flow induced vibrations

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E L S E V I E R Chemical Engineering and Processing 34 I995) 289-298

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Flow induced vibrations in heat exchanger tube bundles

H. Gelbe M. Jahr K. Schr6 der

Inst i tut f i i r Pro zess- und Anlagentechnik, Teck, nische Universit i it Berl in, StraJ3e des I7. Juni I35, 10623 Berlin, Germany

Ded icated to Prof Dr Dietm ar Werner on the occasion o f his 60th bir thday

Abstract

A review is g iven of the m ost imp ortant param eters which have to be evaluated fo r des igning tea1 heat exchangers to w i thsflow-induced vibrations. After a short description of the mechanism of excitation, stability diagrams for fluid elastic instabilitydiscussed. Th e influence of non -un iform velocities in multispan ex chang ers and a sectional calculation of stability relationexplained by an example us ing a f lu id-dynamic computer program. Some recommendat ions for s t ructura l data and design deare offered in conclusion.

ynopse

R o h r e i n R o h r b f i n d e ln , d i e q u e r a n g e s t r 6 m t w e r d e n ,f i i h r en s chon be i k l e i nen F lu idgeschwi nd igke i t e ns t /i nd ige Sc hw i n g bew e g unge n aus . Nac h A r t de r au fd a s R o h r w i r k e n d e n S t r 6 m u n g s k r / i f t e u n t e r s c h e i d e tma n d r e i Gr u pp e n , d i e in Abb . 1 a u fge l is t e t s i nd .

D ie Tu rb u t enz e r r egun g ve ru r s ac h t z e i t l i c h und 6 r-t l ich r ege l lo se Sch w a nkun g en , so da l 3 d i e A mp l i t u denr e l a t i v k l e in b l e ibe n . D ie se G r u ndschwingun g en k6nnenz u Langze i ts ch~ iden de r R o h re i n den Um lenkb l e chend u rch Ma te r i a l ab t r ag f t i h r e n , so dab be i F l f i s s i g ke i t s -u n d Z w e i p h a s e n s t r 6 m u n g e n i n k r it i sc h e n A p p a r a t e n u .U . d e r G renz we r t f~ r d i e m ax im a l zu l~ is si ge Am p l i t uded i e k r i t i s ch e Ge sc h wind igk e i t b e s t immt .

P e r i o d i s c h e A n r e g u n g e n w e r d e n d u r c h i n d e r S t r 6 -m u n g a u f t r e t e n d e D r u c k s c h w a n k u n g e n h e r v o rg e r u f e n ,d i e au f e i nen enge n F r equ enz b e re i c h b e sch r ~ ink t s i nd .D ie w ich t i g s te i s t d ie y o r e E inz e l zy l in de r beka nn t e Wi r-b e l e r r egung . D ie S t rouh a l za h l en na ch G 1 . 1 ) s i n d y o nd e r R o h r a n o r d n u n g u n d v o n d e r R o h r t e i l u n g a b -h ~ ing ig . Abb i ldung 2 ze ig t be i s p i e lh a f t S t r o uha l z ah l enf ~ r d i e ve r s e t z t e Dre i e cks t e i l u n g nach Weave r e t a l .[ 9 ] . D i e se n ehmen m i t abne hmba re r Te i l u ng z u . B e i~ = 1 ,25 be t r /ig t d ie S t rouh alzah l 3 ,6. Dam it i s t d iek r it i sc h e A n s t r 6 m g e s c h w i n d i g k e i t u m d e n F a k t o r 1 8k l e ine r a ls be im E inz e l r oh r. D ie e i n g e b rac h t e k ine t i s cheE n e rg ie is t z u k le i n, u m b e i G a s e n e i n e b e m e r k b a r eR e s o n a n z a m p l i t u d e z n e r z e u g en . A u c h b e i Z w e i p h a s en -s t r 6mungen t r i t t k e i n e Wi rbe l e r r e g ung au f . Da gegen

m u g be i F l t i s s i gke i t s s t r 6mu n g en m i t Wi r be l e r r e gu ngg e r e c h n e t w e rd e n u n d / i h n l i c h w i e b ei d e r T u r b u l e n z err egu n g i s t zu p r i if en , ob d e r G ren zw er t f f i r d i e max im azul / i ss ige Ampl i tude i ibe~schr i t ten wird .

F lu ide l a s t is ch e In s t ab i l i t/ i t en t s t eh t au s s e lb s t e r r eg t eK o p p e l s c h w i n g u n g e n . A b b i l d u n g 3 z e ig t d e n t y p i s c h eA m p l i t u d e n v e r l a u f b e i z u n e h m e n d e r A n s t r 6 m g e s c h w i nd igke i t e i nes B f i n de ls . Wh i r l i ng en t s t e h t du rch s ch wi ngw egpropo r t i ona l e Kr ~ i f t e , d i e d i e A mpl i t u den am k r it i schen Punkt zwar s te i le r ans te igen lassen , es s te l ls ic h a b e r G l e i ch g e w i c h t z w i sc h e n a u f g e n o m m e n e r u ndiss ip ie r te r Energ ie e in . Bei Gal loping dagegen i s t deAns t i eg ab rup t , ve ru r s a ch t du rch s chwing ges chwi nd igke i t sp ropo r t i o na l e Kr f i f t e , d i e de r D f i mpfung e n tgegenwi rken und d i e se add i t i v z u Nu l l ode r n eg a t i v we rden lassen .

Es g ib t e ine Vie lzahl yon Model lans~i tzen f i i r d ieer reg ende n Kr~if te , d ie von C hen [1] besch r ieben werd enDe r f il tes te Ansa tz , G1. 4) , s tam m t yon Con no rs [16]Verwend e t man ande re K ra f t an s / i t z e , un t e r E i n -bez i ehung v o n ge sch wind igke i t s abM n g ig en A n te i l en , se rh~i l t man ff i r F lu ide mi t ger ingen Dichten d ie g le ichAbh / ingigk ei t wie in G1. 4) , f t ir F lu ide mi t grog enDich t en F l f is s i gk ei te n ) dagegen Expo nen t e n P deMa ssend~ imp fungspa rame t e r s , d i e l de i n e r a l s 0 , 5 s i n d

D i e P r f if u n g u n d A n p a s s u n g d e r a u s d e n M o d e l l-ans~i tzen gefundenen Abh/ ingigkei ten e r fo lg t in S tab i l it / i tsd i ag ramm e n . E i n so l c hes D iag r am m f t ir e i n B f in demi t verse tz te r Dre iecks te i lung ze ig t be isp ie lhaf t Abb. 3Zur s i che r en Aus l egung i s t ma n ge zw u ng en , d i e un t e r



29 H. Gelb e et al. ; Chemical Enghwering and Processing 34 1995) 289 -29 8

B egrenzu ng z u b enu t z en , d i e f~ r g r 613ere Wer t e de sM a s s e n d / i m p f u n g s p a r a m e t e r s d u r c h d i e C o n n o r s -G le i chun g gu t b e s ch r i eben w i r d . Im F l t is s i gke it s b e r ei chis t d ie S tab i l i t~ . t sgrenze mi t der f / . i r W irbe ler re gun g( ges t ri che l t ge ze ichn e t ) i d en t i s c h . A n de re A u to r en , z .B .P e t t i g r ew und Ta y l o r [ 15 ], z e i gen k e inen Sp r u n g ode r,z .B. Tro id l [19] , h6 here Grenz wer te .

S t ab i l i t ~ i t s d i ag ram me g es t a t t en d i e Be s t i mmung vonk r i ti s c h en S p a l t g es c h w i n d ig k e i e te n u n t e r d e r A n n a h m eidea ler Verh/ i l tn i sse , das he iBt f i i r homogene Bi indelm i t k o n s t a n t e m Q u e r s c h n i t t ( G l e i c h v e r t e i l u n g d e rR o hre , ke ine Ga s s en o d e r R ands pa l t e n ) sowie f i i r kon -s t a n te A n s t r 6 m - u n d S p a l t g es c h w i n d ig k e i t en in y - u n dz - R i c h t u n g .

In r ea l en W~r me t ibe r t r ag e r n w i rd d i e S t r6m ungd u rch E inba u t en g e s t6 r t , s o dab s i ch ung l e i ch f6 rmigeR o h r u m s t r 6 m u n g e n e i n s t e l l e n , z . B . h i n t e r d e mE i n t r i t t s s t u t z en , h in t e r P r a l l - und Umlenk b l ech en undi n f o lge von B i inde l g a s se n u n d Rand spa l t en sow ied a d u r c h b e d i n g t e r B y - p a s s - S t r 6 m u n g e n . C o n n o r s [ 2 2 ]

ha t se ine S tabi l itS . t sg le ichung (4) au f ungle ichm/ iBigeUm s t r6 m un g e rw e i t e r t und e rM l t GI . ( 7) . G l e i chun g(10) def in ie r t d ie ' / iqu iva len te ' k r i t i sche Geschwin-d igkei t , wie s ie e inem Stabi l i t / i t sd iagramm f i i r g le ich-f 6 rm i g e D u r c h s t r 6 m u n g e n t n o m m e n w e r d en k a n n .

Bei der Aus legung rea ler W~irmei iber t rager i s t o ffe i n e abschn i t t s we i s e Be re chnu n g p ro S t r 6mun g sseg -m e n t zweckmfi l3 ig , G o y d e r [ 2 3 ]. A b b i ldun g 5 v e rdeu t -l ich t d ies f t i r mehrfach ges t i i tz te Bi inde l , wobei mi tv a r i ab l en ode r m i t ' / i qu iva l e n t e n ' kons t an t enG e s c h w i n d i g k e i t e n p r o A b s c h n i t t g e r e c h n e t w e r d e nk ann . E in V or t e i l d ie s e r Vo rge h ens we i se i s t es u . a ., d abauch wei te re yon der Rohr l f inge abhf ingige Gr61?en:

M a s s e n b e l e g u n g , D i c h t e - - z . B . i n K o n d e n s a t o r e n - - ,D / im p f u n gen und dam i t auch S t ab i l it ~ . tskon s t an t en [s .G1. (11 )] be r i i ck s i ch t ig t we rd en k 6nne n , w obe i m an d i ev a r i ab l en Pa r ame t e r z we ckm f i l 3 i g p ro Abschn i t t n k on -s t a n t a nn im mt . D u r ch da s d i f fe r en ti e ll e Ene rg ie v e rh f il t -nis AS~,,, we rde n die BeitrS.ge, die d ie ein zeln enAbschni t te zur Ins tab i l i tS . t be i t ragen , gewichte t .

I n r e a l e n W S x m e i i b e r t r a g e r n k e n n t m a n n u r d i e a u fd en j ewe i l i g en S t r6mu n gsq u e r sc hn i t t b ezogene n Mi t t e l -w e r t e t i ' s . .. und h i e r m e i s t a u ch n i c h t d i e r e i n en Que r-s t r o m a n t e i l e . M o d e r n e F l u i d - D y n a m i k - P r o g r a m m eb i e t en heu t e d i e M 6 g l i chke i t , Gesc hwi nd igke i t s v e r t e i l -u n g e n i n R o h r b { i n d e l n z u b e r e c h n e n u n d R e g e l n

f i i r d i e Wah l von g ee ign e t en ko n s t an t en Ges chwind i g -k e i t en zu e n tw i ck l e n , d i e i n de r P r ax i s e i ne e in f acheD i m e n s i o n i e r u n g o h n e R e c h n e r e i n s a t z e r l a u b e n . B e id e r S c h w i n g u n g s b e r e c h n u n g m i t F l u i d - D y n a m i k - P r o -g r a m m e n m u l 3 m a n z u r B e s t i m m u n g d e s k r i t i s c h e nVo l u me n s t r om s I )'k d en Vo lum ens t rom 12 so l a ngevar i ie ren , b i s f / J r e in kr i t i sches Rohr bzw. f t i r e inenRohrspa l t s d ie in GI . (13) def in ie r te S tab i l i tS . t skenn-zahl K~ = I w i rd . D er Z/ ih le r in G1. (13) h~ingt vo mVo l u m e n s t r o m u n d d e r G e s c h w i n d ig k e i ts v e r te i lu n g , d e r

N e n n e r v o n d e n S t r u k t u r d a t e n u n d d e m K - We r t a u sde m S t ab i l i t ~ . t sd i a g ramm ab . Re chne t man ab -s chn i t t swe i s e , so e rMl t man f t h j e d en Ab sch n i t t d i ed i ffe rent ie l le S tab i l i t f i tskennza hlA K s un d Ks*. na ch Gl.( 16 ) . D urch d i e q uad ra t i s che Mi t t e l u n g do m in i e r e n d iAbsc h n i t t e m i t den g r 6 13 t en Wer t e n yonA K s .

Abb i ldung 6 ze ig t b e r ech n e t e Ge s chwind igke i t sv e r

t e i l ungen f i i r den gemesse nen k r i t i s chen Vo lumen-s t rom in e ine m Ver suchs wf i rme t ibe r t r age r m i t 2 Uml enkun ge n , U rbas e t a l. [ 24 ] . Es wu rd e e in e ab sch n i t t swe i s e Be re chn ung d e r zXK ~ -Wer te du r chge f t i h r t , e i n mami t k on s t a n t e n Ge schw ind igke i t en ~V's.,, u nd zu m ande ren mi t be r e chne t en Ges chw ind igke i t e n . Da s E rge bn i s en tMl t d i e Tabe l l e i n Abb . 7 . D ie k r i t i s c heRoh r r e ihe 2 w i rd mi t K* = 0 ,90 b ewe r t e t , b e i G t i lt i g kede r Re chenvo r seh r i f t Mt t e man 1 ,0 e rwa r t e t . E inGrund f t i r d i e A b we ichung k6n n t e i n d e r r i c h t i ge nBezugs g r6Be f ii r d i e G e sehwind igke i t l i e ge n, D ie be id eSpa l t ge s chw ind igke i t en ft ir e in Ro h r i n 2. Re ih e s i n dun t e r s c h i ed l i ch und d i e Ans t r6 m g e se h w ind igke i t aude m Spa l t de r d a v o r l i egend en Re i h e i st g r613 e r. Ve rs u chswe i se w urde f t i r d i e 2 , Re ihe n u r m i t d e r Sp a l tge schwind igke i t de r da v o r l i egend e n Re ihe , d . h . m ide r A ns t r 6m g eschw in d igke i t , g e r ech n e t . Das E rgebn ii s t K ~= 1,07 , d . h . de r M i t te lw e r t au s den be id enGrenz ff i l l en ze ig t e i ne gu t e Obe r e in s t immu ng vo nM o d e l l r e c h n u n g u n d E x p e r i m e n t .

F t i r i dea l e La g e rb ed ingu ngen l a s s en s ieh E igen -f o r m e n u n d - f r e q u e n z e n y o n E i n z e l r o h r e n i n L u fana ly t i s ch be r echne n . W~ih rend d i e be r echn e t en F r equenz en mi t den i n B{ ind e ln au f t r e t enden r ec h t gui i be r e in s timme n , k 6nn en d i e Ampl i t u d e nv e r lS .u f e we gen i c h t b e k a n n t e r L a g e r d f i m p f u n g e n u n d u n t e r -s ch i ed l i c h e r Las t enve r t e i l un g en i n e i n ze ln en A bsc hn i tten s t~ i rker var i ie ren . Bei Rohren mi t un terschiedl icheS t i it z l /i n g en (Fen s t e r roh r e ) i s t i n a l le n Ab sch n i t t e n mie inem e i nhe i t l ichen We r t f , zu r e ehne n , L eyh [2 9 ] konnt e z e ige n , dab s e l b s t i n Ve r suehe n mi t s eh r k l e i ne nLage r sp i e l en von 0 ,15 mm i n de n S t t it z b l ec he n de r e r s tS c h w i n g m o d e - - a u c h i n d e n A b s c h n i t t e n m i t d e n g er i ngs t en S t i i t zw e i t en - - dom in i e r t e .

Be i d en D / i mpfungen s i nd d r e i An t e i l e zu un t e r s che iden : M a te r i a l d f impfung , v i s kose D ' a m p fung un d S t rukt u rd / im pfung . D ie S t ru k tu rd /hnp f u n g l i e f e r t i n r e a l eA p p a r a t e n d e n g r 6 B t e n A n t e i l u n d w i r d v e r u r s a c h

d u r c h m e c h a n i s e h e u n d v i s k o s e R e i b u n g d e r R o h r e ide n Bohr u n gen d e r S t t i t z b l eehe sow ie du r ch S t6 B e ide n B lec h en . J ah r [ 3 2 ] kon n t e z e i g e n , da b d i e Ma t e r i a ld / imp fun g f i ir f e s t e in geschwe iB te R o h r e un abh f ing ivon de r Amp l i t ude i s t , d ab j edo ch be i a x i a l be -weg l i che n , d u rch Gummi r inge f i x i e r t en L age rn e ins t a rke Ampl i t ude nabh ~ i ng igke i t a u f t r a t . D i e S t ru k tu rd S . m p fu n g z e ig t e si ch s o w o h l v o n d e r L a g e r b r ei t e u n dLage r to l e r anz a l s auch vo n den Ampl i t ud en ab h{ i ng igS i e n im mt au l3e rdem zu , we nn d i e S t t i t zb l eehe n i ch t i



H. G eIbe eta / . / Chemical Engineering and Processing 34 I995) 289- 298 291

den Schwingungsknoten ( f iquid is tan te S t f tzs te l Ien)angebracht werden .

Aus GI. (22) lassen sich die folgenden konstruktivenMal3nahmen entnehmen: Die Rohre inspannl f inge L ha tden sff irksten Einflul3 auf die Stabil i t / i tsgrenze, gefolgtv o m mi t t l e r en Ro h rd u rc hme ss e r m mit e inem Expo-nenten 1 ,5 . Die W anddicke s geht wie der E last izi -

t~itsmodul E m it der W urzel ein. D er Einflul3 der M assem is t ff r P -~ 0 ,5 vernachl/ i ssigbar. Du rch Erh6 hung derD~impfung Ai l~il3t s ich Ws,k erh6h en, wob ei brei tereLager, kleinere Lagerspiele, ungleiche Stftzabstf indeoder zus~itzl iche D/impferelemente, die den Str6mungs-querschnit t m6glichst wenig versperren sollen, helfenk6nnen. G assen im Bfnde l s ind zu vermeiden oderdurch Dichts t re i fen bzw. Verdr~ingungsk6rper zuversperren.

Bei Rohrbfndeln mi t g le ichen St f tz l f ingen i s t derEin t r i t t sbere ich unter dem Stu tzen der kr i t i sche . Daheri s t d ie Gf te der Ein t r i t t sverte i lung entsche idend ff r denGrenzvo lumens t rom . H i l fe s ind Ja lous ie-Ver te ile rb leche

(ke ine P ra ltb leche ) in Verbind ung mi t e inementsprechend gr61?eren f re ien R aum f iber dem Bf indeloder d ie Anbr ingung gr61?erer oder mehrerer S tu tzenam E in t r it t . Der A bs tand zwischen Stu tzenaus t r i t t understem Rohr soll te nicht kleiner als zweimalRohrd urchm esser sein, Jah r und Gelbe [26] . DerEinf lu l? von Pra l lb lechen w urde von Leyh [29] unter-sucht . Als E rgebnis is t festzuhalten, dal3 Prallblechenicht g eeignet sind, die S chwingung sanffi l l igkeit zuverbessern , so ndern in den meis ten F ~i l len d ieseentsche idend verschlechtern .

1 I n t r o d u c t i o n

Tub e bundles subjec ted to a c ross- flow vibrate evenat low fluid velocities. These f lo w-indu ced vibrat ionsare caused by t ime-dependent forces , which can bedetermined by m easur ing the pressure f luc tua tion a t thetube sur face . In order to des ign rea l hea t -exchangertube bundles capable of w i ths tanding cr i t ica l v ibrat ions ,i t i s necessary to obta in inform at ion about a num ber ofinf luencing parameters .

2 E x c i t a t i o n m e c h a n i s m i n t u b e b u n d l e s

Three g ro u p s o f m ec h an i s m hav e bee n advanced byChe n [1], Pa '/doussis [2] , and W eaver a nd Fitzpa tr ick[3]. They are depicted in Fig. 1.

Turbulent exc i ta t ions ex is t even a t low ups t reamveloc i ties. The y are somet imes super im posed by per iod-ical forces, e .g. by vortex excitat ion. The result ingvibrat ions provide the basis for f luid elast ic instabil i tyarising from self-excited forces.

~ b : l c Shed ~c l / TL~b . l. en [ ~ fe fingwlfh a smaLl . [ M th a br~ : l

F P e : ~ n c y S p e c tr um F r ~ q u a ~ S p e c tr umR u ~ a s t l c

~ ~ g

cou st ic esonanceI ~ 1 5 a t t ~ n c j k

Fig 1 V ibration excitation mecha nisms in tube bundles

2 1 Turbulence excit ation

The h igh f low turbulence in a tube bundle i s thecause o f this form of excitat ion [4]. Since the f luctua-t ions are irregular in space and t ime, their ampli tudesare relat ively small . N evertheless they may be the origin

of long- te rm dam age ar i s ing f rom mater ia l abras ion , sot ha t t he m a x imum a l lowab le am p l i tude ma y de t e rminethe cri t ical velocity for l iquid and two-phase f low.Semi-empirical models have been devised to calculatethese ampli tudes [5,6] .

2 2 Periodical excit ation

Per iodica l pressure f luc tua t ions l imi ted to a nar rowfrequency range , as know n f rom s ingle- tube vor texexci ta t ions , a re the reason for the second mechanism.This phe nom enon ma y exc ite v ibra t ion in l iqu id f low oracoustic resonance in gas f low.

The St rouhal num ber [Eq. (1) ] depends on the tubear ray and on the p i tch ra t io of the bundle :

S r = f w ' D (1 )Woo

Undisturbed K~irmfin vortex streets can only developwhen la rge p i tch ra t ios a re involved; wi th smal l p i tchrat ios, i .e . 1.1 < ~ < 2.0, which are of greater interestf rom a technica l po in t of v iew, vor tex format ion i simpeded by ne ighbour ing tubes . S t rouhal numbers de-pend on the pos i t ion of the tube in the bundle and onthe ex is t ing flow condi t ions ( turbulence , Reynolds nu m-ber, acous t ic resonance) . The ex is tence of two S t rouhal

num bers has been d emo nst ra ted in square- in- l ine a r rayswi th la rge p i tch ra t ios [7] , and d i ffe rent S t rouhal num -bers were measured in the f i r s t and second row ofrota ted sq uare arr ays with r > 1.31 by W eaver et al . [8] .Normal ly, wi th smal l p i tch ra t ios , on ly the second andth i rd rows are endangered by vor tex shedding . Beyondthe th i rd row the per iodic i ties a re neglig ib le comparedto broad-band turbulence . S t rouhal numbers for thenormal tr iangular array [9] are depicted in Fig. 2. Theyincrease with decreasing pitch rat io.



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F i g . 2 . S t r o u h a l n u m b e r s f o r n o r m a l t r i a n g u l a r a r ra y s .

3 .5

Measu rem en t s w i t h p r e s su re r e ce ive r s o r ho t w i r e s[ 9 , 10 ] have con f i r med t he va l i d i t y o f t he uppe r cu rv ed e r ived by Zuka u ska s an d K a t i na s [ 11 ] . S ince t heS t r ouha l num b er i s 3 . 6 f o r z - - 1 . 25 , i t f o l l ows t ha t t hecr i t ica l up s t re am veloc i ty ca lcu la ted wi th fw = . / '1 is 18-t i mes sma l l e r t ha n fo r a s i ng l e t ub e . Thus t h e k ine t i ce ne rgy i n ga s f l ow i s t o o sma l l t o c ause an obs e rvab l er e s onance amp l i t ude . Vor t ex exc i t a t i on a l so d o es no ta ppea r i n t wo-ph ase f l o w. On ly i n l i qu i d f l ow ha s i tb e en obse rved . I n t h a t c a s e i t h a s t o be checkedw h e t h e r t h e t h r e s h o l d v a l u e f o r t h e m a x i m u m a m p l i -tude i s exceeded .

2.3. F luid elast ic instabi l i ty

A typ i ca l r e spons e fo r i n c r ea s i ng f r ee - s t re am ve loc i tyin a bundle i s shown in F ig . 3 . At low ve loc i t ies ,v i b r a t i o n s r e s u l t f r o m t u r b u l e n c e a n d s u p e r i m p o s e dvor tex e xc i ta t ion . Ad di t io na l forces , i .e . g3 andg4, o c c u ra t the c r i t ica l ve loc i ty a t which f lu id e las t ic ins tab i l i tyco mmences . T hes e fo r ce s a r e p ro p o r t i ona l t o t he amp l i -t ude s x a n d y, t o t he r e s pec t i v e v i b r a t i on ve loc it i es 2an d ) :,, and t o t he r e spe c ti v e ac ce l e r a ti ons 2 an d / ; :

m j J + d v . f ' + C y . y = g , ( t ) + g 3 ( y , S , , y , x ) (2)

. , <

- Y q x~b~ent [~.¢fet~qg ftuide{astic instaNily

g.O c r i f ~ t

DF L o w V e L o c i t y

F i g . 3 . Ty p i c a l a m p l i t u d e r e s p o n s e .

m • 2 + d , . . 2 + c x x = & ( t ) + /, ' 4 (2 , 2 , x , y ) ( 3 )

Acce l e r a t i on fo r ce s a r e t aken i n to a c coun t by a dd inf l u id m ass [ 1 ] , l e ad ing t o a r educ t i o n i n t he na t u r af requencies .

W h i r l i ng o r so f t s e l f - exc it a t ion [1 2 ] o ccu r s w he n p r eva i l i n g amp l i t ude -p ropo r t i ona l f o r c e s a c t a ga in s t t hs t i f f ne s s . The y a r e a f f ec t ed by t he mo t io n o f t h e t uba n d o f n e i g h b o u r i n g t u b e s l e a d in g t o e v a d i n g m o t i o n sThe e f f ec t o f hyd rodynamic coup l i n g i nc r ea se s w i th igher s ta t ic pressure in a gas as wel l as in l iqu id f low[13 ] . How eve r t he i n f l uence o f damp ing bec omes domnan t f o r l i qu id s [ 1 ] . A t t he s t a r t o f wh i r l in g , an equ il i b r i um ex i s t s be t ween t he a b so rbe d and d i s s i pa t edenergy.

In con t r a s t , t h e amp l i t ude ca u sed by ga l l op ing , a l soca l led hard se l f -exc i ta t ion [12] , increases abrup t ly. S incthe fo r ce s a r e p r o po r t i ona l t o t he v i b r a t i o n ve lo c i t ythe y a c t aga in s t da mp ing and ma k e t h e r e su l t an t f o r ceze ro o r n ega t i ve [2,14 ]. The m o t ion o f a t ube ex c i te d bg a l lo p i n g is i n d e p e n d e n t o f t h e m o t i o n o f n e i g h b o u r i ntub es . Usua l l y wh i r l i ng and ga l l op i n g a r e su pe r imp oseand f l u id e l a st i c v ib r a t ions co mm e n c i n g w i th w h i r li nch ange t o ga l l op in g a t a c e r t a i n p o i n t . F rom expe r ien ce , wh i r l i ng domina t e s i n a ga s f l o w th rough d i sp l a c e d t u b e b u n d l e s a n d g a l l o p i n g o f t e n d o m i n a t e s il i qu id f l ow th rough i n - l i n e t ube bund l e s . Th us , f o r apa r t i cu l a r d e s ig n , i t is im p o r t an t ( a ) t o k no w th e c r i t i cave loc i ty for f lu id e las t ic ins tab i l i ty and avoid i t , (b) tde mons t r a t e t he admi s s ib i l i t y o f t he am p l i t u de s r e acheby t u rbu l e n t bu ff e t i ng ( fo r l i qu i d a n d tw o-p ha se f l o w)(c) to ca lcu la te and check the c r i t ica l ve loc i ty and thma t ch ing amp l i t ude a t t he s t a r t o f vo r t ex e xc i t a t i o n(bu t on ly fo r l i qu id s ) a n d (d ) t o dem o n s t r a t e t he po s s ib l e i nvo l vem en t o f a co u s t i c r e son anc e ( bu t on ly fogases) .

3 . Es t im a t ing the c r i t ica l ve loc i ty fo r f lu id e l a s t i cins t ab i l i t y

A n u m b e r o f m o d e l s a n d t h e o ri e s h a v e b e e n p r o -p osed , a s d e sc r i bed b y Chen [ 1 ] and Pe t t i g r e w a ndTay l o r [ 15 ]. Th e f i rs t wa s advanc ed b y Co nno r s [ 16] . Has sumed an equ i l i b r i um be twe en t he e x c i t i ng f o r c e , t hfo r c e coe ff i ci en ts depe nd in g on t he am p l i t ud es x an d y

an d t he dam p ing fo r ce . F r om th i s he de r i ve d t h e f o llowing s tab i l i ty c r i te r ion for whi r l ing :

WS.k __ K . _I/@ 'A (4)w ~. = f . D ' 4 P D -

wh ich i nc l udes t he d im ens i o n l e s s c r i t i c a l ve lo c i t y w ~=,t h e m a s s d a m p i n g p a r a m e t e r ( m . A ) / ( p . D 2 ) a n d t h ef ac to r K wh ich i s o f t e n ca l l ed t he s t ab i l i t y c on s t a n t .

T h r o u g h t h e u s e o f o t h e r f o r c e m o d e l s f o r t h ef u n c t i o n s & a n dg4, inc luding the force coeff ic ien t



H. Gelb e et al. / Chemical Engineering and Processing 34 1995) 289 -29 8 2 9 3

O i m e n s i o n t e s s E r i t iz a { V e t o c i t yw~

1 0 0 . . . . . . . . I . . . . . . . . i . . . . . . . . L• ~ h ~ AV E R ~ 8 1 ) o v

V K~RTI~1974 ~ l~- ~R t~80t

* HBJCR VII:~4T 1S8¢1

f l 0 - i ~ , , m - n a ~ e , t a ~ m ~ I ° / .

F o p ' r = 1 , 3 7 5 : - • ,y r o i d t

lVu d e x S h e d d , - , . -

S t = 0 , : 8 1 , , , . . . . I . . . . . . . . I . . . . . . . . ] i

0 , 1 1 1 0 I 0 0

M a s s D a m p i n g P a r a m e l e P m ._ AAp D 2

Fig. 4. Stability map obtained by Chen [1] using the C onnors equa-tion.

depe nd on the vibr at ion velocity (galloping) [17], thesame dependence as expressed in Eq. (4) for f luidswi th a low dens i ty has been obta ined . However forf luids wi th a h igh dens i ty, the exponent o f the massdamping parameter in Eq. (4) becomes smal le r than0.5. For high f luid densit ies, correlat ions are often usedwhich make i t poss ib le to cons ider the damping sepa-rately:

= K . • A P 5 )

The s tab i li ty cons tan t K i s a func t ion of the model , thebundle geometry, the p i tch ra t io and the d imens ionlessvelocity. A l is t summarizing the published values ofKc~ and /? can be fo und in the pa pers of Ch en [1] andAndjeli6 [12].

F i t t ing the exper imenta l da ta and ver i f ica t ion of themodels m ay be achieved by mean s of s tab i l ity :maps , in

which va lues for we are p lo t ted aga ins t the mass dam p-ing parameter. Chen [1] has publ i shed maps for fours tandard bundle a r rays . The two s taggered ar raysdem onst ra te a dependence on p i tch [18] . As an exam-ple , a map for a bundle wi th a normal t r iangular a r rayis shown in F ig . 4 . The measured va lues a re va l id for

= 1 .375 and were conver ted us ing the equat ion:

w ¢ Owe (r = 1.375) = (6)

2.105 (z - 0.9)

The s t ruc tura l da ta m, A a nd f ma y be de l :e rminedeither in air or in a non-turbulent f luid. Chen hasrem arked tha t the va l id ity of Eq. (6) has only been

dem onst ra ted for gases.From Fig. 4 i t wil l be seen that variat ion in the

measurement va lues i s qu i te ex tens ive and can be ex-p la ined by s t ruc tura l in f luences , unknown boundarycondi t ions , non-cons is ten t use or inexac t va lues in thes t ruc tura l da ta, o r incom ple te docum enta t ion . For apractical design, i t is necessa ry to use th e low l imitwhich i s wel l descr ibed by the Connors equat ion forh igher mass dam ping param eters . C hen [1] has ver i fiedtha t a d iscont inui ty occurs in the s tab i l i ty curve on

changing f rom a compress ib le to an incompress ib lemedium. I t can be seen f rom Fig . 4 tha t the lowerstabil i ty threshold in the l iquid range is identical withtha t for vor tex exc i ta t ion (dot ted l ine) . Other au thors ,e.g. Pett igrew and Ta ylor [15], have n eglected this dis-cont inui ty or have measured h igher thresholds , e .g .Tro idl [19]. I t is assum ed that the extensive scattering of

measurement va lues in the l iqu id reg ion resu l t s f romthe overlapping l imits for vortex excitat ion and fluidelast ic instabil i ty. Thus, for a safe design, the lowestcurve should be used . Turbulence exc i ta t ion and f lu idelast ic instabil i ty in two-phase f lows have been investi-gated b y Jatzlau [20] and Che n [21].

Cri t ical velocit ies determined from stabil i ty maps areonly va l id for idea l condi t ions , i . e . homogeneous bun-dles with a constant overall cross-section, with no side-passages and wi th cons tan t ups t ream and gap ve loc i tiesin any cross-section. Such condit ions can only beachieved in wind tunnels . The use of da ta obta inedfrom a s tab i l ity map in the des ign of a rea l hea t

excha nger will be described in the following sections.

4 . C o n s i d e r a t i o n o f a n o n - u n i fo r m v e l o c i ty d i s t r i b u ti o n

In rea l hea t exchangers , the f low through the bundleand around the tubes i s d i s t r ibu ted behind the in le tnozz les , by impingement p la tes, ba nes and by-passgaps . This leads to ax ia l and rad ia l s t ream componentswhich cause a n on-un i form f low field a roun d the tubes .

Con ners [22] imp roved his stabil i ty equation (4) fornon-u ni form f low fields , bu t m ain ta ined the assum pt ionof whi r ling . The ex tended Conno rs eq uat ion i s then:

~ L 1 / 2

j m ( z ) ' @ [ ( z ) d zw* =f..~)S'kD = K ' ~-7'Ai 0

/ ~ z~ p ( z ) V s ( z ) @ } ( z )dzd

7)

In this e quation, ~)S,k is the cri t ical gap velocity of th ebundle normal to the tube , @i(z) the i th v ibra t ionmode , Ss(Z ) the re la t ive ve loc i ty func t ion and L thetube length.

I f the tube mass and f lu id dens i ty a re cons tan t , i tfol lows from Eq. (7) that:

~)S.k= K . A/A-~'mf .D JD2.pwith the energy rat io:

z) • @7° . )d z-

S i ~

0L@ (z) dz

and w he r e th e p rodu c t

8 )

(9)



294 H . G e l b e e t a l . / C h e m i c a l E n g i n e er i n g a n d P r o c e s s h l g 3 4 1 9 9 5 ) 2 8 9 - 2 9 8

rl=

9 z) O z)" / r ' - I "

------ ki_ImLk

1 , l 1 10 l~ % t~, t~=t~, ~ l~ l~I i 3 4

, _ A l l _,

Fig. 5. Sectional calculat ion undertake n in stages.

~?'S,k ~ , = ff'S,k= WS,k (10 )i s the ' equiva len t ' c r i t ica l ve loc i ty ~?S,k , wh ich s hou ld b et he s ame a s t ha t ob t a i n ed f r o m t h e s t ab i l i t y map fo runi form f low, due to Eq. (4) .

When de s ign in g a r e a l h ea t exc hange r, i t i s r e com-m e nded t ha t eve ry s ec t i on n be c a l cu l a t e d s ep a ra t e ly[ 23 ] . Th i s i s sho wn in F ig . 5 f o r a mu l t i span b u nd l e .Th e ca l cu l a t i on ma y b e ba sed on va r i ab l e o r on ' equ i v -a len t ' cons tan t ve loc i t ies per sec t ion .

T h e a d v a n t a g e o f th i s m e t h o d i s t h a t a d d i t i o n a lp a r ame te r s wh ich a r e dep ende n t o n z , e . g . mas s pe ru n i t l eng th , d ens i t y, da mp ing a nd s t ab i l i t y cons t an t s ,c a n be cons id e r ed . F o r t h i s pu rpos e , i t i s u se fu l t oa s sume mean v a lue s w i t h i n a g iven s ec t i o n n :

w * = / , ,~ , K '2 • (11):7D 2 jw h e r e P i s t h e e x p o n e n t o f th e m a s s d a m p i n g p a r a m e -t e r i n t he s t ab i l i t y equa t i o n . Th e d i f f e r en t i a l ene rgyra t io :

~ ~ s ( Z ) ' * } ( z ) d zAS i.,, _ l~

foL ¢ ~ z )dz

i 2 (z) dz ~ (z) dz~ . z . ~ J g .

= V ~ , . g = R J ~ , . ( 1 2 )

f ~ @ ~ ( z ) d z f ? O ~ ( z ) d z

d e f ine s t he p ropo r t i on o f i n s t ab i l i t y i n e v e ry s e c t i o n .T he quan t i t y q s , , , i s t he ' equ iv a l en t ' c o n s t a n t ve loc i t yr a t i o f o r s ec t i on n , kno w l edg e o f wh i ch i s nece s s a ry fo ra s a f e appa ra t u s d e s ig n , a t l e a s t f o r s e c t i o ns w i t h t helo nges t span l e n g t h . Norma l l y i n r e a l hea t ex change r s

o n ly t he m ean v a lue R Js, ,,, d e t e rm i ne d f r om t he vo l ums t r eam d i v ided by an a s sumed c ro s s - s e c t i on , i s know nHow ev e r, w i th mod e rn f l u id -dynamic p r og rams , i t iposs ib le to ca lcu la te ve loc i ty f ie lds in tube bundles . Othe ba s i s o f t he se ca l c u l a t i ons and u s ing t he eq ua t i o ng ive n above , i t s hou ld b e pos s ib l e t o de r i ve ru l e s f osugge s t i n g ' e qu iva l en t ' co n s t an t ve l o c i t i e s wh ic h m ak

s imp l e d e s ig n s p os s i b l e w i t h ou t t he h e l p o f co m pu te r sUsing Eq. (8) or (11) , i t i s poss ib le to ca lcu la te the

m ax imum gap ve l o c i t y. H o we ve r, t he d e s i gn e r n eed s t oknow the c r i t i c a l vo lume s t r e am fo r a g iven hea t exchan g e r. Us ing f l u id -dyna mi c p ro g r am s , t h e vo lum est ream has to be var ied unt i l for a c r i t ica l gap re la t ivto a c r i t ica l tube the s tab i l i ty re la t ion K~ def ined in Eq(13) beco mes equ al to 1 .0 . I t fo l lows f rom Eqs . (4) and(8) tha t th e pote nt ia l r i sk of ins tab i l i ty re la tive to ths t a b i l i t y c o n s t a n t K can be exp re s s ed a s :

K = __ ~ s~ = f ( f O 1 3 )Ws.k f (O , p , A i , f , m , K)

The numer a to r i n Eq . ( 13 ) de pends o n t h e i n l e t v o l umst ream I: and the ve loc i ty d is t r ibu t ion , whi le the den o m i n a t o r d e p e n d s o n t h e s t r u c t u r a l d a t a a n d t h e Kva lue e x t r a c t ed f rom t h e s t ab i l i t y ma p . T he s t ab i l i t ythre sho ld va lue i s K* = 1 . In tha t case I~ '= ~ ;'k and~¢'s = *¢'s.k.

I n t he s imp l e s t c a se , i f a ll p a r am e t e r s a r e cons t an tover z , Eq . (8) can be used wi th :

NS ,= Z AS,-.,, (14)

I x

When ca l cu l a t i ng s e c t i o n pe r s ec t i o n , one ob t a in s ad i ffe rent ia l s tab i l i ty re la t ionship for each sec t ion:

a x e , , , = 1 : )WS ka

wi th

K * = ~,~,=, AK~.,~ (16)

Bec ause o f t he sq u a re r o o t va lue , se c t i o ns w i th t heh i g h e s t A K * , v a l u es d o m i n a t e .

The v e loc i ty d i s t r i bu t i on ca l cu l a t e d f o r t he measu redc r i t i c a l vo l ume s t r e am in an exp e r im en t a l h ea t ex -cha nge r wi th tw o baff les i s shown in F ig . 6 [24] . Also

shown wi th do t t ed l i ne s a r e t he a s s u m ed c ons t a n t veloc i t ies . In th i s f igure , the f i r s t th ree rows behind then ozz l e a r e w indow tubes an d a r e t h e r e f o r e n o t sup -p o r t e d be tween s ec t i ons 4 a nd 5 . S e c ti o ns 1 and 2 a r en o t on - s t r e a m.

Sect ion a l ca lcu la t io ns of the AKs*. va lues have beenma d e , ba s ed f i rs t o n t he a s sum pt io n o f con s t an t ve loci t ies ~ :' s, , for the ra t io b e twee n the v olum e s t ream andt he f r ee c ro s s- s ec t ion o f t he bu n d l e ( in Se c t i on 3 t h e f r ec ro s s - s ec t i o n be low the nozz l e ) . S e cond ly, c a l cu l a t ed



H. Ge lbe e t a I . / Chemic a l Eng inee r ing and ProcessO*g34 I995)2 8 9 - 2 9 8 295

g a p v e l o c i t y i n m / sI O 0

50

0

5 0

1 0 0

1 5 0

n = 1

,,, , I,,

t

2 3 4 5

Fig. 6. Calculated gap veloci t ies and mo de shape.

. . . . . . . . . 1 . r o w

. . . . 2 r o w- - 3 r o w

Fig. 7. Stability relationships for a real heat exchanger.

Gap K~ fo r assumed K~ fo r ca lcula tedconstant veloci t ies gap veloci t ies

10 4-I 05 0.58 1.14 1.28 0.23 0.68 0.72

205 -20 6 0.58 1.04 1.19 0.19 0.88 0.90305 -30 6 0.58 0.95 1.11 0.17 1.13 1.14

with K = x/zXK 2 + AK~,~+s and K = 2.48.

velocit ies have been used. The results are shown in Fig.7 [25].

The cons tan t ve loc i ty model predic ts va lues whichare too high because:

1. The a ssum ed velocity behin d the nozzle in section 3was too high (AK*,s = 0.53 instead of 0.23, 0.19 and0.17, respectively); how ever, this has no ma jor effectdue to the dominat ing inf luence of the~ K ~ , 4 + 5

values.2. For the f irst three tubes in the window, the full

volum e f low was ass umed to be a c ross- flow. In fac t ,for the f irst row the potential r isk of instabil i ty isleast because of the greater axial f low (0.68 insteadof 1 .14) whi le the h ighes t i s found in the th i rd row(1.13 in comparison to 0.95).

For cri t ical row 2, the value K* is 0.90 instead of 1.0 aswould have been expected . There i s an impor tan t rea-son for th i s devia t ion: for non-uni form f low around atube , the qu es t ion ar i ses which i s the r ight da tum valuefor the velocity. Thus, the two gap velocit ies for a tube

in a row can d i ffer. Addi t iona lly, the ups t ream veloc ityof a tube in the gap of the preceding row can be h igherthan tha t in the same row. This was the case for thesecond row in the sections 3, 4 + 5. For this reason, ame an va lue should be es tab l i shed to a l low the force onthe tube to be modelled exactly. Stabil i ty relat ions forthe second row were ca lcu la ted us ing the gap ve loc i tyof the row before , i . e . wi th the ups t ream veloc i ty. Theresul t was K* = 1 .07 . Thus , i t may be concluded tha tthe model approximate ly f i t s wi th the measurements .

Fur th er va l ida t ions of the ex tended Conno rs equat ionhave been d escr ibed by J ahr and Gelbe [25 , 26].

The h igher va lues of K* ( th i rd row) re la t ive to thesecond row are surpr is ing . This can be expla ined by thefac t, tha t the ca lcu la t ion model ignores by-pass s t reamsin the holes of the tube suppor t p la te . These lower theveloc ity ma xim um in the th i rd row and hence wil l

reduce the K* value in practice.

5 Inf luence of st ructure and design

5 1 Structure data

For the de terminat ion of c r i t ica l ve loc i ties f rom s ta -b i l ity d iagrams, a good knowledge of the fo l lowings t ruc ture da ta i s requi red : v ibra tion f requency ~ . ) ;damping (A;) ; and mass per uni t l ength (m) . Amongsto ther th ings , they depend on the des ign de ta i l s of thehea t exchanger, par t icu lar ly on the suppor t condi t ions

of the tubes and o n the num ber and des ign of thebaffles.The f requen cy and m ass per uni t l ength ma y be

ca lcu la ted wi th a reasonable degree of exac tness . How-ever, a goo d es t imat ion of the dam ping va lues is no tposs ib le and they have to be d e termined af te r cons t ruc-t ion of the appara tus . An other problem is , tha t depend -ing on the s tab i l i ty equat ion used , the s t ruc ture da tament ioned above are needed for d i ffe ren t boundarycondit ions , i .e. ( i ) in the absence o f the influence ofa fluid (vacuum), (ii) in a static fluid for (a) a singletube (wi thout in te rac t ions) and (b) a tube bundle (cou-p led v ibra t ion m odes) , and ( i i i ) wi th f lu id-coupled

forces, al l under real ist ic support condit ions. Becausebo un dar y con dit ion ( i) is the easiest to real ise, Che n [1]referre d his stabil i ty diagram s to this state. S tate ( i i)(a)is also often used as a boundary condit ion (e.g. Pett i-grew an d T aylo r [15]) , while condit ions ( i i )(b) and( i ii ) a re se ldom used . Chen [27] dem onst ra ted th a ti t i s permiss ib le to choose one or o ther of boundarycon ditio ns (i) or (ii)(a) if this is also done for all threeparameters .

5 I i Frequency and vibration mo deI f a very smal l deform at ion is assumed, the fo l lowing

equat ion can be der ived f rom the par t ia l d i ffe ren t ia l

equat ion for the f ree v ibra t ion of a homogeneous rod[28]:

22 x / -~ (17)= 27cL2

In th is equat ion , the2i quant i t ies a re the e igenvaluesfor the v ibra t ion modes @i and d epend on theboundary condi t ions . Trac t ive forces (e .g . for f ixedsuppor ts ) increase the na tura l f requency, while pressureforces (e.g. due to heat extension) lower i t .



296 H. Gelbe et al. / Chem ical Engineerhlg and P rocesshlg 34 (I995) 289-29 8

For ideal support condit ions, i t is possible to calcu-la te the na tura l modes and f requencies of s ingle tubesin a i r ana ly t ica l ly. Al thou gh the ca lcu la ted f requenciesf i t very wel l wi th the measured va lues , the ca lcu la tedampl i tude curves d i ffe r as a resu l t o f unkn own suppor tdamping and different loads in the bundle section [29].The na tura l f requency for f lu ids of h igher dens i ty de-

creases due to the addi t iona l f lu id mass to be moved.The inf luence of dens i ty and v iscos i ty has been de-scribed by Stockmeier [30].

The same va lue off . has to be used in a l l sec t ions fortubes wi th d i ffe rent span length fu l l on s t ream. Leyh[29] showed that , even in experiments with a very smallbaff le c learance of 0 .15 mm , the f i r s t mode dom inatedfor sections with the shortest span length. The f irstmode usually leads to the lowest cri t ical velocity. How-ever, i f other modes are excited due to the f low field, ah igher mode can become cr i t ica l . This may happen i fmo re than one in le t nozz le is used or i f the bundle i sonly part ial ly subjected to cross-flow.

Baffle clearances are also very important . The f irstmo de i s the only one wi th the lowes t f requency whenthe clearances are sufficiently small and t he sup ports areac t ive . Whether tubes may v ibra te in the mode of aninac t ive suppor t wi th a cor responding low f requency,which could reach the ins tab i l i ty a rea before the sup-por t becomes ac t ive wi th increasing ve loc ity and ampl i -tude , needs to be demonst ra ted . A repor t aboutv ibra t ions in a condenso r due to inac tive suppor ts wi thc learances in the a rea of technica l use be tween 0 .4-0 .6mm has been publ i shed by Yeh and Chen [31] .Long- te rm damag e occurred caused by increased ampl i -tudes at relatively low velocities.

5 .1 .2 . DampingDamping may be def ined by the logar i thmic decre-

ment :

A = 2 n ( (18)

or by the damping ra t io :

d(19)

~ = 2,v/-

Three d i ffe rent types of dam ping need to be d is tin-guished:

1 . Mate r ia l damping ,which occurs main ly in the sup-ports and is not significant.

2 . Viscos i ty o r f lu id dam ping ,which cannot be neglec tedfor f luids with higher density and viscosity.

3 . S t ruc tu ra l damping ,which i s the main type in rea lhea t exchangers . I t i s caused by mechanica l andviscous fr ict ion of the tubes in the baffles and also byimpact forces.

Dam ping genera l ly depends on am pl i tude and hence onthe v ibra t iona l mode . Impact forces due to la rger sup-

por t c learances can cause non- l inear behaviour andhys teres i s pheno men a. Jahr [32] dem onst ra ted tha t m a-te r ia l dam ping fo r weld ing fixed tubes does not dependon the ampli tude. In contrast , axial f i 'eely supportedtubes , which are only suppor ted by O-r ings, have a h ighampl i tude dependence . S t ruc tura l damping var ies wi ththe width an d the c learance to le rance of the suppor t . I t

increases if the baffles are not f ixed in the vibrat ionnode poin ts (no equid is tan t suppor ts , see Jendrze jczyk[33] and Chen [1]).

Values for s t ruc tura l damping have been g iven byPett igrew et al . [34]. Fo r gase ous med ia, A = 0.044 hasbeen recom men ded as a safe va lue , whereas for l iqu idsA = 0.062 if f > 100 Hz; for low er frequencies, high erva lues a re recomm ended. However, the damping canalso be up to f ive-t imes higher and c an differ by a facto rof ca . 2 wi th in the same bundle .

5 .1 .3 . M as s pe t un i t l eng thThe m ass damp ing pa r a me te r and t he f r equency a r e

usual ly ca lcu la ted us ing the mass per un i t l ength of thetube plus the f luid mass inside. The effect of the hydro-dynam ic mass on a s ingle tube in inf in i tely expandedmedia has a l so to be cons idered:

1m ~ = -~ zrpD 2 2 0 )

Fo r air a nd l ight gases, this effect can be n eglected. Th eaddit ional mass increases for tubes near to the wallsand for tubes vibrat ing in a bundle. Chen [1] hasdefined a coefficient C,, for th e effective addit io nal mass:

c.m , = -- ~ n p D (21 )

and has suggested l imit ing values as a function of thepitch rat io. For low ampli tude values, C, usually isassumed to be uni ty.

5 .2 . h ~ u e n c e o f d e s ig n

I f the mom ent of inert ia, which depends on thediameter D and the wal l th ickness s of the tube , i sintroduced into Eqs. (17) and (4), an expression for thecri t ical velocity for f luid elast ic instabil i ty can be ob-tained:

D2 AfWS,k K('c) • 2~ • E 0'5' s 0'5'L2 m O , 5 _ i , 2 2 )

The fo l lowing des ign sugges tions can be der ived f romthis equation. The span length L has . the greatest influ-ence on the s tab i l ity of the bundle and the longes t spanlength is the most imp or tan t (window tubes) . In c r i t ica lhea t exchangers , window tubes must be avoided . Theinf luence of the mean d iameter Dm of the tubes i s ofnext impor tance . Tubes wi th smal l d iameters (conden-sors ) cause more v ibra t iona l problems. The wal l th ick-



H. Gelbe et al. / Chemical Engineering and Processing 34 1995) 289 -29 8 297

h e s s s a n d t h e m o d u l u s o f e l a st i c it y E h a v e a n i n f lu e n c ew h i c h i s p r o p o r t i o n a l t o t h e i r s q u a r e r o o t . T h e i n f lu e n c eo f t h e m a s s m c a n b e n e g l e c t e d f o r P ~ - 0 . 5 . I f t h edam ping A,. i s inc reased , the c r i t i ca l ve loc i ty can beh igher. To ach ieve th i s e ffec t , b roader suppor t s , lowers u p p o r t c l e a ra n c e s , n o n - e q u i d i s t a n t s p a n l e n g t h s o r a d -d i t i o n a l d a m p e n e r s , w h i c h s h o u l d h a v e n o i n f l u e n c e o n

the f low f ield, are useful .F o r n o r m a l t r i a n g u l a r a r r a y s a n d r o t a t e d s q u a r e

a r rays , Soper [18] d i scovered th a t the c r i t i ca l ve loc i tyWS,k inc reases w i th inc reas ing p i t ch ra t io ~ . Ga ps wi th inthe bundle mu s t be avo ided o r c losed by us ing sea l s t r ipso r d i s p l a c e m e n t b o d i e s . O n t h e o n e h a n d b y - p a s ss t reams reduce the ve loc i ty in the bundle , whi le on theo t h e r h a n d t h e f l u i d v e l o c i ty i n t h e g a p s c a n b e c o m e s oh igh tha t a d jacen t tubes beg in v ib ra t ing , e specia] ily whe nthe f low i s fo rced back in to the bundle a t obs tac les .

F o r t u b e s w i t h e q u i d i s t a n t s p a n l e n g t h s , t h e i n f l o warea beh ind the nozz le i s c r i t i ca l . The c r i t i ca l vo lumes t r e a m c a n b e i n c r e a s e d i n p r o p o r t i o n t o t h e r a t i o o f th e

nozz le to the f ree bundle c ross - sec t ion . Hence the l imi t -i n g v o l u m e s t r e a m d e p e n d s o n h o w w e l l f l o w i s d i s -t r i b u t e d b e f o r e e n t r a n c e i n t o t h e b u n d l e . T h i s c a n b ea c h i e v e d b y u s i n g f l o w d i s t ri b u t o r s b e h i n d t h e n o z z l e o rby us ing severa l o r b igger in le t nozz les . The d i s tanceb e t w e e n t h e n o z z l e o u t l e t a n d t h e f i r s t t u b e r o w s h o u l dno t be smal le r than two tube d iamete rs [26] .

The in f luence o f impinge me nt p la tes has been inves t i -ga ted by Le yh [29] . In co n t ras t to f low d i s t r ibu tors ,p l a t es a r e n o t c a p a b l e o f i m p r o v i n g t h e v i b r a t i o n r e si s-t ance . In fac t , the c r i t i ca l ve loc i t i e s were lower in mos tof the cases s tud ied . I f i t i s no t poss ib le to avo id p la tes ,t h e i r d i a m e t e r s h o u l d b e b i g g e r t h a n t h e d i a m e t e r o f t h e

i n l e t n o z z l e . T h e d i s t a n c e b e t w e e n t h e i m p i n g e m e n tp la te and the she l l shou ld be o f a s i ze su ff i c ien t Lo avo idh i g h r a d i a l v e l o c i t y c o m p o n e n t s f r o m t h e e d g e o f t h ep la te ac t ing on the tubes .

6 Conclusions

An overv iew is g iven o f the pa ram ete rs which a ffec tt h e v i b r a t i o n a l e x c i t a t i o n i n a t u b e b u n d l e h e a t e x -changer as we l l a s ru les fo r avo id ing the i r nega t ivein f luence on the c r i t i ca l ve loc i ty. The app l icab i l i ty o fs tab i l i ty maps wi th rea l a ppa ra tus has bee n inw,~s tiga tedin de ta i l , a s we l l a s the in f luence o f the ve loc i ty f i eld andt h e s t r u c t u r a l d a t a . I t h a s b e e n d e m o n s t r a t e d t h a t aconserva t ive sa fe des ign i s poss ib le , bu t tha t someim por tan t p rob lem s e .g. dam ping va lues ) s t il l r equ i reresearch .

Nomenclature

c spr ing cons ta n t , kg s -2Cp forc e coef ficien t, -

dDE

Sfwgl ...4I

KK *L177

n

NPS

SiASi,,,S rt

9k

W ~

W S

WS,k

}VS,k

WS,kW

X, y

2 , 22 , 2Z

CA

P

21A i

PC

~ s

v e l o c i t y - p r o p o r t i o n a l d a m p i n g , k g s - td i a m e t e r o f tu b e , mm o d u l u s o f e l a s t i c it y, k g m - I s - 2n a t u r a l f r e q u e n c y o f t u be s , s - 1v o r t e x s h e d d i n g f r e q u e n c y, s -ex te rna l fo rces , kg m s -2m o m e n t o f i n e r t i a ,m 4

s tab i l i ty cons tan t , -s t ab i l i ty re la t ion , -span l eng th , mmass pe r un i t l eng th , kgm - 1

num ber o f a span sec t ion , -num ber o f a l l span sec t ions , -e x p o n e n t o f m a s s d a m p i n g p a r a m e t e r, -tube wal l th ickness , menergy f rac t ion , -d i ffe ren t ia l energy f rac t ion , -S t r o u h a l n u m b e r, -t ime, sv o l u m e s t r e a m ,m 3 s -~

cr i t i ca l vo lum e s t ream, m 3 s -f ree in f low ve loc i ty, m s -1gap veloci ty, m s -1c r i t i ca l gap ve loc i ty, m s -m axi mu m cr i t ica l gap ve loc i ty, m s -~equ iva len t c r i t i ca l gap ve loc i ty, m s -1d imens ion less c r i t i ca l ve loc i ty, -a m p l i t u d e s , mvibra t iona l ve loc i t i e s , m s -v ib ra t io na l acce le ra t ions , m s -2c o o r d i n a t e a l o n g t h e t u b e , m

expo nen t in s t ab i l i ty equa t io n , -d a m p i n g e x p o n e n t i n s t a b i l i t y e q u a t i o n , -d a m p i n g r a t i o , -e igenvalues, -l o g a r i t h m i c d e c r e a s e i n d a m p i n g , -dens i ty, kgm - 3

pi tch ra t io , -n o r m a l i z e d a m p l i t u d e f u n c t i o n , -normal ized ve loc i ty d i s t r ibu t ion func t ion , -

ndices

ikn

S

n u m b e r o f m o d ecr i t i ca l va luessec t ion o f a bundleg a p

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flow induced vibrations

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