a review of heat exchanger tube bundle vibrations in two-phase cross-flow

8/11/2019 A Review of Heat Exchanger Tube Bundle Vibrations in Two-phase Cross-flow

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Nuclear Engineering and Design 230 (2004) 233251

A review of heat exchanger tube bundle vibrationsin two-phase cross-ow

Shahab Khushnood , Zaffar M. Khan, M. Afzaal Malik,Zafar Ullah Koreshi, Mahmood Anwar Khan

College of Electrical & Mechanical Engineering, National University of Sciences and Technology, Rawalpindi, Pakistan

Received 8 May 2003; received in revised form 8 October 2003; accepted 21 November 2003

Abstract

Flow-induced vibration is an important concern to the designers of heat exchangers subjected to high ows of gases or liquids.Two-phase cross-ow occurs in industrial heat exchangers, such as nuclear steam generators, condensers, and boilers, etc. Undercertain ow regimes anduid velocities, theuid forces result in tube vibration anddamagedue to fretting andfatigue.Predictionof these forces requires an understanding of the ow regimes found in heat exchanger tube bundles. Excessive vibrations undernormal operating conditions can lead to tube failure.

Relatively little information exists on two-phase vibration. This is not surprising as single-phase ow induced vibration; asimpler topic is not yet fully understood. Vibration in two-phase is much more complex because it depends upon two-phase owregime, i.e. characteristics of two-phase mixture and involves an important consideration, which is the void fraction. The effectof characteristics of two-phase mixture on ow-induced vibration is still largely unknown. Two-phase ow experiments aremuch more expensive and difcult to carry out as they usually require pressurized loops with the ability to produce two-phasemixtures. Although convenient from an experimental point of view, airwater mixture if used as a simulation uid, is quitedifferent from high-pressure steamwater. A reasonable compromise between experimental convenience and simulation of steamwater two-phase ow is desired.

This paper reviews known models and experimental research on two-phase cross-ow induced vibration in tube bundles.Despite the considerable differences in the models, there is some agreement in the general conclusions. The effect of tube bundlegeometry, random turbulence excitations, hydrodynamic mass and damping ratio on tube response has also been reviewed.Fluidstructure interaction, void fraction modeling/measurements and nally Tubular Exchanger Manufacturers Association(TEMA) considerations have also been highlighted. 2004 Elsevier B.V. All rights reserved.

1. Introduction

Two-phase cross-ow induced vibration in tubebundles of process heat exchangers and U-bend regionof nuclear steam generators can cause serious tube

Corresponding author. Tel.: +92-51-2873423;fax: +92-51-2824132.

E-mail address: [email protected] (S. Khushnood).

failures by fatigue and fretting wear. Tube failurescould force entire plant to shutdown for costly repairsand suffer loss of production. Such vibration prob-lems may be avoided by thorough vibration analysis.However, these require an understanding of vibra-tion excitation and damping mechanism in two-phaseow. An important parameter that characterizes thetwo-phase ow is void fraction, which is the ratio of the volume of gas to the volume of the liquidgas

0029-5493/$ see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.nucengdes.2003.11.024


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234 S. Khushnood et al. / Nuclear Engineering and Design 230 (2004) 233251

Nomenclature

a tube gap

Av upstream ow areab exponent B Chisholm correlation parameterC connement factorC 1, C 0 drift ux model parametersC I coefcient of interactionC1 non-dimensional liquid force

coefcientCg non-dimensional gas force

coefcientCap capillary numberd , D tube diameter De equivalent diameter Eu Euler number EuG0 Euler number for gas EuL0 Euler number for liquid f tube natural frequency in two-phase

mixture f g tube natural frequency in airF instantaneous normal forceFr Schrage correlation parameterFIV ow induced vibrationg acceleration due to gravityGP pitch mass ux, mass owHEM Homogeneous Equilibrium Modeli mode number jg supercial gas velocity jl supercial liquid velocityK constant fraction in Smith correlation,

Connors critical factorK 0, K 1 drift ux model parameters L, l tube length, limit span subjected to

cross-ow L drift ux model parameter

m mass, mass per unit lengthm0 massmgeq gas phase effective massmh hydrodynamic mass/added massmleq liquid phase effective massm t mass of tube alone

m mass ow ratemp mass ux of mixturen number of points discretized/index in

Chisholm correlation

N gamma count of experimental (trial) N G gamma count for 100% gas N L gamma count for 100% liquidNPSD Normalized Power Spectral Density pD dynamic pressure pS static pressureP pressure, tube pitchP cr two-phase ow critical pressure

p pressure drop across tube row pG0 Pressure drop for gas pL0 Pressure drop for liquid pTPF two-phase ow pressure drop

PDF probability density functionQ mass uxr drift ux model parameter Re Reynolds number Ri Richardson numberRAD Radiation Attenuation MethodR-11, R-12 refrigerant typesRMS root mean squareS velocity ratio/slipS F( f ) power spectral density of random

excitation forceS sliding distance

T time period

T s sample durationTEMA Tubular Exchanger ManufacturersAssociation

U g , U G gas velocityU l , U L liquid velocityU h homogeneous velocityU p , V p pitch velocityU c critical velocityU cl critical ow velocity of liquid phaseU cg critical ow velocity of gas phaseV eq equivalent two-phase velocityU gj averaged gas phase drift velocityW work rate x location X quality of ow/mass qualityY Chisholm parameterY (1),Y ( x), Y tube response(Y 2) mean square tube response(Y 2)0.5 root mean square tube response


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S. Khushnood et al. / Nuclear Engineering and Design 230 (2004) 233251 235

Greek symbols void fraction

average value of (s)T (time length of gas phase)H homogeneous void fraction non-dimensional parameter non-dimensional parameter average damping in two-phaseg homogeneous void fraction total damping ratio, modal damping

ratioeq non-dimensional effective ow

velocity of gas phase ratio of non-dimensional effective

masses per unit length of tube inliquid and gas phases

L liquid phase absolute viscosity viscosity of mixture in McAdams

equation l viscosity of liquid phaseg viscosity of gas phase eq non-dimensional effective ow

velocity of liquid phase two-phase mixture density g , G gas mass density

l, L liquid phase density H HEM uid density density difference between phases liquid surface tension2L0 two-phase friction multiplierL0 function of two-phase ow

parameters i(S ) shape function (S ) ow distribution

Table 1Types of ow in two-phases

Flow type Average void fraction Specication

Bubble 0.3 Some bubbles are present in liquid ow and move with the same velocitySlug 0.30.5 Liquid slugs ow intermittentlyFroth 0.50.8 More violent intermittent owAnnular 0.80.9 Mainly gas ow, liquid adheres to the tube surfaceMist 0.9 Almost gas ow. Mist sometimes causes energy dissipation

mixture. A number of ow regimes ( Table 1) canoccur for a given boundary conguration, dependingupon the concentration and size of the gas bubbles and

on the mass ow rates of the two-phases. Two-phaseow characteristics greatly depend upon the type of ow occurring.

Tube vibration in two-phase ow displays differentow regimes, i.e. gas and liquid phase distributions,depending upon the void fraction and mass ux. It isknown that four mechanisms are responsible for theexcitation of tube arrays in cross-ow ( Pettigrew et al.,1991). These mechanisms are turbulence buffeting,vortex shedding or Strouhal periodicity, uid-elasticinstability and acoustic resonance. Table 2 presentsa summary of these vibration mechanisms for sin-gle cylinder and tube bundles for liquid, gas andliquidgas two-phase ow, respectively. Of these fourmechanism, uid-elastic instability is the most dam-aging in the short term because it causes the tubes tovibrate excessively, leading to rapid wear at the tubesupports. This mechanism occurs once the ow rateexceeds a threshold velocity at which tubes becomeself excited and the vibration amplitude rise rapidlywith an increase in ow velocity.

Typically, researchers have relied on the Homo-geneous Equilibrium Model (HEM) to dene impor-

tant uid parameters in two-phase ow, such as den-sity, void fraction and velocity. This model treats thetwo-phase ow as a nely mixed and homogeneousin density and temperature, with no difference in ve-locity between the gas and liquid phases. This modelhas been used a great deal because it is easy to im-plement and is widely recognized which facilitatedearlier data comparison. Other models include SmithCorrelation ( Smith, 1968 ), drift-ux model developedby Zuber and Findlay (1965) , Schrage Correlation(Schrage, 1988 ) which is based on empirical data, and

Feenstra model ( Feenstra et al., 2000 ) which is givenin terms of dimensionless numbers.


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Table 2Vibration excitation mechanisms ( Pettigrew et al., 1991 )

Flow situation Fluid-elastic instability Periodic shedding Turbulence excitation Acoustic resonance

Cross owSingle cylinder

Liquid a b b a

Gas a c c a

Two phase a a b a

Tube bundleLiquid b c c a

Gas b a c b

Two phase b a b a

a Unlikely.b Possible.c Most important.

Dynamic parameters such as added mass and damp-ing are very important considerations in two-phasecross-ow induced vibrations. Hydrodynamic massdepends upon pitch-to-diameter ratio and decreasewith increase in void fraction. Damping is very com-plicated in two-phase ow and is highly void fractiondependent.

Tube-to-restraint interaction at the bafes (loosesupports) can lead to fretting wear because of out of plane impact force and in-plane rubbing force. Frick

et al. (1984) has given an overview of the developmentof relationship between work-rate and wear-rate.

Another important consideration in two-phase owis the random turbulence excitation. Vibration re-sponse below uid-elastic instability is attributed torandom turbulence excitation ( Pettigrew et al., 2000;Mirza and Gorman, 1973; Taylor et al., 1989; Papp,1988; Wambsganss et al., 1992 ) to name some whohave carried potential research for RMS vibration re-sponse encompassing spatially correlated forces, Nor-malized Power Spectral Density (NPSD), two-phaseow pressure drop, two-phase friction multiplier,mass ux, and coefcient of interaction between uidmixture and tubes.

Earlier reviews on the topic are provided byPaidoussis (1982) , Weaver and FitzPatrick (1988) ,and Price (1995) . More recently researchers haveexpanded the study to two-phase ow which occurin nuclear steam generators and many other tubularheat exchangers, a review of which was last given byPettigrew and Taylor (1994) . The aim of present at-tempt is to review all known models and experimental

research on two-phase cross-ow induced vibrationsof tube bundles. It is intended to provide designguidelines to the heat exchanger designers, to give aninsight to the process designers and the maintenancepersonnel.

2. Modeling two-phase ow

Most of the early experimental research in this eld

relied on sectional models of tube arrays subjected tosingle-phase uids such as air or water, using rela-tively inexpensive ow loops and wind tunnels. Thecheapest and simplest approach to model two-phaseow is by mixing air and water at atmospheric pres-sure. However, airwater ows have a different den-sity ratio between phases than steamwater ow andthis will affect the difference in the ow velocity be-tween the phases. The liquid surface tension, whichcontrols the bubble size, is also not accurately modeledin airwater mixtures. Table 3 gives the comparison

of liquid and gas phase of refrigerants R-11, R-22 andairwater mixtures at representative laboratory con-ditions with actual steamwater mixture properties attypical power plant conditions ( Feenstra et al., 2000 ).This comparison reveals that the refrigerants approxi-mate the liquid surface tension and liquid dynamic vis-cosity of steamwater mixtures more accurately thanairwater mixtures.

In the outer U-bend region, of typical nuclear steamgenerators such as those used in the CANDU de-sign, the tubes are subjected to two-phase cross-ow


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Table 3Comparison of properties of airwater, R-22, and R-11 with steamwater at plant conditions ( Feenstra et al., 2000 )

Property R-11 Airwater R-22 Steamwater

Temperature ( C) 40 22 23.3 260Pressure (kPa) 175 101 1000 4690Liquid density (kg/m 3) 1440 998 1197 784Gas density (kg/m 3) 9.7 1.18 42.3 23.7Liquid kinematic viscosity ( m2 /s) 0.25 1.0 0.14 0.13Gas kinematic viscosity ( m2 /s) 1.2 1.47 0.30 0.75Liquid surface tension (N/m) 0.016 0.073 0.0074 0.0238Density ratio 148 845 28.3 33Viscosity ratio 0.20 0.70 0.47 0.17

of steamwater. It is highly impractical and costly

to perform ow induced vibration experiments ona full-scale prototype of such a device so thatsmall-scale sectional modeling is most often adopted.R-11 simulates the density ratio, viscosity ratio andsurface tension of actual steamwater mixtures betterthan airwater mixtures and it also allows for local-ized phase change which airwater mixture does notpermit. While more costly and difcult to use thanairwater mixture, R-11 is a much cheaper modelinguid than steamwater because it requires 8% of theenergy to evaporate the liquid and operating pressureis much lower, thereby reducing the size and cost of the ow loop (Feenstra et al., 2000 ).

3. Representative published tests on two-phaseow across tube arrays

Table 4, an extension of period beyond 1993(Nakamura et al., 1993 ) presents summary of salientfeatures of the experimental tests performed on thethree possible geometric tube arrangements.

4. Thermal hydraulic models

Considering two-phase ow, homogeneous ow as-sume that the gas and liquid phases are owing at thesame velocity, while other models for two-phases ow,such as drift-ux assume a separated ow model withthe phases allowed to ow at different velocities. Gen-erally the vapor ow faster in upward ow because of the density difference.

4.1. The homogeneous equilibrium model (HEM)

A general expression for void fraction , is givenby (Feenstra et al., 2000 ):

= 1 +S GL

1X 1

1(1)

where G and L are the gas and liquid densities, re-spectively and S is the velocity ratio of the gas and liq-uid phase (i.e. S =U G /U L). The quality of the ow, X is calculated from energy balance, which requiresmeasurement of the mass ow rate, the temperatureof the liquid entering the heater, the heater power, andthe uid temperature in the test section. The HEMvoid fraction H , is the simplest of the two-phase uidmodeling, whereby the gas and liquid phases are as-sumed to be well mixed and velocity ratio S in Eq. (1)is assumed to be unity. The average two-phase uiddensity, is determined by:

= G +(1 ) L (2)The (HEM) uid density, H , is determined using

Eq. (2) by substituting H in place of . The (HEM)pitch ow velocity, V P is determined by:

V P =G PH

(3)

The pitch mass ux, GP , is determined from owmeasurements obtained from the orice plate readingby:

G P =( m/A v)P (P D)

(4)

where m is the mass ow rate, Av is the upstream owarea, P is the tube pitch, and D is the tube diameter.


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Table 4Representative published tests on two-phase ow

Researchers (published) Fluid Tube array Voidfraction

Tubelength(mm)

Naturalfrequency(Hz)

Dampingratio (%)

Pettigrew and Gorman(1973)

Airwater Triangular/parallel,square/rotated square

1020%(quality) 50.8

17, 30 2.52.7

Heilker and Vincent(1981)

Airwater Triangular/rotatedsquare

0.50.87910

5662 0.84

Hara and Ohtani (1981) Airwater Single tube 0.020.6160

Rigid

Remy (1982) Airwater Square 0.650.85 1000 56.6 0.61.75Nakamura et al. (1982) Airwater Square/rotated square 0.20.94

190142 1.31.7

Pettigrew et al. (1985) Airwater Triangular/square 0.050.98600

2632 0.98.0

Axisa et al. (1984) Steamwater Square 0.520.98 1190 74 0.23.0Nakamura et al. (1986) Steamwater Square 0.750.95

17415.216 4.08.0

Hara (1987) Airwater Single/row 0.010.558

6.08.4 2.915.6

Goyder (1988) Airwater Triangular 0.50.8360

175

Gay et al. (1988) Freon Triangular 0.580.84 1000 39.8 0.891.7Nakamura and Fujita

(1988)Airwater Square 0.020.95

60052

2.1Funakawa et al. (1989) Airwater Square/triangular 0.00.6

10012.8

3.3Nakamura et al. (1990) Steamwater Square 0.330.91

17494137

Axisa et al. (1990) Steamwater Square/triangularparallel/triangular

0.520.99 1190 72 0.24

Papp and Chen (1994) Normaltriangular/normalsquare/paralleltriangular

2598%

Pettigrew et al. (1995) Freon Rotated triangular 4090%,1090% 609

281500.15

Noghrehkar et al.(1995)

Airwater Square fth and sixthrow

090%200

1525 0.33.9

Taylor et al. (1995) Airwater U-bend tube bundlewith 180 U-tubesparallel triangularconguration

090% U-tuberadii0.60.7

23114 1.52

Marn and Catton (1996) Airwater Normal triangular,parallel triangular,and rotated square

599%

0.721

Taylor and Pettigrew(2000)

Freon Rotated triangularand rotated square

5098%609

0.25

Pettigrew et al. (2000) Airwater Normal 30 androtated 60 triangular,normal 90 androtated 45 square

0100%600

30160 15

Inada et al. (2000) Airwater Square 070% 198 15 2.5 1.6Eq. addeddampingcoefcient


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Table 4 ( Continued )

Researchers (published) Fluid Tube array Voidfraction

Tubelength(mm)

Naturalfrequency(Hz)

Dampingratio (%)

Nakamura et al. (2000)(summary of Takai)

Freon 46 5 U-bend tubes,specication of actualWestinghouse type-51series steam generator

(Notconsidered)based onConnorssingle-phaserelation

1626 Damping

ratio < 1

Feenstra et al. (2000) R-11 Parallel/triangular 00.99

0100 1.12.9

Results 19731993 ( Nakamura et al., 1993 ).

The main problem with using the (HEM) is that it

assumes zero velocity ratio between the gas and liquidphases. This assumption is not valid in the case of vertical upward ow, because of signicant buoyancyeffects.

4.2. Homogeneous ow ( Taylor and Pettigrew, 2000 )

This model assumes no relative velocity betweenthe liquid velocity U l and the gas velocity U g:

Slip

=S

=1 : U

h =U

g =U

l,

g =j g

j g + j l(5)

where U h is the homogeneous velocity, g is the ho-mogeneous void fraction, jg is supercial gas velocityand jl is the supercial liquid velocity.

4.3. Smith correlation

Smith (1968) assume that kinetic energy of the liq-uid is equivalent to that of the two-phase mixture anda constant fraction K of liquid phase is entrained withthe gas phase. The value K =0.4 was chosen to cor-respond with the best agreement to experimental datafor ow in a vertical tube. Using the smith correlation,the slip is dened as follows:

S =K +(1 K)(X l/ g) +K( 1 X)

X +K( 1 X)1/ 2

(6)

where X is the mass quality, g is the density of thegas phase and l is the density of the liquid phase.

4.4. Drift-ux model

The main formulation of drift-ux model was devel-oped by Zuber and Findlay (1965) . This model takesinto account both the two-phase ow non-uniformityand local differences of velocity between the twophases. The slip is dened as follows:

S =(1 g)

(1/(C 0 +(U gj /j)) g)

=(C 0 +(U gj /j)) (X/(X( 1 g/ l))+( g/ l))

(1

(X/(X( 1

g/ l))

+( g/ l)))

(7)

Out of the four unknown values of this relation,two are determined experimentally: X directly and jindirectly from mass ux measurement m :j =j g + j l = m

Xg +

(1 X) l

(8)

The remaining two unknowns are empirical andLellouche and Zolotars Correlation ( Lellouche andZollotar, 1982 ) is used to estimates these. This corre-

lation is well adapted for tube bundles in two-phasecross-ow. An averaged gas phase drift velocity is de-ned as follows:

U gj =1.41 l g

2l

1/ 4(1 g)1/ 2

(1 +g)(9)

where is the liquid surface tension.

C0 =L

K 1 +(1 K 1) rg(10)


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with

K 1

=K 0

+(1

K 0)

g

l

1/ 4

,

K 0 =(1 +e( Re/ 105) )1

r =1 +1.57( g / l)

(1 K 0 ), L =

(1 eC1g )(1 eC1 )

,

C1 =4P 2cr

P(P cr P)where Pcr is the two-phase ow critical pressure. Re =UD / , where D is the tube diameter and is the mix-ture viscosity which can be evaluated by McAdamsequation:

= l

1 +g( l/ g 1)(11)

where l and g are the viscosities of liquid and gasphases, respectively.

4.5. Schrage correlation

The correlation by Schrage (1988) is based on em-

pirical data from an experimental test section, whichmeasures void fraction directly. This test section hastwo valves capable of isolating a part of the ow al-most instantaneously.

The correlation is based on physical considerationsand assumes two hypotheses:

1. For very low qualities, the gas phase appears onlyas very small bubbles. The ow behaves as a homo-geneous one and the minimum reduced void frac-tion, g / gh value is 0.1.

2. For very high qualities, when X

= 1, ow is con-

sidered homogeneous and we obtain the limit con-dition g / gh =1.

Then, the Schrage correlation is as follows:

ggh =1 +0.123Fr

0.191 ln X with Fr =m

l gD(12)

This correlation was established with an airwatermixture but it remains valid for any other phase ow.

4.6. Feenstra model ( Feenstra et al., 2000 )

In this model predicted velocity ratio of the phases

is given by:

S =1 +25.7( Ri Cap )0.5 P

D1

(13)

where the velocity ratio S is used in conjunction withEq. (1) for void fraction. The Richardson number Riis calculated by:

Ri = 2ga

G 2P(14)

where a is the gap between tubes, ( = l g)is the density difference between the phases, g is thegravitational acceleration and GP is the mass ow. Thecapillary number ( Cap ) is calculated by:

Cap = LU G

(15)

where L is the liquid phase absolute viscosity, isthe surface tension of liquid and U G is the gas phasevelocity.

4.7. Comparison of void fraction models

The (HEM) greatly over-predicts the actual gammadensitometer void fraction measurement and the pre-diction of void fraction model by Feenstra et al., issuperior to that of other models. It also agrees withdata in literature for airwater over a wide range of mass ux and array geometry ( Feenstra et al., 2000 ).

5. Dynamic parameters

5.1. Hydrodynamic mass

Hydrodynamic mass mh is dened as the equivalentexternal mass of uid vibrating with the tube. It isrelated to the tube natural frequency f in two-phasemixture as discussed in ( Carlucci and Brown, 1983 )and is given by:

mh =m tf gf

2

1 (16)

where mt is the mass of tube alone and f g is the natu-ral frequency in air. The tube frequency is measured at


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6. Flow regimes

Many researchers have attempted the prediction of

ow regimes in two-phase vertical ow. As yet, a muchsmaller group has examinedow regimes in cross-owover tube bundles. Some of the rst experiments werecarried out by Grant ( Collier, 1979 ) as it was the onlyavailable map at the time. Early studies in two-phasecross-ow used the Grant map to assist in identify-ing tube bundle ow regimes ( Pettigrew et al., 1989b;Taylor and Pettigrew, 2000 ). More recently ( Ulbrichand Mewes, 1994 ) performed a comprehensive anal-ysis of available ow regime data resulting in owregime boundaries that cover a much larger range of ow rates. They found that their new transition lineshad an 86% agreement with available data. Their owmap is shown in Fig. 3 by Feenstra et al. (1996) withthe ow regime boundary transitions in solid lines andthe ow regimes identied with upper-case text. Thedotted lines outline a previous ow regime map basedon Freon-11 ow in a vertical tube, from ( Taitel et al.,1980).

Almost every study of ow regimes in tube bundleshas concluded that three distinct ow regimes exist.In fact, several studies have shown that these regimescan easily be identied by measuring the probability

density function (PDF) of the gas component of the

Fig. 3. Flow regime map for vertically upward two-phase ow from(Feenstra et al., 1996 , 1980). Data symbolize: square ( Pettigrewet al., 1989a,b ), upward triangle ( Axisa, 1985 ), downward triangle(Pettigrew et al., 1995 ), and circle (Feenstra et al., 1995 ).

ow (Ulbrich et al., 1997; Noghrehkar et al., 1995;Lian et al., 1997 ).

7. Tube to restraint interaction (wear work-rate)

Signicant tube-to-restraint interaction can lead tofretting wear. Large amplitude out-of-plane motionwill result in large impact forces and in-plane motionwill contribute to rubbing action. Impact force andtube-to-restraint relative motion can be combined todetermine work-rate. Work-rate is calculated using themagnitude of the impact force and the effective slid-ing distance during line contact between the tube andrestraint ( Taylor et al., 1995 ). The work rate is givenin Eqs. (19) and (20) :

W =1T s

n

i=0F i dS i (19)

W =1T s

n

i=0F i S i =

1T s

n

i=0F i +F i+1

2 S i (20)

where F is the instantaneous normal force, S is thesliding distance during line contact and n is the num-ber of points discretized over the sample duration T s.

As the work-rate increases, the effective wear rate in-creases and the operational life of the U-bend tubedecreases. Implementation of the technology is de-scribed in detail by Fisher et al. (1991) . Measured val-ues of wear work-rate for pitch velocity and mass ux(Taylor et al., 1995 ) are presented in Fig. 4a and b,respectively. The effect of uid-elastic forces is veryevident in the measured work-rates.

It is interesting to note that at higher pitch veloci-ties and/or mass uxes, the wear work-rate does notincrease. Further study is required to understand whythe ow-rates do not affect the work-rates. This maybe related to the fact that at high void fractions andhigh ow rates the random excitation forces are con-stant with increasing ow rate ( Taylor, 1992 ).

8. Fluid-elastic instability

Pettigrew and Gorman (1973) established thefundamental treatment of two-phase ow inducedvibration. That was based on the method used for


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Fig. 4. (a) Measured work-rate vs. pitch velocity ( Taylor et al., 1995 ). (b) Measured work-rate vs. mass ux ( Taylor et al., 1995 ).

single-phase ow-induced vibration. It was con-cluded that uid-elastic instability seems possible intwo-phase ow as well as in liquid ow. The same in-

stability criterion seemed to apply to both liquid andtwo-phase ow according to the following equationin terms of dimensionless ow velocity U p/ fD anddimensionless mass-damping term [2 m/D 2]1/ 2 .

U p fD =K

mD 2

1/ 2

(21)

where K is the instability factor, U p the average owvelocity, the average uid density, the averagedamping in two-phase ow and f is the tube naturalfrequency in two-phase ow mixture at mass ux mp,with u p = mp/ and is the total damping ratio.Fluid-elastic instability is possible when the uiddynamic forces on the tubes are proportional to thetube motion. When energy absorbed form the uid bythe tubes exceeds the energy dissipated by damping,uid-elastic instability results in very large vibrationamplitudes. A complete, stability boundary estimationmethod has not yet been established, at least not in for-mal theoretical manner ( Nakamura et al., 2000 ). A de-sign procedure has already been proposed by Connors(1970) and it is widely believed to be useful for de-

signers, although there have been some modicationsof his parameters such as the critical factor K , and theexponents in Eq. (21).

The distribution of density and uid ow velocityis estimated using the following equation proposed byConnors (1978) :

U cf i D =K

m0 i 0 D 2S i

1/ 2

(22)

where

S i = L

0 ((S)/ 0)(S)2 i (S) 2 ds

L0 (m(S)/m 0) i (S)

2 ds

where (S ) is the ow distribution, i (S ) is the corre-sponding shape function, L is the limit span subjectedto cross-ow and i refers to mode number.

Eq. (22) is theoretically derived from the hypoth-esis that the uid-elastic force has a linear relationto the tube displacement; this implies the so calleddisplacement mechanism ( Chen, 1987 ). However, itwas late established in the 1980s that the mechanismof uid-elastic instability is not limited to displace-ment mechanism but also that a velocity mechanismexists. Eq. (22) therefore cannot be applied to every


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case. This is true for single-phase ow. Pettigrew et al.(1978) considered the effect of two-phase ow, butConnors derived the analysis from the same equation.

Some papers have been published for the estimationof the critical factor K in two-phase ow conditionbased on Eq. (21) by Antunes et al. (1985) . Nakamuraet al. (1997, 2000) have tried to develop a new methodwith different approaches to this problem. This methodis based on the insight into the physical aspect of the two-phase phenomenon. The method is essentiallymacroscopic in nature being based on Connors dis-placement mechanism.

Eq. (23) gives the stability boundary for the case of homogeneous two-phase ow.

2eq + 2eq =1 + (23)where parameter = (m geq /m leq ) is dened to bethe ratio of non-dimensional effective masses per unitlength of tube in liquid phase and gas phase, respec-tively. Eq. (23) is the energy balance in simple form. eq and eq are the non-dimensional effective ow ve-locities of the liquid phase and gas phase, respectively,where =( / 1 ) (non-dimensional parameter), (s). T is the time length of gas phase, the aver-age value of (s), a non-dimensional parameter andT is the time period. =(C g/C l)( g/ l)(U cg/U cl)2 ,where Cl , C g are non-dimensional uid force coef-cients, U cl and U cg are critical ow velocities for dif-ferent phases.

Fig. 5. Vibration response at 80% void fraction: exible vs. rigid tube bundle ( Pettigrew et al., 1995 ).

Fig. 6. Instability results in two-phase cross-ow: comparison of

Freon vs. airwater ( Pettigrew et al., 1995 ).

In two-phase Freon, the critical mass ux for insta-bility is reasonably well dened as shown in Fig. 5.Instability results ( Pettigrew et al., 1995 ) are presentedin Fig. 6. Generally, the results show two regions of in-stability. The rst is a region where the mass-dampingparameter exponent is roughly 0.5 as expected. Theseregions occur at void fractions below approximately80% for airwater mixtures, and below approximately65% for Freon two-phase ow. For this region, the in-

stability factor K is above the recommended guidelineK = 3.0. There appears another instability region atvoid fractions above 80% for airwater and above65% for Freon, where the slope of instability curve


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is considerably lower. In fact, the change in behavioris dramatic for Freon two-phase ow. At 90% voidfractions the instability factor K =1.36, which is lessthan the recommended design guidelines of K =3.0.

9. Random turbulence excitation

For a given void fraction, the vibration responseof the tube bundle generally increase with mass uxuntil uid-elastic instability is reached as shown inFig. 7ad , which gives typical comparative (exibleversus rigid tube bundle) vibration response (RMS)at the tube free end versus mass ux. The differ-ence in vibration response between exible bundle and

Fig. 7. (ad) Typical RMS vibration response: comparison between exible and rigid tube bundle ( Pettigrew et al., 2000 ).

rigid bundle is small at mass uxes below instabil-ity, which means that random turbulence excitation isnot much effected by surrounding tubes. On the other

uid-elastic instability occurs at lower mass uxes forexible tube bundle ( Pettigrew et al., 2000 ). The vibra-tion response below instability is attributed to randomturbulence excitation. The approach is to deduce therandom turbulence excitation forces from the vibrationresponse ( Pettigrew et al., 2000 ). The relationship be-tween the mean square amplitude of the tube vibrationresponse (Y 2) and the power spectral density S F(f) of the random excitation forces can be developed fromrandom vibration theory. For the fundamental mode of a cantilevered tube subjected to uniformly distributedand spatially correlated forces over its entire length,


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Y 2(l) by Mirza and Gorman (1973) is given by:

Y 2(l) =0.613S F (f)/( 16 2f 3m 2) (24)where m is mass per unit length including that of tubemt and hydraulic mass mh , is modal damping ra-tio, f is natural frequency of tube, and C = 0.613 isthe coefcient of uid tube interaction which varieswith the tube boundary conditions and axial location x.For Taylors experimental conditions (Cantilever tube)maximum displacement occur at x = l, where l isthe tube length. Thus, it is easy to deduce S F( f ) fromthe vibration response with Eq. (24). A relationship(Inada et al., 2000 ) between the vibration responseand the mass ux mp was established by doing a lin-

ear regression analysis on the experimental data. Bychoosing only the data points corresponding to massuxes below the threshold for uid-elastic instabilitya mass ux exponent b was determined for the rela-tion (Y 2)0.5 mbp . The scatter in the results was sig-nicant, but a tendency towards a value of b = l wasfound in the void fraction range of 2590%. At voidfraction below 25% the plots were extremely difcultto interpret, due to the presence of some periodic wakeshedding peaks, as discussed in Taylor et al. (1989) .

Pettigrew et al. (2000) uses an exponent of b = l tocollapse the random turbulence data. This allowed forthe calculation of a simplied normalized power spec-tral density NPSD of the excitations forces formulated

Fig. 8. (a and b) Normalized power spectral density of random turbulence excitation ( Pettigrew et al., 2000 ).

by:

NPSD =S F(f)

( mpD) 2 (25)

The normalized power spectral densities ( Nakamuraet al., 1982) were plotted against void fraction inFig. 8a and b. A comparison between P / D = 1.22and 1.47 indicates that random turbulence forces arenot greatly affected by P / D. This is not entirely un-expected since the response is not much affected bythe motion of surrounding tubes as shown in Fig. 7,which gives typical RMS vibration response. Fig. 8aand b illustrates that the random turbulence levels arepractically the same for triangular and square bundlesfor the worst orientation. The curve ( Taylor et al.,

1989) is dened by:NPSD =10

(0.03g5) , for 25 < g < 90% (26)Total ow pressure is a sum of dynamic pressure pD,

and static pressure pS. Flow-induced vibration mustdissipate energy from this ow. Therefore, tube vi-bration must require a decrease in total ow pressure.When other ow parameters remain the same, the mea-sure of dissipated energy is a decrease in static pres-sure. Papp (1988) proposed the following relationship:

(Y 2)0.5 pd (27)

where p is the pressure drop across a tube row, andd is the tube diameter. Of course, there is energy dissi-


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10. Measurement of void fraction

In general, the surveyed research indicates two

types of void fraction measurements ( Feenstra et al.,2000), the HEM void fraction and Radiation Atten-uation Method (RAD) void fraction. HEM refers toHomogeneous Equilibrium Model and RAD refersto Radiation Attenuation Method. The determinationof uid parameters (uid density and ow velocity)are quite different when these two methods are used(Feenstra et al., 2000 ). In RAD method ( Feenstraet al., 2000; Chan and Bannergee, 1981 ) gamma uxfrom radiation source which penetrates the test sectionwill be attenuated by different amounts dependingupon the average density of the two-phase ow. Voidfraction can be determined by interpolating theaverage density of the uid between the benchmark measurements for 100% liquid and gas according tothe following equation:

=ln(N/N L)

ln(N G /N L)(34)

where N represents the gamma counts obtained dur-ing an experimental trial, N L and N G are the referencecounts obtained prior to the experiment for 100% liq-uid and 100% gas, respectively. U G and U L can be

Table 6Results of example FIV analysis (TEMA, 8th ed.)

calculated by:

U G = xGP G

(35)

U L =(1 x)G P(1 ) L

(36)

A logical measure of an equivalent two-phase veloc-ity, V eq determined from averaging the dynamic headof the gas and liquid phases is given by Eq. (37):

V eq = GU 2G +(1 ) LU 2L (37)Local void fraction measurements have been pub-

lished by Haquet and Gouirand (1995) , Lian et al.(1997) , and Mann and Mayinger (1995) .

11. Application of TEMA (FIV) code

Table 6 give the results summary of application of TEMA (2003) (FIV) software to a U-tube span subjected to two-phase cross-ow. Summary of input ge-ometric and process parameters is given. Analysis re-sults for various mechanisms, logarithmic decrementand acceptable amplitudes are also presented.


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12. Conclusions

The current paper has reviewed the two-phase

cross-ow induced vibration in tube bundles. Follow-ing topics have been considered.

Types of two-phase ow and ow regimes. Choice of modeling uids, their comparison fromexperimental standpoint and dynamic parameters

such as damping and added mass.

Various void factors models, their comparison andvoid fraction measurement. Tube vibration excitation mechanisms induced byuid-elastic instability and random turbulence. Wear work rates for tube-to-restraint interaction.

There is a strong need for establishing reliabledesign procedures for two-phase cross-ow tubebundle vibrations. This could be achieved by carry-ing out modeling and simulation of the system withuidstructure interaction focusing on void fraction,and reliable experimental data. Test data on highpressure and temperature conditions are insufcient,therefore a potential challenge lies ahead.

Acknowledgements

The authors are indebted to College of Electrical &Mechanical Engineering, National University of Sci-ences & Technology (NUST), Pakistan, and PakistanScience Foundation (PSF) for the completion of thisresearch work. We also gratefully acknowledge thehelp provided by Pakistan Scientic & Technical In-formation Center (PASTIC) in acquiring the relatedliterature.

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a review of heat exchanger tube bundle vibrations in two-phase cross-flow

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