Turbulent-Flow Hydro Static Bearings Analysis and Experimental Results

8/14/2019 Turbulent-Flow Hydro Static Bearings Analysis and Experimental Results

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Int. J. Mech. Sci. Vol. 37, No. 8, pp. 815-829, 1995Copyright 1995 Elsevier Science LtdPrinted in G reat Britain. All rights reserved0020-7403/95 $9.50 + 0.00

T U R B U L E N T - F L O W H Y D R O S T A T I C B E A R I N G S:A N A L Y S I S A N D E X P E R I M E N T A L R E S U L T S

LUIS SAN A NDR ES *, DARA CHILDS* and ZHOU YAN G*Mechanieal Engineering Department, Texas A&M University, College Station, Texas 77843, U.S.A. and

Cummins Engine Company, Inc., Charleston, South Carolina 29405, U.S.A.(Received 29 June 1994; and in revised form 8 November 1994)

Abs trae t--A bulk-f low thermohydrodynamic (THD) analys is for predic t ion of the s ta t ic anddynam ic performance characteristics of turbulent-flow, process-liquid hydros tatic jou rnal bearings(HJBs) is presented. The film-averaged momentum transport and energy equations replace thelubric ation Reynolds equation, and fluid inertia on film lands and at recess edges are preserved in theanalysis . Flow turbulence is accounted through turbulence shear parameters based on frictionfactors derived from M oody 's formulae. Numerical predictions are comp ared successfully to experi-men tal results from a five-recess, turbulent-flow, water-lubrica ted hydrost atic bearing o perating a ta h igh rota t ional speed. HJBs opera t ing in a hydrob mode ( i .e . wi th journal ro ta t ion) provide nobetter s tability characteristics than hydro dyna mic jour nal b earings and are likely to show half-speedwhirl.

AAoArb

C, C ,C~jCdCpDdoex, ey

A , AFx, FyH ,H ,

K i jkx, k~k j , k B

L , lMi irh~t

nNrecP

P~, P~, P~Px, PYR

RecRjRB

r j , rBTA TtU ,VU

N O T A T I O NnDL, journal or bearing surface area I-m2]C~nd2/4, equivalent orifice area [m 2]bl, recess ar ea [m 2 ]recess circumferential length [m ]radial clearance, characteristic clearance ( = {c(y)}min) I-m]dam ping force coefficients [Ns m -2 ]empirical orifice discharge coefficientspecific hea t [J kg - 1 K - 1journal d iameter [m]orifice diameter [m]displacements of the journal [m]am[l+(cm rj, B/H +bm /Rj, B)~m], turbulent friction factors based onam = 0.001 375; b m = 5 105; Cm = 104; em = 2~5fluid film forces IN]film thickness, recess depth [m]stiffness force coefficients [ N m -1 ](kj + kB)/2f j , R j , fB , R B , turbulent shear parametersbearing and recess axial lengths I-m]inert ia force coefficients [kg ]flow rate ove r differential segments [kg s- l]bearing mass f low ra te [kg s -1 ]normal vec tor to recess boundarynum ber of bearing recessesfluid pressure [N m - 2]externa l supply , ambient and recess pressures [ N m -2 ]Perturbed (dynamic pressures) IN m - a]journal radius [In]p, R f2c , /# , , nominal c i rcumferent ia l f low Reynolds numb erpH x / (U - f2R ) 2 + V2/#, Reynolds numb er re la t ive to journal surfacepHx/-~ + V2/1~,Reynolds number relative to bearing surfacemean roughness depth a t journal and bearing surfaces [m]bulk fluid-film temperature [K]Texl ,- Ts [K]time Is]z~=ArR, torqu e over a recess [ N m ]mean velocities [ms - 1U i Jr- Vj

815

Moody 's equat ion,


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8 1 6 L . S a n A n d r e s e t a l .

~t r

V~X , Y

X ~ y , ZQ(.0P#

~ x , y~ x u , d

"C"Cxz~ "~yz

Scr ip ts :a

r

s

JB

i , j

( H r + H ) A r + V s , r e c e s s v o l u m e [ m 3v o l u m e o f o r if i ce s u p p l y l i n e [ m 3inertial coordinates( 0 , ri D ) , ( 0 , L ) ,( O , H ( x , y , t ) )( U l r = o ) / ( ~ ) R ) , circumferentialvelocityentranc e swirl factorrotational speed of journal [rad s- 1]excitation or whirling frequency[rad s - l ]fluid density [kg m- a]fluid viscosity [N s m- 2empirical entra nce loss coefficients~x at up- , down-strea m recess edgese / c . , dimensionless ournal eccentricity~ot dimensionless im e coordina tewall sh ear stresses

refers to amb ient or discharge conditionsrefers to recess cond itionsrefers to supply conditionsrefers to journa lrefers to bushingrefers to first-o rder pertu rbatio ns (i, ~ X, Y directions)refers to chara cteristic (supply) values

1. I N T R O D U C T I O NO n e o f t h e m o s t s i g n i fi c a n t i n d i c a t o r s o f h i s to r i c a l c h a n g e i n t r i b o l o g y h a s b e e n t h e u s e o fp roces s f lu ids as lub r ican t s in f lu id - fi lm bea r ing sys tem s [1 ] . P roces s - l iqu id o r p ro d-u c t - l u b r i c a t e d h y d r o s t a t i c j o u r n a l b e a r i n g s ( H JB s ) a r e n o w u s e d i n l i q u e f ie d n a t u r a l g a s( L N G ) p u m p s , a n d c o n s e q u e n t l y o v e r h a u l i n t e r v a l s a r e e x t e n d e d t o s e v e r a l t i m e s t h o s e o fL N G p u m p s s u p p o r t e d o n c o n v e n t i o n a l b a l l b e a r i n g s [ 2 ] . H J B s h a v e a l so b e e n s e le c t e d a ss u p p o r t e l e m e n t s i n f u t u r e c r y o g e n i c h i g h - s p e e d t u r b o m a c h i n e r y s u c h a s th e H i g h P r e s s u r eF u e l T u r b o p u m p ( H P F T P ) a n d t he H i g h P re s su r e O x y g e n T u r b o p u m p ( H P O T P ) o f t h eS p a c e S h u t t l e M a i n E n g i n e ( S S M E ) [ 3 ] .

A s y s t e m a t ic re s e a rc h p r o g r a m o n H J B s f o r p o t e n t ia l c r y o g e n i c t u r b o p u m p a p p l i c at i o n sh a s b e e n c a r r i e d o u t a t t h e a u t h o r s ' U n i v e r s i t y s in c e 1 9 89 . A t e st f a ci l it y w a s d e s i g n e d a n db u i l t to m e a s u r e b o t h s t a t ic a n d d y n a m i c p e r f o r m a n c e c h a r a c t e r i st i c s o f h y b r i d ( h y d r o s -t a t i c / h y d r o d y n a m i c ) b e a r i n g s f o r t h e a p p l i c a t i o n d e s c r i b e d a b o v e . P u r i f i e d , h e a t e d ( 55 C )w a t e r i s u s e d a s t h e l u b r i c a n t i n t h e f a c i l i t y t o a c h i e v e c o m p a r a t i v e l y h i g h R e y n o l d sn u m b e r s i n t h e t e s t b e a r i n g w i t h o u t u s i n g c r y o g e n i c li q u id s . A d e s c r i p t i o n o f t h e t e s t f a c il i tya n d p r o g r a m a s w e l l a s s o m e o f t h e t e s t r e su l t s is g i v e n i n [ 4 ] .

A l o n g w i t h t h e e x p e r i m e n t a l i n v e s t i g a t i o n , S a n A n d r e s [ 5 , 6 ] i n t r o d u c e d a t u r b u l e n t -i n e r t ia l b u l k f l o w a n a l y s i s fo r p r e d i c t i o n o f t h e i s o t h e r m a l p e r f o r m a n c e c h a r a c t e r is t i c s o fo r i fi c e - co m p e n s a t e d H J B s w i t h i n c o m p r e ss i b le l iq u id s . F i l m - a v e r a g e d m o m e n t u m e q u a -t i o n s r e p l a c e t h e l u b r i c a t i o n R e y n o l d s e q u a t i o n t o k e e p f l u i d i n e r t i a l t e r m s t y p i c a l l yn e g l e c t e d i n c o n v e n t i o n a l m o d e l s . F l u i d i n e r t i a a t t h e f i l m l a n d s r e d u c e s f l o w r a t e s a n de n h a n c e s h y d r o d y n a m i c ef fe ct s. F o r l a m i n a r f l o w H J B s , r e c e s s - v o l u m e f l u id c o m p r e s s i b i l it yi s s h o w n t o d e t e r i o r a t e t h e b e a r i n g d y n a m i c s t a b i li t y c h a ra c t e r i st i c s [ 7 ] .

T o a v o i d t h e c o m p l e x i t y o f a f u ll T H D a n a l y s i s b u t s ti ll p a r t i a l l y a c c o u n t i n g f o r t h e f l u i dp r o p e r t i e s v a r i a t i o n , S a n A n d r e s [ 8 ] e x t e n d e d t h e i n c o m p r e s s i b l e l i q u i d m o d e l t o a b a r o t -r o p i c f l u i d m o d e l f o r a n a l y s i s o f c r y o g e n i c l i q u i d H J B s . T h e f l u i d p r o p e r t ie s a r e c o n s i d e r e dt o d e p e n d s o l e l y o n t h e l o c a l p r e s s u r e a n d a m e a n o p e r a t i n g ( u n i f o r m ) t e m p e r a t u r e .N u m e r i c a l r e s u l t s s h o w t h e e f f e ct s o f v a r i a b l e p r o p e r t i e s t o b e s i g n i f i c a n t f o r a L H z (l i q u idh y d r o g e n , h i g h l y c o m p r e s s i b l e ) h y d r o s t a t i c b e a r i n g , b u t s h o w n o s i g n i f i c a n t d i f f e r e n c eb e t w e e n t h e t w o m o d e l s f o r a L O 2 ( l iq u i d o x y g e n , le ss c o m p r e s s i b le t h a n L H z ) b e a r i n g .

H e r e , a b u l k -f l o w t h e r m o h y d r o d y n a m i c ( T H D ) a n a l y si s i s i n t r o d u c e d t o d e t e r m i n e t h es t a t ic a n d d y n a m i c p e r f o r m a n c e c h a r a c t e r i s ti c s fo r t u r b u l e n t f l o w H J B s . N u m e r i c a l p r e d i c -t i o n s o f f l o w a n d r o t o r d y n a m i c f o r c e c o e ff i c ie n t s a r e c o m p a r e d w i t h e x p e r i m e n t a l r e s u l t sf r o m a w a t e r - l u b r i c a t e d h y d r o s t a t i c b e a r in g . I n t h e a n a l y s i s , p o i n t w i s e e v a l u a t i o n o f


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Turbu le n t- f low hydrosta t ic bear ings 817

t em pera tu re and hence liqu id p rope r t ie s is ach ieved th roug h the so lu t ion o f the ene rgyt ransp or t eq ua t ion in the flu id f ilm wi th an ad iaba t ic b oun dar y a s sum pt ion ju s t i fi ed fo rHJBs wi th la rge mass f low ra tes . The s ta t ic cha rac te r i s ti c s o f a H JB inc lude the f ilmpressure , f lu id veloci ty and temperature f ie lds , mass f low ra te , f lu id-f i lm forces or bear ingload capac i ty , f r i c tion to rque , and pow er d i s s ipa t ion . The dynam ic fo rce cha rac te r i s ti c srefer to the s t if fness (Ku) , dam ping (Cu), and add ed m ass (M u) coeff ic ients requ ired forro to rdyn am ic ana lysi s . These coe f fic ien t s a re de f ined by the fo l lowing re la t ionsh ip :

F ,, = L F , , o J - L K ~ /< ,,,~ JL A Y J L q x q ~ J Ai-LMyx M yr J A]? (1 )

w h e r e A X (t ) a n d A Y( t ) a r e c o m p o n e n t s o f t h e j o u r n a l - c e n t e r d y n a m i c d i s p la c e m e n t a b o u tan equ i l ib r ium pos i tion . T he dynam ic - fo rce coe f fic ien t s de f ined by E qn (1 ) a re im por ta n tmeasu res o f dyn am ic bea r ing pe r fo rman ce s ince they in f luence the sys tem c r i t i ca l speeds ,the re son an t am pl i tude re sponse , and s tab il i ty o f the ro to r -be a r ing sys tem.

2. M A T H E M A T I C A L M O D E LThe gene ra l type o f bea r ing cons ide red a s a suppor t e lemen t fo r c ryogen ic l iqu idtu rbo pum ps i s a 360 -deg ree hydros ta t ic jou rn a l bea r ing , o r i fi ce -compensa ted , wi th a va r i-ab le num ber o f f eed ing recesses o r po cke ts ma ch ined in the su r face o f the bea r ing [3 ] . Thef low i s con f ined to the th in annu la r r eg ion be tween an inne r ro ta t ing jou rna l and a s ta t ion -

ary bushing (Fig . 1) .2.1. Governing equations for turbulent f luid-film flow sLarge p ressu re g rad ien ts typ ica l in c ryogen ic HJBs cause h igh ax ia l tu rbu len t f lowReyno lds num bers , an d the e f fect o f tu rbu len t mix ing fa r o u twe ighs mo lecu la r d if fu sivi ty . Inconsequence , the tem pera tu re r i se p roduce d by v i scous d i s s ipa t ion tends to be d i s t r ibu tedun i fo rm ly ac ross the f i lm th ickness and thus tem pera tu re g rad ien ts in the c ross - fi lm

xI=OF (~. . Y

X Orifice andupply kine

a ~ ~ h=c+ex osO+evsinO

0 ~ D

Fig . 1 . Ge om etry o f a hydro sta t ic jou rna l bear ing: (a) Ax ial v iew and coo rd inate system,(b ) Unwrapped bear ing su rface.


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818 L. San And reset al.co o rd in a t e (z ) a re co n f in ed t o t u rb u l en t f l o w b o u n d a ry l ay e rs ad j acen t t o t h e b o u n d in g(b ea r in g an d j o u rn a l ) su r faces [9 , 1 0 ]. Fu r th e rm o re , i n t h e ab sen ce o f reg io n s o f rev e rsedflow or rec i rcu la t ion , th e f lu id ve loci ty fie ld presen ts the sam e charac ter i s t ics as d iscusseda b o v e .

Th e co n s id e ra t i o n s p re sen t ed a l l o w th e t h ree -d imen s io n a l co n t i n u i t y , mo men tu m an den e rg y eq u a t i o n s t o b e i n t eg ra t ed ac ro ss t h e f i lm th i ck n ess t o d e t e rmin e t h e two -d imen -s io n a l b u lk - f l o w g o v e rn in g eq u a t i o n s fo r t h in f lu id -f i lm f l o ws [1 1 , 1 2 ]:

C o n t in u i t y eq u a t io na ( p H ) a ( p H U ) a ( p H V )

- - + - - + - - - 0 ( 2 )a t a x a yC ircu mferen t ia l - mo men tu m eq u a t io n

a(pHU ) a (pH U 2) O(pHUV) - H OPa ~ -f a x + a y = ~ x + ~ z l ~ (3 )A x i a l - m o m e n t u m e q u a t i o n

a ( p H V ) a ( p H U V ) a ( p H V 2 ) _ H O Pa ~ + ax + a ~ = ay + ~ '= [~ (4)En erg y - t ra n sp o r t eq u a t io n

~ ( p H V T ) ~ ( ~ P a P V a P ']O( p _ _ _ HT) a ( p HUT)Cp ~ t~ t -{- a x+ R f ~ z = l n - Uv=z g - v z , = I o ( 5 )

wh ere t h e b u lk - f l o w p r imi t i v e v a r i ab l e s (U, V, P , an d T) a re d e f i n ed a s av e rag e q u an t i t i e sac ro ss t h e f i lm th i ck n ess , an d Q, rep re sen t s t h e h ea t f l u x f ro m th e f l u id f ilm to t h e b o u n d in gso l i d s . No te t h a t t h e mo men tu m f l u x es i n Eq n s (3 -5 ) a re a s su med t o b e a l i g n ed wi th t h emas s m ean v e lo ci ti e s . Th i s s imp l i f ica t i o n i s fu l ly j u s t if i ed fo r l a rg e R ey n o ld s n u m b er f l o ws[1 3 ,1 4 ] .

Th e wa l l sh ea r s t re sse s a re ca l cu l a ted acco rd in g t o t h e b u lk - f l o w th eo ry fo r t u rb u l en ce i nth in f i lm f low s [12 , 13]:

# (k, V) (6)

z~=ln H O P IX [ U k B - - ( U - Rf~)k j ]wh ere t h e t u rb u l en t sh ea r p a ram e te rs (k= , k x) an d (k , kB) a re l o ca l fu n c t io n s o f t h e R ey n o ld sn u m b e r s a n d f r i c t i o n f a c t o r s b a s e d o n M o o d y ' s f o r m u l a e [ 1 5 ] . T h e m o d e l c h o s e n t orep re sen t t h e w a l l sh ea r s t re s se s a s fu n c t i o n s o f t h e ro t a t i o n a l sp eed an d b u lk - f l o w v e lo c it ie si s b ased o n i t s s imp l i c i ty o f imp lem en ta t i o n , i ts ab i l i ty t o ch a rac t e r i ze d i rec t l y ro u g h su r faceco n d i t i o n s , an d mo s t imp o r t an t l y , o n i t s accu racy wh en co mp ared t o o th e r c l a s s i ca lt u rb u l en t l u b r i ca t i o n mo d e l s [1 6 , 1 7 ].2.2. M a ss co n serva t io n a t a recess

Th e co n t i n u i t y eq u a t i o n a t t h e recess i s d e f i n ed b y t h e g lo b a l b a l an ce b e tween t h e f l o wth ro u g h t h e o r i f ice re s tr i c t o r, t h e recess o u t f l o w in to t h e f ilm l an d s (Q,) an d t h e t em p o ra lch an g e o f fl u id m ass wi th in t h e recess v o lu me (Vr). Th e recess f l o w co n t i n u i t y eq u a t i o n i sex p ressed a s:

A o ~ / 2 p , ( P s - P , ) = Q r + p , - - ~ - + p , V , f l - ~ - - - f l t - ~ r ( 7 )


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T u r b u l e n t - f l o w h y d r o s t a t i c b e a r i n g s 8 1 9

w h e r e

a r e t h e l i q u i d c o m p r e s s i b il i t y f a c t o r a n d v o l u m e t r i c e x p a n s i o n c o e f fi c ie n t , r e s p ec t iv e l y ; a n dO r = ~ r P H ( U " n) dF (9)

i s the mas s f low r a te ac ro s s the r eces s edges (F r ) and en te r ing the f i lm l ands .2.3. Gl oba l e ne rgy ba l anc e e qua t i on a t a re c e s s

A g l o b a l e n e r g y b a l a n c e e q u a t i o n a t a b e a r i n g r e c e ss i s d e r i v e d , re f l e c ti n g t h e h e a t~ a r r y - o v e r ( a d v e c t i o n ) a n d m i x i n g e f fe c ts , a n d t h e f r i c t i o n h e a t g e n e r a t i o n ( d i s s i p a ti o n ) inthe recess (F ig . 2) :

C p ~ - - ' - V "1 - C p ~ g h d T d q - 2 2 f f l s i d e T s i d e ) = C p 2 ? h u T u -~- O r T s -1 - T orr~ '~ ( 1 0 )w h e r e

7 ~r = r ~ z A r R (11)i s the to rque ove r the r eces s a r ea , Qr i s the to ta l mas s f low r a te th rough the s upp ly o r i f i ce ,V~ i s the r eces s vo lume , and the s ub s c r ip t s " u" , " d" a nd " s ide" r e fe r to the up s t r eam ,d o w n s t r e a m , a n d s i d e e d g e s o f a r e c t a n g u l a r r e c e ss , r es p e c ti v e ly .

T h e t e m p e r a t u r e s a t t h e d o w n s t r e a m a n d s id e e dg e s o f t h e r ec e ss a r e a p p r o x i m a t e l y e q u a lt o t h e r e c e s s t e m p e r a t u r e :

Td = Ts ide = Tr = co ns ta nt (12)w h i l e th e t e m p e r a t u r e a t t h e u p s t r e a m o f t h e r e c es s is g i ve n b y :{ Tr i f (U -n ) > 0 ;T u = u p s t r e a m v a l u e s o t h e r w i s e . (13)

Q r , s

I i lU U , T u I I U d , d

F i l m L a n d \ R o t o r S u r f a c e

U p s t r e a m AU U , U

S id e I V s i d e , s i d e- \D o w n s t r e a m/

U d , T d

Side / I V s i d e "s i dFig . 2. C onc eptu a l desc r ip t ion of g loba l energy ba lanc e a t a recess .


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820 L. San Andr es e t a l .

3. B OUND AR Y C ON DITIO NS

Th e b o u n d a ry co n d i t i o n s fo r t h e f l o w v a r i ab l e s a re ex p ressed a s :(a) O n t h e 3 6 0 ex t en d ed f i lm l an d , t h e p re ssu re , v e lo c i t y an d t em p era tu re f i el d s a re

co n t i n u o u s an d s i n g l e -v a lu ed i n t h e c i rcu mfe ren t i a l (x ) d i rec ti o n .(b) D u e t o g eo m et r i c sy m me t ry an d n o j o u rn a l m i sa li g n men t , t h e ax i a l v e lo c i t y (V) an dthe ax ia l g rad ie n ts (d /dy) of a ll the f low variab les are zero a t the c i rcum feren t ia l cen ter line

(y = 0) o f the bearing .(c) At t h e b ea r in g ex i t p l an e (y = L) , t h e f lu id p re ssu re t ak es a co n s t an t v a lu e eq u a l t o t h e

d i sch a rg e o r amb ien t p re ssu re (Pa ) fo r su b so n i c f l o w co n d i t i o n s .(d ) Th e recess -ed g e t emp era tu re s a re o b t a in ed a s d esc r i b ed ab o v e . F lu id i n e r t i a a t t h e

recess ed g es is t r ea t ed t h r o u g h a Be rn o u l l i -t y p e re l a t i o n sh ip [8 ] , wh i l e th e v e lo c i t y v ec to r i sco n s id e red t o b e n o rma l t o t h e recess ed g es .(e ) At t h e f l u id / j o u rn a l an d t h e f l u id /b ea r in g i n t e r faces , t h e h ea t f l u x t o t h e b o u n d in g

surfaces Qs i s assu m ed to be zero . This ap par en t overs im pl i f ica t ion i s fu l ly jus t i f ied in lieu oft h e e x te n s iv e n u m e r i c a l w o r k p e r f o r m e d b y Y a n g e t a l . [18] .

4. P E R T U R B A T I O N A N D N U M E R I C A L A N A L Y S E S

Fo r sma l l amp l i t u d e mo t io n s o f t h e j o u rn a l a b o u t an eq u i l i b r i u m p o s i t io n , a l l f l o wv ar i ab l e s a re ex p ressed a s t h e su p e rp o s i t i o n o fze r o th - an d f i r s t -o rd e r f ie ld s rep re sen t in g t h es t ead y s t a t e an d d y n am ic mo t io n co n d i t i o n s , r e sp ec ti v e ly . Ex p a n s io n o f t h e g o v e rn in geq u a t i o n s i n t h e p e r t u rb a t i o n v a r i ab l e s y ie ld s t h e ze ro th - an d f i r s t-o rd e r f l o w eq u a t i o n s .Re fe ren ces [8 ] an d [1 1 ] p ro v id e co m p le t e d esc r ip t i o n s o f t h e an a ly s is an d t h e n u m er i ca lme th o d u sed . F lu id - f i lm fo rces an d ro to rd y n amic co e f f i c i en t s a re fo u n d b y i n t eg ra t i o n o fthe ca lcu la ted pressure f ie lds on the journal surface , i . e . ,

= L 2 ~ p o | s i n o i d O d y (14)F ~ j owh ere Po co r re sp o n d s t o t h e ze ro th -o rd e r p re ssu re f i e l d , an d

K i j - o ) 2 M i j + i c ~ C i j = - R P j h i dO dy (15)j o . ) o

w i th i , j = X , Y h x = c o s O h y = s i n 0a n d P x , P Y a re t h e d y n a m ic p re ssu re f ie ld s fo r j o u rn a l m o t io n s i n t h e X an d Y d i rec ti o n s ,respect ive ly [8] .

A ce ll fin i te -d i fference schem e is im plem ented to so lve the nonl inear d i fferen t ia l equat io nso n t h e f i l m l a n d s [ 1 3 ] , a n d a N e w t o n - R a p h s o n s c h e m e i s u s e d t o u p d a t e t h e r e c e s sp re ssu re s an d t o sa t i s fy t h e mass co n t i n u i t y co n s t ra in t a t each b ea r in g recess [8 ] . Th en u m e r ic a l p r o c ed u r e u s es th e S I M P L E C a l go r it h m i n tr o d u c ed b y V a n D o o r m a a l a n dRa i th b y [1 9 ] . Th i s a l g o r i t h m i s we l l k n o wn in t h e l i t e ra tu re , an d d e t a i l s o n i t s su p e r io rco n v e rg en ce ra t e , g r i d re f i n emen t sen s it iv i ty , an d accu racy a re w e l l d o c u m en ted [2 0 , 2 1 ] .

P a s t s im p l e r m o d e l s fr o m t h e s a m e a u t h o r [ 5 , 8 ] h a v e e v o lv e d to t h e c u r r e n t T H D m o d e lan d p ro v id e a mo re accu ra t e y e t e f f i c i en t co mp u ta t i o n a l t o o l . Th e co mp u ta t i o n a l an a ly sesh av e b een v a l i d a t ed wi th ex ten s iv e co r re l a t i o n s t o ex p e r imen ta l m easu rem en t s i n t u rb u l en tf l ow, wa t e r - l u b r i ca t ed h y d ro s t a t i c b ea r in g s [4 , 2 2 ] . Fu r th e r v a l i d a t i o n s t o ex p e r im en ta lfo rce co e f f i c i en t d a t a fo r LH2 HJBs a re g iv en b y Yan g e t a l . [2 3 ] . Ku r t i n e t a l . [ 4 ] a n dF r a n c h e k e t a l . [2 2 ] a l so rep o r t sen s i ti v it y an a ly ses o f th e n u mer i ca l p red i c t i o n s re l a t i v e t oexpe rim enta l v a lues for a _+ 10% varia t ion in the input em pir ica l para m eters (ori ficed ischarg e coeff ic ien t Ca, in le t losses ~x ,y, and re la t ive surface roughn ess) .In g en e ra l , c a l cu l a t io n s sh o w th a t a re l a t iv e ly sma l l n u m b er o f g ri d p o in t s fo r d i sc re t iz -a t i o n o f t h e b ea r in g su r face is t y p i ca l ly req u i red t o g e t g r i d i n d ep en d en t re su lt s . Less t h an3 % d i f fe ren ce i n b ea r in g s t a t ic an d d y n am ic p e r fo rman c e ch a rac t e r i st i c s a re o b t a in e d w h enco m p ar in g t h e re su l ts f ro m a 4 9 b y 8 g ri d (n u mb er o f c i rcu mfe ren t ia l x ax i al p o in t s ) wi thth o se f ro m a 7 9 b y 1 6 g r i d fo r t h e t e s t b ea r i n g rep o r t ed i n t h i s p ap e r .


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Turbulent -f low hydrosta t ic bear ings 8215. R E S U L T S A N D D I S C U S S I O N

The numer ica l example re fe r s to a HJB a r t i c le t e s ted by Moshe r e t a l . [24] . The te s tbea r ing i s a f ive -recess, o r i f i ce -compensa ted , smo oth- sur face hy dros ta t i c bea r ing wi thcharac ter is t ics out l ined in Table 1. A comple te descr ipt ion of the tes t fac i l i ty, exper imenta lprocedure and pa rame te r ident i f i ca t ion t echnique i s g iven by Chi lds and Ha le [25] . Theope ra t ing condi t ion for the bea r ing inc ludes :(a) 3 rot a t io na l speeds: 10 000, 17 500, an d 25 000 rpm(b) 2 supp ly pressures : 4.0, 5.5, an d 7.0 M Pa (600, 800, 1000 ps i)(c) 6 jour na l e ccen tr ic i ty ra tios : 0 .0, 0.1, 0.2, 0.3, 0.4, and 0.5(d) 1 sup ply tem pera ture : 55 C (130 F).

Em pir ica l p aram eter s l ike the or i f ice discharge coeff ic ients (Cd), the pre-swir l fac tor (c0,and the en t rance coe f fic ien t s a t the r ecess edges (~xu, ~xd, and Cr) a re need ed for num er ica lca lcu lat ions . Table 2 pre sent s the va lues of these pa rame te r s which a re de te rmin ed bymatching measured f low ra te s w i th the ca lcu la ted ones for the concent r ic ca ses . Theresu l t ing pa ram e te r s a re then u sed for the num er ica l ca lcu la t ions of al l non-ze ro-eccent r i -c i ty ra t io cases .The v i scos i ty and dens i ty of wa te r a re e s t ima ted f rom the fo l lowing formulae g iven byS he r m a n [ 26 ] :

( T ' ~ 8 '9# = 1.005 x 10 -3 \ 2 ~ J e t47~l /r -1/293)] (16)p = 1 0 0 0 e - e . s s 10 4 [ ( r - e 9 3 ) - ( e - o . 1 ) ] (17)

w h e r e t h e t e m p e r a t u r e ( T ) is i n K a n d t h e p r e s s u r e ( P ) i s i n M P a . A l l t h e o t h e r p r o p e r t i e s o fw a t e r a r e t a k e n a s c o n s t a n t .

T h e r o t a t i o n a l R e y n o l d s n u m b e r ( R ec = p , f ~ R c , / l ~ , ) b a s e d o n t h e s u p p l y p r o p e r t i e s a n dt h e n o m i n a l c l e a r a n c e i s e q u a l t o 2 . 5 x 1 04 f o r 2 5 0 0 0 r p m , t h u s s h o w i n g a n a p p l i c a t i o nw h e r e h y d r o d y n a m i c e ff ec ts a n d f l o w t u r b u l e n c e a r e s i g n i fi c a n t.

T h e m e a s u r e d a n d p r e d i c te d b e a r i n g d y n a m i c c h a r ac t e ri s t i c s , s u c h a s s ti ff ne ss , d a m p i n g ,a n d a d d e d m a s s c o e ff i ci e nt s , t h e w h i r l f r e q u e n c y r a t i o a s w e l l a s s t a ti c l o ad , f l o w r a t e a n dt e m p e r a t u r e a r e p r e s e n t e d a s f o l lo w s .

Tab le 1. Characteristics of water H JB [24, 25]Diam eter (D)Length (L)No. of recesses (Nrec)Recess volum e (Vr)Recess area ratio ( A J A )Orifice diameter (do)Orifice su pply lin e volum e (V~)Lan d roughness (peak-peak) ( r j and rB)Square recess (At x Br)Nominal clearance (at zero speed) (c ,)Supply f lu id tempera ture (Ts)

76.441 mm (3.0095 in)76.2 m m (3 in)50.185 x 10 _6 m a (0.0112891 in 3)0.22.49 m m (0.098 in)0.129 10 -6 m 3 (0.00787173 in 3)0.33/~m (13 pin)27 x 27 m m 2 (1.064 x 1.064 in 2)0.127 mm (0.005 in)328 K (130 V)

Table 2 . Empir ica l pa ramete rs for wate r HJBsf] (rpm) Ps (MP a) Cd ~ ~xu ~ x d ~ r17400 4.0 0.9035 0.5 0.25 0.5 0.57.0 0.8578 0.5 0.25 0.5 0.524600 4.0 0.8812 0.5 0.25 0.5 0.57.0 0 .8984 0.5 0.25 0.5 0.5


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8 2 2 L . S a n A n d r e s e t a l .

5.1. Static performance characteristicsStatic load capacity. F ig . 3 s h o w s t h e e x p e r im e n t a l a n d t h e o r e t i c a l e c c e n t r i c i t y r a t io s a sa f u n c t io n o f t h e s t a t i c l o a d f o r t h e h ig h e s t s p e e d t e s t e d ( 2 4 6 0 0 r p m ) . N o t e t h a t s o l ids y m b o l s in t h e f i g u r e s r e p r e s e n t e x p e r im e n t a l r e s u l t s , w h i l e h o l lo w s y m b o l s r e p r e s e n tn u m e r ic a l p r e d ic t io n s . T h e j o u r n a l d i s p la c e m e n t in t h e b e a r in g in c r e a s e s a lm o s t l in e a r lyw i t h t h e s t a t i c l o a d , w h ic h i s a c o m m o n f e a t u r e f o r in c o m p r e s s ib l e f lu id h y d r o s t a t i cb e a r in g s a n d a n n u la r s e a l s . T h e b e a r in g l o a d c a p a c i t y a l s o in c r e a s e s w i t h s u p p ly p r e s s u r ea n d r o t a t io n a l s p e e d , s in c e a h ig h e r s u p p ly p r e s s u r e p r o v id e s a l a r g e r h y d r o s t a t i c f o r c e a n din c r e a s in g r o t a t io n a l s p e e d g e n e r a t e s a gr e at er h y d r o d y n a m ic fo r ce . T h e n u m e r ic a l p r e d ic -t i o n s c o r r e la t e v e r y w e l l w i t h e x p e r im e n t a l m e a s u r e m e n t s ( m a x im u m d i f f e r e n c e : 7 . 4 %) .N o t e t h a t t h e e x p e r im e n t s d o n o t s t a r t a t z e r o s t a t i c lo a d , t h a t i s , t h e t e s t b e a r in g i s s l i g h t lye c c e n t r i c f o r z e r o a p p l i e d l o a d .M assf low rate . F ig . 4 s h o w s t h e e x p e r im e n t a l a n d t h e o r e t i c a l m a s s f l o w r a t e a s a f u n c -t i o n o f th e e cc e n tr i ci ty r a t io f o r s u p p l y p r e s su r e s eq u a l t o 4 M P a a n d 7 M P a . N o t e t h a t th es y m b o l s d o n o t c o in c id e w i t h e a c h o t h e r o n t h e h o r i z o n t a l a x i s s in c e t h e e c c e n t r i c i t y r a t io sa r e a c t u a l ly f u n c t io n s o f t h e g iv e n e x t e r n a l s t a t i c l o a d s . T h e m a s s f l o w r a t e o f t h e b e a r in gd e c r e a s e s s l o w ly w i t h t h e e c c e n t r i c it y r a t io . A s e x p e c t e d , a h ig h e r s u p p ly p r e s s u r e (i .e . h ig h e rp r e s s u r e d r o p a c r o s s t h e o r i f i c e ) p r o d u c e s a l a r g e r m a s s f l o w r a t e . T h e m a s s f l o w r a t ed e c r e a s e s w i t h r o t a t io n a l s p e e d s d u e t o t h e flu id v i s c o u s f o r c es g e n e r a t e d b y j o u r n a lr o t a t io n a n d t h e r e d u c t io n o f t h e r a d ia l c le a r a n c e f r o m t h e c e n t r if u g a l g r o w t h o f t h e s h a ft .

0 . 6 -

0 , 5 -o o . 4 -e~U o , ~ -

~ 0.2-

0.1

0.0

_ . Q _ _ ~ T : s t ( 4 M P a )T ~ e o r y (4 - M P a )

2030 4000 500 0 ~.OOO 10000S T A T I C L O A D C N )

F i g . 3 . E c c e n t r i c i t y r a t i o v s s t a t i c l o a d ( W a t e r H J B ) (P s = 4 a n d 7 M P a , P a = 0 . 1 M P a , T s = 5 5 C ,f ~ = 2 4 7 0 0 r p m ) .

o ,

2 .0

1 .81 .6

T e s t ( 4 M P a )O T h e o r y C 4 M P a) - T e s t (7 M P a )

I I

0 ,0 0.1 0 .2 0 .5 0 . 0 .5E C C E N T R I C I T Y R A T I O

F i g . 4 . M a s s f l o w r a t e v s e c c e n t r i c i ty r a t i o ( W a t e r H J B ) ( P s = 4 a n d 7 M P a , P ~ = 0 .1 M P a ,Ts = 55 C, fl = 24 700 r p m ) .


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Turbulent-flow hydro static bearings 823Th e l o wes t f l o w ra t e o ccu rs a t t h e l o w su p p ly p re ssu re (4 M Pa) , h ig h eccen t r i c it y (0 .5 ) an dh ig h sp eed (24 6 0 0 rp m) co n d i t io n . T h e n u m er i ca l p red i c t i o n s m a tch t h e ex p e r imen ta l d a t av e ry we ll (max imu m d i f fe ren ce < 3 %) .

Flu id ex i t t emp era tu re . Fig . 5 sh o ws t h e ex p e r imen ta l an d t h eo re t i ca l t emp era tu re s n ea rth e ex i t r eg io n o f t h e b ea r in g v e rsu s t h e eccen t r ic i t y ra t io . T h e su p p ly t em p era tu re i s a l sopresen ted in the f igures (dashed l ine) . The ex i t t empera ture increases wi th the eccent r ic i tyra t io . Th e m ax im u m t emp era tu re r ise ac ro ss th e b ea r in g l en g th (AT) is ab o u t 4 C a t th eh ig h es t sp eed (2 4 6 0 0 rp m) an d eccen t r ic i t y ra t i o (0 .5 ), b u t t h e l o wes t su p p ly p re ssu re(4.0 M Pa ) co n d i t i o n . Th i s i s ex p ec t ed s in ce t h e t emp era tu re r i se ac ro ss t h e b ea r in g l en g th i sp ro p o r t i o n a l t o t h e ro t a t i o n a l d rag p o we r (i n c reas in g w i th j o u rn a l eccen t r ic i ty ) , b u ti n v e rse ly p ro p o r t i o n a l t o t h e ma ss f l o w ra t e wh ich i n c reases wi th th e su p p ly p re ssu re . No teth a t t h e co n t r i b u t i o n o f t h e rad i a l-c l ea ran ce red u c t i o n d u e t o j o u rn a l r o t a t i o n t o t h e f ilmtemp era tu re r i se co u ld b e imp o r t an t s i n ce a sma l l e r c l ea ran ce p ro d u ces a l a rg e r f r i c t i o nto rq u e a lo n g w i th a sma l l e r b ea r in g f l o w ra te . M o s t o f t h e p red i c t ed ex it t emp era tu re s a reh ig h e r t h an t h e measu red v a lu es p re su mab ly d u e t o t h e ad i ab a t i c su r faces co n d i t i o nimp o sed o n t h e an a ly s is . Th e max im u m d i f fe ren ce b e twee n th e p red i c t ed an d m easu re d ex i tt emp era tu re s i s l es s t h an 2 % an d o ccu rs a t t h e l a rg es t eccen t r ic i t y ra t i o (0 .5 ), ro t a t i o n a lsp eed (2 4 6 0 0 rp m) , an d su p p ly p re ssu re (7 M Pa) co n d i t i o n . I f o n ly t h e t em p era tu re r ise(AT) i s co n s id e red , t h e max imu m d i ffe rence o f p red i c t i o n i s ab o u t 2 7 % . H o w ev er , a s t oa p o in t-wi se ma tch , t h e n u m er i ca l p red i c t i o n s a re g o o d , an d t h e ad i ab a t i c f l o w assu m p t io nis fu l ly jus t i f ied for th e be aring s tud ied .

Ex p e r im en ta l d a t a fo r wa t e r H JB s wi th sma l l e r c l ea ran ces (c , = 0 .0 7 62 m m an d0 .1 0 16 ram) a re a l so av a i l ab l e b u t n o t p re sen t ed h e re. Y an g et al. [ 1 8 ] s h o w t h a t t h ead i ab a t i c f l o w assu m p t io n i s ad eq u a t e fo r f lu id -f ilm f l o ws wi th l a rg e m ass f l o w ra t e s (~ / ) .Th i s a t y p i ca l f l o w co n d i t i o n s fo r an n u l a r p re ssu re sea ls an d H JB s wh ere ax i a l h ea tad v ec t i o n d o m in a t e s t h e t h e rm a l p ro cess . As th e b ea r in g c l ea ran ce d ec reases , t h e mass f l o wra t e d ec reases b u t t h e v i sco u s d i s s i p a t i o n i n c reases . Tab l e 3 p re sen t s t h e t h eo re t i ca l an dex p e r ime n ta l ex i t t emp era tu re s o f wa t e r H JB s wi th t h ree d i f fe ren t c l ea ran ces an d fo r t h el a rg es t sp eed (2 4 6 0 0 rp m) an d su p p ly p re ssu re (7 M Pa ) t e s t ed . P red i c t i o n s o f f l u id t emp er -a tu re s fo r th e sma l l c lea ran ce (c , = 0 .0 7 62 mm ) wa te r H JB a re n o t a s g o o d a s t h o se fo r th el a rg e (c , = 0 .1 2 7 ram) o r t h e med iu m (c , = 0 .1 0 16 ) c l ea ran ce wa t e r HJB s . P red i c t i o n s o f a l lt h e o th e r b ea r in g p e r fo rman ce ch a rac t e r i s t i c s l i k e mass f l o w ra t e , l o ad cap ac i t y , an dro to rd y n amic fo rce co e f f i c i en t s , a re n o t a f fec t ed b y t h e sma l l t emp era tu re v a r i a t i o n s(6 T < 1 0 C) in t h e th ree w a t e r H J B s .5.2. Dyn a mic p er fo rma n ce ch a ra c te r i s t i c s

Th e n u m er i ca l r e su l t s fo r t h e d y n am ic fo rce co e ff i ci en ts d e f i n ed in E q n (1) a re ev a lu a t edfo r sy n ch ro n o u s o p e ra t i o n (co = f~) an d co m p ared wi th t h e ex p e r imen ta l d a t a .

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Turbulent-Flow Hydro Static Bearings Analysis and Experimental Results

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