , s i m p l y b e c a u s e t h e t y p e a n dcharac te r i s t i cs o f the bear ing res t r i c to r i s an impo r tan tp a ra m e t e r a n d m u s t b e i n c l u d e d i n t h e a n a l y s i s . Af i rs t t r i a l to co rre la te the theo re t i ca l resu l t s wi th thes u p p l y p r e s s u re w a s m a d e b y K a s s a b 6 . H e o b t a i n e dt h e o re t i c a l l y t h e p r e s s u re d i s t r i b u t i o n a l o n g t h e e x t e r -n a l l y p r e s s u r i z e d r e c t a n g u l a r r e c e s s e d g a s b e a r i n g b yt w o m e t h o d s . Th e f i r s t u s e d t h e m e a s u re d i n l e t

No t a t io nmoBoB~doHLL *LoL.,MP aP re~Pxp *Tx,y,zUPA

B e a r i n g o u t e r a r e aB e a r i n g o u t e r w i d t hR e c e s s w i d thS u p p l y h o l e d i a m e t e rF i lm th icknessLo a d -c a r ry i n g c a p a c i t yD i m e n s i o n l e s s l o a d -c a r ry i n g c a p a c i t y[ = L/(Ps-Pa)Ao]B e a r i n g o u t e r l e n g t hR e c e s s l e n g t hMass f low ra teA t m o s p h e r i c p r e s su r eR e c e s s p r e s s u reS u p p l y p r e s s u reP re s s u re a t a n y d i s t a n c e x

( P - _ P p ~D i m e n s i o n l e s s p r e s s u re [= (P s ) ]A b s o l u t e t e m p e r a t u r eC a r t e s i a n c o o rd i n a t e s y s t e mLi n e a r v e l o c i t y c o m p o n e n t i n x d i r e c t i o nG a s d e n s i t yGas v i scos i tyR e c e s s d e p t hMechanical Engineering Department, Faculty of Engineering, Alexan-dria University, Alexandria 21544, EgyptPresent address: Mechanical Engineering Department, Kuwait Univer-sity, KuwaitReceived 6 April 1992; revised 29 May 1993; accepted 23 November1993

p re s s u re a s t h e r e c e s s p r e s s u re a n d t h e s e c o n d t h em a x i m u m p re s s u re o b t a i n e d a f t e r d e p re s s i o n a s t h er e c e s s p r e s s u re . B o t h p r e s s u re s w e re fu n c t i o n s o f t h es u p p l y p r e s su re . Th e r e s u l ts o b t a i n e d u s i ng t h e s e c o n dm e t h o d s h o w e d b e t t e r a g r e e m e n t w i t h e x p e r i m e n t a l

TRIBOLOGY INTERNATIONAL 030 1-6 79X /94 /03 /015 9-0 9 1994 Butterworth-Heinemann Ltd 15 9


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S. Z. Kassa b--Externally pressurized rectangular gas bearingresul ts th an tho se using t he firs t. Sh aw ky ~2 use dK a s s a b ' s 6 r e s u lt s t o o b t a i n a n e m p i r i c a l f o rm u l a w h i c hre l a te s t h e m a x i m u m v a l u e o f p r e s s u re r e c o v e re d a f t e rd e p re s s i o n ( e f f e c t i v e r e c e s s p r e s s u re , P r ) w i t h t h eo u t e r s u p p l y p r e s s u re , P s , f o r d i f f e r e n t o p e ra t i n gc o n d i t i o n s a n d o u t e r d i m e n s i o n s a s :

( e r - e a ) ( ~ ) ( 8 t2(P s - P a ) - 3 . 95 - 2 . 5 \ ~ /- 1 3 8 6 ( L ~ ) - 0" 1 6 (P-z"p,S-Pa) - 0.29

b e a r i n g i n m i n d t h a t e m p i r i c a l c o r r e l a t i o n s , a l t h o u g hval id fo r the t es t da ta used , a re no t necessar i lya p p l i c a b l e o v e r t h e v e ry w i d e r a n g e o f p o s s i b l eo p e ra t i n g c o n d i t i o n s . Th e re fo re a d e t a i l e d d i s c u s s i o no f S h a w k y ' s e m p i ri c a l e q u a t i o n i s i m p o r t a n t b e fo rep resen t ing the resu l t s .A l t h o u g h S h a w k y ' s 12 e m p i r i c a l e q u a t i o n t o o k i n t oc o n s i d e ra t i o n t h e e f f e c t o f t h e o p e ra t i n g c o n d i t i o n s( f il m th i c k n e s s , H , a n d s u p p l y p r e s s u re , P s ) o n t h er e c t a n g u l a r b e a r i n g p e r fo rm a n c e , i t d i d n o t i n c l u d et h e v a r i a t io n s o f a l l t h e b e a r i n g g e o m e t r i c a l p a r a m e t e r s .Th e o n l y g e o m e t r i c a l p a r a m e t e r i n c l u d e d i n S h a w k y ' se q u a t i o n i s t h e b e a t i n g o u t e r w i d t h t o l e n g t h r a t i o ,B o / L o . T h e r e a r e o t h e r g e o m e t r i c a l p a r a m e t e r s w h i c hw e re h e l d c o n s t a n t ( L r / L o ) = 0 .9 , ( B d B o ) = 0.1 and( d o / L o ) = 0 .02 . I t i s impor tan t to no te tha t th i s doesn o t m e a n t h a t t h e v a r i a t i o n s o f t h e s e g e o m e t r i c a lp a r a m e t e r s h a v e n o e f f e c t o n t h e e m p i r ic a l f o rm u l au n d e r c o n s i d e ra t i o n . I n c o n t r a s t , t h e e x p e r i m e n t a lr e s u lt s p r e v i o u s l y o b t a i n e d s h o w e d s u b s t a n t i a l e f f e c t sb y t h e s e p a r a m e t e r s o n p r e s s u re d i s t r ib u t i o n a n dc o n s e q u e n t l y o n t h e e f f e c t i v e r e c e s s p r e s s u re . Th efo l lo w i n g p a ra g ra p h s s h e d s o m e l i gh t o n t h e e f f e c t o ft h e s e p a r a m e t e r s o n t h e r e c e s s p r e s s u re a l o n g w i t ht h e c o n d i t i o n s u n d e r w h i c h S h a w k y ' s f o rm u l a c a n b eused e f fec t ive ly .S a l e m a n d S h a w k y 5 d e m o n s t r a t e d e x p e r i m e n t a l l y t h a ta s t h e r e c e s s s i z e i n c r e a s e s , s o d o e s t h e p r e s s u red e p re s s i o n d u e t o t h e i n c r e a s e i n m a s s f l o w r es u l ti n gf ro m r e d u c t i o n i n t h e b e a r i n g ' s h y d ra u l i c r e s i s t a n c e .Th e y a l s o s h o w e d t h a t t h e a s s u m p t i o n o f o n e -d i m e n -s i o na l f l ow c a n b e u s e d a s t h e r e c e s s le n g t h a p p ro a c h e st h e b e a r i n g o u t e r l e n g t h . C o n s e q u e n t l y , S h a w k y ' se m p i r i c a l e q u a t i o n fo r b e a r i n g s w i t h ( L r / L o ) = 0.9m a y b e v a l i d u n d e r t h e a s s u m p t i o n o f a o n e -d i m e n s i o n a lf lo w b e a r i n g . M o r e o v e r , B o f f e y et al. 13 fo u n d t h a tincreas ing recess l eng th in the s ing le-o r i f i ce rec tangu larg a s b e a r i n g i n c r e a s e s m a x i m u m l o a d c a p a c i t y a n dmax imum f i lm s t i f fness .O n t h e o t h e r h a n d , S h a w k y a n d K a s s a b 7 s t u d i e de x p e r i m e n t a l l y t h e e f f e c t o f r e c e s s w i d t h t o b e a r i n go u t e r w i d t h r a t i o , B J B o , o n r e c t a n g u l a r b e a r i n gp e r f o rm a n c e . T h e y f o u n d t h a t t h e m a x i m u m p r e s su r ea t t a i n e d a f t e r d e p re s s i o n ( i . e . t h e e f f e c t i v e r e c e s sp ressu re) i s a func t ion o f ( B J B o ) . F u r t h e r , t h e i r r e su l t ss h o w e d t h a t t h e m a x i m u m d i m e n s i o n l e s s lo a d a n d t h em i n i m u m m a s s f lo w r a te a r e a l w a y s a t ( B d B o ) =0 .1. M o re o v e r , E I -G a y e r 9 s u p p o r t e d t h e i r f in d in g s.A c c o rd i n g l y , t h i s p a r t i c u l a r v a l u e o f ( B r / B o ) = 0.1w a s c h o s e n a n d h e l d c o n s t a n t b y S h a w k y ~2.Th e o r i f ic e ' s d i a m e t e r a n d c o n s t ru c t i o n h a v e a p ro fo u n de f f e c t o n g a s b e a r i n g p e r fo rm a n c e . S h a w k y4 s tud ied160

e x p e r i m e n t a l l y t h e e f f e c t o f s u p p l y h o l e d i a m e t e rt o b e a r i n g o u t e r l e n g t h r a t i o , ( d o / L o ) , o n b e a r i n gp e r fo rm a n c e . H i s r e s u lt s s h o w e d t h e d e p e n d e n c eo f p r e s s u re d i s t r ib u t i o n a n d d e p re s s i o n o n ( d o / L o ) .F u r t h e r , S a l e m a n d S h a w k y s s t u d ie d t h e e f f e c t o fro u n d i n g t h e s u p p l y -h o l e e d g e . Th e y t e s t e d o n e p a dw i t h a s h a rp -e d g e d s u p p l y h o l e a n d r e t e s t e d t h e s a m ep a d a f t e r r o u n d i n g t h e h o l e . Th e y fo u n d t h a t t h ep re s s u re d e p re s s i o n d e c re a s e d a n d t h a t t h e p r e s s u rea t t a i n e d a f t e r d e p re s s i o n i s i n c r e a s e d . Th i s i m p ro v e -m e n t i n p r e s s u re d i s t r ib u t i o n l e a d s t o a m a rk e d i n c r e a sein the load -car ry ing capac i ty o f the bear ing a t thee x p e n s e o f i nc r e a s in g t h e m a s s f l o w r a t e . U n fo r t u -n a t e l y , t h e m e a s u re m e n t s o f S h a w k y 4 a n d S a l e m a n dS h a w k y 5 w e re m a d e u n d e r c o n s t a n t i n l e t p r e s s u re a n dt h e re i s n o r e c o rd o f th e s u p p l y p r e s s u re . Th i s m a d ei t imposs ib le to inc lude the i r da ta in Sha wky ' s empi r ica le q u a t i o n . Th e re fo re t h e e m p i r i c a l f o rm u l a m a y b eva l id on ly fo r a sharp -edged supp ly ho le wi th a cer ta ind i a m e t e r .F ina l ly , the bas ic geomet ry o f a long i tud ina l recess ina rec tangu lar pad i s a spec ia l case . Fo r an ex te rna l lyp re s s u r i z e d r e c t a n g u l a r b e a r i n g w i t h a c ro s s - s h a p e dre c e s s , B o f f e y et a l . 14 foun d tha t th e e f fec t o f p rov id ingthe bear ing wi th a recess and o f increas ing the dep thof the recess i s to increase the load a t a g iven a i r gap .Th e i m p ro v e m e n t i n l o a d c a p a c i t y i s b ro u g h t a b o u tb y a n i m p ro v e m e n t i n p r e s s u re d i s t ri b u t io n w i th i n t h ebearing fi lm.F ro m t h e p r e v i o u s d i s c u s s i o n , S h a w k y ' s e m p i r i c a le q u a t i o n m a y b e u s e d e f f e c t i v e l y u n d e r t h e fo l l ow i n gdes ign and opera t ing cond i t ions :(1 ) W hen the bear ing has a long i tud ina l recess wi tha l e n g t h c l o s e t o t h e b e a t i n g ' s o u t e r l e n g t h(2 ) F o r t h e c o n d it i o n o f m a x i m u m d i m e n s i o n l e s sl o a d a n d m i n i m u m m a s s f lo w r a te a t c e r t a i no p e ra t i n g c o n d i t i o n s(3 ) W h e n u s in g a s h a rp -e d g e d s u p p l y h o l eThe p rese n t inves t iga t ion a t t em pts to f il l the gapb e t w e e n t h e t h e o re t i c a l a n d e x p e r i m e n t a l r e s u l t so b t a i n e d fo r a n e x t e rn a l l y p r e s s u r i z e d r e c t a n g u l a rr e c e s s e d g a s b e a r i n g . I t m a k e s u s e o f S h a w k y ' se m p i r i c a l f o rm u l a t o d e d u c e b e a r i n g p e r fo rm a n c eu n d e r v a r i o u s g e o m e t r i c a l a n d o p e ra t i n g c o n d i t i o n sand d i scusses the ranges and l imi ta t ions o f the empi r ica lfo rm u l a e m p l o y e d . I n a d d i t i o n , a c o m p a r i s o n b e t w e e nt h e t h e o re t i c a l r e s u l t s a n d p r e v i o u s l y o b t a i n e d d a t a i sg iven to check the va l id i ty and usefu lness o f thetheore t i ca l ana lys i s .T h e o r e t i c a l a n a l y s i sF o r a n e x t e rn a l l y p r e s s u r i z e d r e c t a n g u l a r r e c e s s e d g a sb e a r i n g (F i g 1 ) t h e p r e s s u re d i s t ri b u t i o n m a y b eo b t a i n e d fo r s t e a d y , l a m i n a r , o n e -d i m e n s i o n a l f l o w i nt h e d i r e c t i o n o f t h e b e a r i n g w i d t h , x , u n d e r t h ea s s u m p t i o n s o f n e g l e c t in g b o t h b o d y a n d i n e r t i a f o r c e sas :

d 2 u 1 d Pdz 2 - bt dx (1 )In teg ra t ing equat ion (1 ) twice wi th respec t to Z us ingt h e fo l l o w i n g b o u n d a ry c o n d i t i o n s :

1 9 94 V O L U M E 2 7 N U M B E R 3


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S. Z . K a s s a b - - E x t e r n a l l y p r e s s u r i z e d r e c t a n g u l a r g a s b e a r i n g

~ . . . . _ k H

L o

J

YR e c e s s

| , ,I '!!j , !!Fig 1 R ecessed rectangular bearing

( i) a t Z = 0 u = 0(ii) a t Z = H u = 0gives

1 d Pu - d x ( z 2 - z / - / ) ( 2 )

For com press ib le f low, the mass f low ra te , 2~ /, i s g ivenbyf : - p L o H 3 d P/~/= pud A = 61~ dx

th e nd P _ -61J, /~/ (3)d x L o l l 3 p

F o r i s o th e r ma l f l o w ,PP - R T (4 )dP -61J .R T ~ td x - L o l l P (5 )

I n t e g r a t i n g e q u a t i o n ( 5 ) u s in g t h e f o l l o w in g b o u n d a r ycondi t ions :( i ) a t x = x P = P xB0(i i) a t x - P = P a2gives

P ~ = P ~ + L o l lFor the r ecessTRIBOL OGY INTERNATIONAL

nrx = ~ - P x = er

p 2 = p 2 + L o l l 3 (7 )= LoB3 (p2_ e2) (8 )1 2 ~ R T [ ( B o / 2 ) - ( B r /2 ) ]

F r o m e q u a t i o n s ( 6 ) a n d ( 7 )( p ~ _ p 2 ) _ [ (Bo/2) - x ]( p E _ p ~ ) [ ( B o / 2 ) - ( B r / 2 ) ]

E q u a t i o n ( 9 ) g iv e s t h e p r e s s u r e d i s t r i b u t i o n i n t e r msof the r ecess pressure , P r , and the r ecessed bea r ingd ime n s io n s .S ince , in prac t ica l appl ica t ions , the supply pressure ,P ~ , i s t h e k n o w n p a r a me te r f o r a b e a r in g d e s ig n e r ,then i t wi l l be use fu l to have a r e la t ion be tween thesupply pressure and the r ecess pressure a s a func t iono f t h e o p e r a t i n g c o n d i t i o n s a n d b e a r in g d ime n s io n s .T h e e mp i r i c a l e q u a t i o n o b t a in e d b y S h a w k y 12 i s u s e dto r e la te the r ecess pressu re , P r , to the supply pressure ,

R e s u l t s a n d d i s c u s s i o nT h e e f fe c t s o f t h e o p e r a t i n g c o n d i t i o n s ( f i lm th i c k n e s s ,H , a n d s u p p ly p r e s s u re , P s ) o n t h e p r e s s u r e d i s t r i b u t i o na lo n g t h e b e a r in g w id th a r e s h o w n in F ig s 2 a n d 3 .F igure 2 shows tha t , a t ce r ta in Ps and ( B r /B o ) ,inc reas ing the f i lm th ickness , H , r e su l t s in a dec reasein t h e d ime n s io n l e s s p r e s s u r e P * = ( P - P a ) /( P s -P a ) . T h i s b e h a v io u r i s d u e t o t h e d e c r e a s e i n t h ee f fec t ive r ecess pressure a s the f i lm th ickness inc reases .The e f fec t can be expla ined phys ica l ly a s fo l lows .Inc reas ing the f i lm th ickness a t a ce r ta in supplyp r e s s u r e me a n s d e c r e a s in g t h e b e a t i n g ' s h y d r a u l i c

1 . o _ _ , ' ' ' ' I

& o . 6 -

- I

0-0 - " "' . . . . . . 'GO 0-2 0-4 O.G 0 .8 1.02 X / Bo

Fig 2 Dimen sionless pressure d ist r ibu tion a long bear ingwidth a t d i f fe ren t va lues o f f i lm th ickness1 6 1


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S. Z. Kassab--External ly pressurized rectangular gas bearingresistance to the flow. Consequently the mass flowrate increases followed by an increase in the pressuredepression inside the recess and a decrease in theeffective recess pressure, which represents the startingpressure to the land region. It is important to notethat the empirical equation detects this. Figure 3presents the effect of Ps on the dimensionless pressure,P*, at certain values of Bo/Lo, Br /Bo and H. As Psincreases, the effective recess pressure (the constantpressure inside the recess region) decreases. Conse-quently, the pressure inside the land region, P*,also decreases. This behaviour is physically correct.Increasing Ps at a certain film thickness (i.e. a certainbearing hydraulic resistance) means increasing thepressure difference between the supply pressure to thebearing, Ps, and the exit pressure (the atmosphericpressure, Pa)- Increasing this difference (Ps - Pa)results in a rise in the mass flow rate through thebearing, which is usually followed by increasing thepressure depression after the supply hole and insidethe recess region. This results in a reduction in theeffective recess pressure, which is the inlet pressureto the bearing land region. Meanwhile, Fig 3 showsthat at a certain value of Ps the pressure decreases asthe film thickness increases. This is in agreement withthe results shown in Fig 2.On the other hand, there are many geometricalparameters affecting the performance of externallypressurized rectangular recessed gas bearings. One ofthese is the bearing outer width to length ratio, B o/Lo. A study by Shawky~2 concerning the relationbetween the effective recess pressure and the rectangu-lar bearing outer dimensions under different filmthicknesses and supply pressures clarified that thisrelation is parabolic, with a peak value at ( B o /L o ) =0.8. This is in agreement with the experimentalresults obtained by Shawky and Kassab8. ThereforeShawky's~2 empirical equation and, consequently, thepresent study, which makes use of Shawky's equation,took this effect into consideration. To shed some light

1.0 I I I T- [Ps:2"0ba rabs~ Br =0.1 O - 0 - 8 ~ p H = 3 0 P m

,,oT%o.4-1

/ I _ \ ' \ ' % . ,E 0 . 2 ~ .i5 ~..

0.00 . 0 0 - 2 0 - 4 0 . 6 0 " 8 1 -02 X / B o

Fig 4 Effect of bearing outer width to length ratio (Bo/Lo) on the dimensionless pressure distribution

on this behaviour, Fig 4 shows that the dimensionlesspressure distribution, P*, for ( B o /L o ) = 2/3 has higherpressure values than the corresponding distribution for( B o /L o ) = 1. In order to understand this behaviour itis important to note that the value 2/3 is closer to 0.8(the position of the apex of the parabola) than 1.The effect of the dimensionless recess width, ( B , /B o ) ,on the dimensionless pressure distribution, P*, isshown in Fig 5. For a certain value of ( B o /L o ) andH, as ( B r /B o ) increases, the pressure at the beginningof the bearing land region remains constant in magni-tude but shifts to the right, according to the correspond-ing value of ( B J B o ) < ( 2 x l B o ) < 1. Increasing ( B JBo) results in raising the dimensionless pressure, P*.This is mainly due to the following reasons. First,

n

1.0

0 '8

0 . 6Q.

~ 0.4t-O

~ 0 . 2

0 - 00 .0

I I ~ I IB o _ 1 I B r 0 1

=

. = , 0 . m

-_.- -

0 - 2 0 . 4 0 . 6 0 . 8 1 .02 X l B oFig 3 Dimen sionless pressure d is t r ibu t ion a long bear ingwidth a t d i f fe ren t va lues o f supply pressure1 6 2

1 . o I I I I IP s = 2 . 5 b a r a b s . , Bo_ IL o 2

o __ 0 . 8 - - H = 1 0 p m S y m b o l B r l B oI.

* 0 - 6

" 0 . 4 0 2 0

o . q I I I I "~0 - 0 0 - 2 0 - 4 0 . 6 0 - 8 1 . 0

2 X / B oFig 5 E ffect o f the recess width to the bearing outerwidth ratio (Br/Bo) on the d imens ionless pressuredistribution

1 9 9 4 V O L U M E 2 7 N U M B E R 3


5/9

1 2 0 0

1 0 0 0 -t -O* " 8 0 0 -QZ

J 6 0 ( -400, J 401 -20C -

0 0

S. Z . Kassa b--Externally pressurized rectangular gas bearing

I I I o l b : : =r . . . . . I . . . " - - .~ x p . o . I o o

~ ' - - , ~ l L o 3 1 8 oo ~ . "


6/9

S. Z. Kassab--Externally pressurized rectangular gas bearing0 - 6 0 . 6

0 .5

..J 0 . 4 -"oo~ , o . 3 -o~r-o03 0 . 2 -E~5

0 . 1 -

0.( 0.0

- 1

/ / [batPlbsly m p o L7

2 . 53 - 0

I I I t t0 . 2 0 . 4 0 . 6 0 " 8 1 " 0

B o / L o

0 . 5

0.40 . 3

0.2--

0'I0.00.0

Ps[ b a r a b s ] S y m b o l1 . 52 . 02 . 53 . 0

H =30 p.m

. barabso,,~,- % _ -" , " 3~ 5 " ' , \

I I I I I0"2 0 "4 0"6 0"8 1"0B o / L o

Fig 7 Va riation o f load-carrying capacity w ith width/lengthdependent on the pressure distribution results aspresented in Figs 2 to 4.The effect of changing the supply pressure, P~, on theload is shown in Fig 6. It is clear from Fig 6(a) thatfor ( B o / Lo ) = 2/3 and at a certain film thickness, H,increasing the supply pressure has the effect ofincreasing the load. Further, at a certain supplypressure, as the film thickness increases, the loaddecreases. The results shown in Fig 6(b) for ( B o / Lo )= 1 are in qualitative agreement with those in Fig 6(a)for ( B o / Lo ) = 2/3.On the other hand, for better presentation and foreasier comparison between different conditions, theload results are plotted in a dimensionless form, L*= L / {P ~ - P a ) A } , in Figs 6(c) and 6(d). Figure 6(c)shows that increasing the supply pressure, at a certainfilm thickness, has the effect of decreasing the dimen-sionless load, L*. This is in ag reement with the resultsof dimensionless pressure, P*, shown in Fig 3. Further,the differences between values of the load L* atvarious supply pressures are almost constant withrespect to the change in film thickness. In addition,increasing the film thickness at a certain supplypressure results in a reduction in the dimensionlessload, L*. The results shown in Fig 6(d) for (Bo /Lo)= 1 are in qualitative agreement with those presentedin Fig 6(c). They are also in agreement with thepressure distribution results shown in Fig 2. Thus, thesame physical reasons previously given for the pressuredistribution results are equally effective in explainingthe load-carrying capacity results.The effect of varying the bearing outer dimension on theload-carrying capacity is shown in a non-dimensionalrepresentation ( B o / Lo ) versus L* in Fig 7. At H =10 ~m Fig 7(a) shows that at certain supply pressure,

ratio (Bo/Lo) at di f ferent values of supply pressurethe load L* increases as ( B o / Lo ) increases up to avalue 0.8 and then starts to decrease. The location ofmaximum load is a result of the empirical equation asexplained previously in connection with the pressuredistribution presented in Fig 4. In addition, Fig 7(a)shows that, for a certain value of (Bo /Lo) , increasingthe supply pressure, Ps, results in a decrease in non-dimensional load L*. This behaviour is in agreementwith the results shown in Fig 6. Moreover, the resultspresented in Fig 7(b) for H = 30 Ixm supported theabove findings.The effects of changing the operating conditions andbearing outer dimensions on the bearing mass flowrate, M, are presented in Figs 8 and 9. Figure 8(a)shows that for ( B o / Lo ) = 2/3 and for a certain valueof Ps as the film thickness increases so does the massflow rate. This behaviour is physically correct since asH increases, for a certain value of ( B o / Lo ) andP~, the bearing hydraulic resistance decreases andconsequently the mass flow rate increases. In addition,Fig 8(a) shows that for a certain value of H, as Psincreases so does M. This behaviour has been physicallyexplained in connection with the results presented inFig 3. Moreover, the results presented in Fig 8(b) for( B o / Lo ) = 1 are in qualitative agreement with thosein Fig 8(a) and strongly support the characteristicsdescribed above.Figure 9 demonstrates the effect of the bearing outerdimensions ( B o / Lo ) on the mass flow rate, ForH = 10 i~m, ~ind at a cert ain supply pressure, as (Bo /L0) increases, M first increases up to a certainmaximum (at (Bo /Lo) = 0.55 to 0.65) and then startsto decrease. Further, for certain values of Bo/Lo, it isclear f rom Fig 9 tha t as the suppl]( pressure , Ps,increases, so does the mass flow rate, M. These results

1 6 4 1 9 9 4 V O L U M E 2 7 N U M B E R 3


7/9

S. Z . Kassab--External ly pressurized rectangular gas bearing

5 0

4 0

tn 30

= ~ 2 0 '-: 1 0

0 0

( ~ ) I I I I I 5 0 lB o . 2 , B ri +1 +O , / ,o+ , + f /

- - ~ ~ / / " l / l 3 0/ / , , , , "/ / . . ,

OI1 0 2 0 3 0 4 0 5 0 0H [ J a m ]

I I I 1 I

B o = 1 1 , B r = 0 . 1Lo B'- '~

1 0 2 0 3 0 4 0 5 0H [ J a m ]

F i g 8 E f f e c t o f f i l m t h i c k n e s s o n m a s s f l o w r at e, 1~1, a t d i f f e r e n t s u p p l y p r e s s u r e s

u1 O

1 .0

0 "~

0 - 6

m 0 " 4w" 7 .O '

I Hi

= 1 0 p mI I I I

%,...-- . . . . . ~ - s

0 - 0 I I I I I0-0 0 ,2 0 .4 0 .6 0"8 1 -0B o / L o

F i g 9 E f f e c t o f b .e a ri ng o u t e r d i m e n s i o n s (Bo /Lo) o nm a s s f l o w r at e, M , a t d i f f e r e n t s u p p l y p r e s s u r e s

a re i n a g re e m e n t w i t h t h o s e s h o w n i n F i g 8 a n d c a nb e e x p l a i n e d u s i n g t h e s a m e p h y s i c a l r e a s o n s .C o m p a r i s o n w i t h e x p e r i m e n t a l d a t aTh e r e s u l t s p r e s e n t e d a b o v e w i l l b e c o m p a re d w i t hs o m e o f t h e e x p e r i m e n ta l d a t a o b t a i n e d b y K a s s a b 6a n d S h a w k y a n d K a s s a b 7 '8 f o r a n e x t e rn a l l y p r e s s u r i z e drec tangu lar recessed a i r bear ing us ing an o r i f i cer e s tr i c to r . F i g u re 1 0 s h o w s a s a m p l e o f t h e p r e s s u reresu l t s p resen ted in F igs 2 to 4 as wel l as thec o r r e s p o n d i n g e x p e r i m e n t a l d a t a r e p o r t e d b y S h a w k yand Kas sab 7 '8 . I t i s impo r tan t to no te tha t thee x p e r i m e n t a l d a t a u s e d a r e t h e p r e s s u re d a t a m e a s u re dT R I B O L O G Y I N T E R N A T I O N A L

a l o n g t h e b e a r i n g c e n t r e l i n e . Th e fo l lo w i n g o b s e r -v a t i o n s c a n b e d e d u c e d f ro m t h e f i g u re :(1 ) Th e p r e s s u re i n s id e t h e r e c e s s i s n o t c o n s t a n t, a sa s s u m e d t h e o re t i c a l l y . Th e p r e s s u re d i s t r i b u t i o ni s c h a ra c t e r i z e d b y a p r e s s u re d e p re s s i o n z o n efo l l o w e d b y a n i n c r e a s e t o a c e r t a i n m a x i m u m .(2 ) A t t h e o u t e r e d g e o f t h e r e c e s s a n d t h e b e g i n n in go f t h e b e a r i n g l a n d r e g i o n , i . e . ( 2 x / B o ) = 0.1 ,t h e t h e o re t i c a l r e s u l t s a r e i n g o o d a g re e m e n tw i t h t h e e x p e r i m e n t a l o n e s i n a ll p r e s e n t e d c a s e s .(3 ) In the bear ing l and reg ion , i .e . ( 2 x / B o ) > - 0 .1 ,b o t h t h e o re t i c a l a n d e x p e r i m e n t a l r e s u l t s h a v et h e s a m e t r e n d ; a s t h e d i s t a n c e i n c r e a s e s , t h ep re s s u re d e c re a s e s u n t i l t h e y b o t h r e a c h a t m o s -p h e r i c p r e s s u re a t t h e b e a r i n g e x i t .( 4 ) Th e t h e o re t i c a l a n d e x p e r i m e n t a l r e s u l t s p r e -sen ted in F ig 10 (a) f o r ( B o / L o ) = 2 /3 a re ing o o d a g re e m e n t w i t h e a c h o t h e r , w h i l e F i g 1 0 (b )s h o w s a r e m a rk a b l e d e v i a t i o n b e t w e e n t h e o re ti c a la n d e x p e r i m e n t a l r e s u l t s .(5 ) In genera l , the theo re t i ca l resu l t s a re h igher thant h e c o r r e s p o n d i n g e x p e r i m e n t a l o n e s .O n t h e o t h e r h a n d , a c o m p a r i s o n b e t w e e n t h e l o a d -c a r ry in g c a p a c i t y , L , o b t a i n e d u s i ng t h e p r e s e n t a n a l y s isa n d t h e c o r r e s p o n d i n g e x p e r i m e n t a l l o a d r e s u l t so b t a i n e d b y S h a w k y a n d K a s s a b 8 i s s h o w n i n F i g 6 .Th e fo l l o w i n g o b s e rv a t i o n s c a n b e d e d u c e d f ro m t h i sfigure:(1 ) B o t h t h e th e o re t i c a l a n d e x p e r i m e n t a l r e s u l tsh a v e t h e s a m e t r e n d ; t h e l o a d d e c re a s e s a s t h ef i lm th ickness increases o r the supp ly p ressu red e c re a s e s .(2 ) Th e d e v i a t i o n b e t w e e n t h e t h e o re t i c a l a n d t h ec o r r e s p o n d i n g e x p e r i m e n t a l r e s u l t s i n c r e a s e s a sthe f i lm th ickness increases .F i n a l l y , t h e d i f f e r e n c e b e t w e e n t h e p r e s e n t t h e o re t i c a lresu l t s and the exper imen ta l resu l t s , shown in F igs 6

1 6 5


8/9

S. 7,. Kassab- -Ex terna l l y p ressur i zed rec tangu lar gas bear ing

1 .0 ~ I I IB -~ = 2 1 " S - l: = ' 1 ' P s = 2 b a r a b s "o I 3 o

~_. 0 . 8 - ~ i 1 ~ 30 5 0 H [ p m ]- ~ - ~ - - - T h eo re tica l

0 ' 6 - - A ~ A E x P e r i m e n t a l.

z -

~ 0 . 2 -i5

0 .0 I I I I " %0.0 0 -2 0"4 0.6 0.8 1.02 X / B o 0 . 0

I ~ I I IB r = 0 .1 P s = 2 b a r a b s .' " B o

1 0 3 0 5 0 H [ l . t m ]

-

0 . 2 0 . 4 0 . 6 0 - 8 1 .02 X / B o

Fig 10 Comparison between the present theoretical results and the experimental results of Sha wk y a nd Kassab 8a n d 1 0 , c a n b e c o n s i d e re d a d i r e c t c o n s e q u e n c e o f th efo l lowing reasons :(1 ) Th e s i m p l i fy i n g a s s u m p t i o n s u s e d i n o rd e r t oo b t a i n t h e t h e o re t i c a l r e s u l t s .( 2 ) Th e e x p e r i m e n t a l r e s u lt s u s e d fo r t h e c o m p a r i s o nh a v e a n a m o u n t o f u n c e r t a i n t y a n d a r e s u b j e c t

to some d i f f i cu l t i es .( 3 ) O n e s h o u l d n o t e t h a t t h e t w o r e a s o n s m e n t i o n e da b o v e m a y a c t i n a n o p p o s i t e d i r e c t i o n a n d t h i se n h a n c e s t h e d i f f e r e n c e b e t w e e n t h e t h e o r e t i c a la n d e x p e r i m e n t a l r e s u l t s .

Conc lus ionsTh e e f f e c t s o f t h e o p e ra t i n g c o n d i t i o n s a n d t h e b e a r i n gd i m e n s i o n s o n th e p e r fo rm a n c e o f a n e x te rn a l l yp re s s u r i z e d r e c t a n g u l a r r e c e s s e d g a s b e a r i n g a r e s t u d i e dt h e o re ti c a l ly . T h e r e s u lt s w e r e o b t a i n e d u s i ng s i m p l ify -ing assumpt ions ( i . e . Lr/Lo = 0 .9 , m a x i m u m d i m e n -s i o nl e s s l o a d a n d m i n i m u m m a s s f l o w r a t e , u s i n g as h a rp -e d g e d s u p p l y h o l e ) . Th e r e s u l t s o b t a i n e d a r ec o m p a re d w i t h a s a m p l e o f a v a i l a b l e e x p e r i m e n t a lr e s u l t s . Th e fo l l o w i n g c a n b e c o n c l u d e d :(1 ) F o r a b e a r i n g w i t h c e r t a i n d i m e n s i o n s , i n c re a s i ngt h e f i l m t h i c k n e s s o r t h e s u p p l y p r e s s u re r e s u l t si n a r e d u c t i o n i n t h e d i m e n s i o n l e s s p r e s s u re a n dload-car ry ing capac i ty wh i le i t increases the massf low ra te .( 2 ) Th e m a x i m u m d i m e n s i o n l e s s l o a d -c a r ry i n gcapac i ty fo r a l l opera t ing cond i t ions i s a t (Bo/Lo) = 0 .8 .(3 ) F o r a c e r t a i n fi lm t h i c k n e s s a n d s u p p l y p r e s s u re ,

as (Bo/Lo) increases , the mass f low ra te in i t i a l lyi n c r e a s e s t o a m a x i m u m v a l u e a n d t h e n s t a r t s t od e c re a s e .(4 ) Th e re i s q u a l i ta t i v e a g re e m e n t b e t w e e n t h ep re s e n t t h e o re t i c a l r e s u l t s a n d t h e c o r r e s p o n d i n g16 6

a v a i l a b l e e x p e r i m e n t a l d a t a . I n g e n e ra l , t h et h e o re t i c a l r e s u lt s a r e h i g h e r t h a n t h e e x p e r i m e n -ta l da ta . Th is d i f fe rence i s be l i eved to be main lyd u e t o t h e s i m p l i fy i n g a s s u m p t i o n s u s e d i n t h ep resen t s tudy , in add i t ion to poss ib le l imi t s o fu n c e r t a i n t y o f t h e e x p e r i m e n t a l d a t a e m p l o y e d .

AcknowledgementsT h e a u t h o r w o u l d li ke t o t ha n k D r K a m e l A .E l s h o r b a g y a nd D r M e d h a t A . S h a w k y o f t h e M e c h a n -i c al En g i n e e r i n g D e p a r t m e n t , A l e x a n d r i a U n i v e r s i ty ,fo r r e a d i n g t h e o r i g i n a l m a n u s c r i p t . Th i s p a p e r w a so r i gi n a ll y p r e s e n t e d a t t h e 4 t h In te rn a t i o n a l C o n fe r e n c eo f F l u i d M e c h a n i c s , 2 8 -3 0 A p r i l 1 9 92 , A l e x a n d r i a ,Eg y p t .

References1. K amal W.A. Effec t o f d i f fe ren t parameters on t he per formanceof external ly pressurized bearing. MSc thesis, Facul ty ofEngineering, Alexandria Universi ty, 19722. Salem E,A. and Slmwky M.A. Effect of fi lm profi le on theperformance of external ly pressurized rectangular sl iding gasbearings. Bulletin Facultyof Engineering, Alexandria University,1976, 15 , 925-9453. Salem E.A. and Slmwky M.A. L ubrican t inert ia effect inexternal ly pressurized rectangular gas bearings. Wear,1979, 54,201-2094. Slmwky M.A. Analysis and appl icat ions of gas bearings, PhDthesis, Facul ty of Engineering, Alexandria Universi ty, 19765. ~ E .A . t oNI S imwky M.A. An exper imenta l i nves t iga t i onin to t he per formance of ex t e rna l l y pressur i zed rec t angul ar ai rbearings. W ear 1978, .~0,237-2576. Kammb S.Z. Gas bearings. MSc thesis, Facul ty of Engineering,Alexandria Universi ty, 19807 . S lmwky M.A. a~l Klmab S .Z . Exper imenta l op t imiza t i on ofexternal ly pressurized rectangular gas beat ings dimensions.

1 9 94 V O L UME 2 7 NU MBE R 3


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S . Z . K a s s a b - - E x t e r n a l l y p r e s s u r i z ed r e c t a n g u l a r g a s b e a r in g

Bulletin Facu lty of Engineering 198 1, Alexandr i a Univers i t y ,20, 205-2258 . Shawky M.A. and Kassab S .Z . Performance charac t e r i s t i cs o fex t erna l l y pressur i zed rec t angul ar gas bear ing compensa t edwith orifice restrictor. Bulletin Facu lty of Engineering 1981,Alexandr i a U nivers i t y , 20 , 227-2489 . EIGayar M.S .A. Th e perform ance of ex t e rna l l y pressur i zed a ir

t h rus t bear ings . MSc thes is , Facul t y of Enginee r ing , MansouraUnivers i t y , 198610. Kandi l H .A. Opt imiza t i on and perform ance improv ement s ofex t erna l p ressur i zed rec t angul ar gas bear ings . MSc thes i s ,Facul t y of Engineer ing , A lexandr i a Univers i t y , 198711. Khalil M.F. , Sha wky M.A. and Kandil H.A . Elastohydrostat icl ubr i ca t i on of ex t e rna l l y pressur i zed rec t angul ar gas bear ing :

12 .

13 .

14 .

Exper im enta l ver if i ca ti on . Presen t ed a t t he Egypt i an Socie ty ofTr ibo logy F irs t Tr ibo logy Conference , 20-21 Decem ber , C ai ro ,Egypt , 1989Shawk y M.A. A n emp irical form ula for effect ive recess pres surein rec t angul ar gas bear ings . Alexandria Engineering J 1990, 29 ,No. 4, Sec. A, 217-221Boffey D.A. , Waddel l M. and Deardent J .K . A theore t ica land exper imenta l s t udy i n to t he s t eady-s t a t e per formancecharacterist ics of indust rial ai r lubricated thrust b earings. Tribol-ogy International 1985, 18 , No. 4 , 229-233Boffey D.A. , Barrow A.A. and Deardend J.K. Exper imenta li nves t iga t i on i n to t he per formance of an aeros t a t ic i ndus t r i a lt h rus t bear ing . Tribology International 1985, 18, No. 3, 165-168

TRIBOLOGY INTERNAT IONAL 167

rectengular gas bearing

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