Flow Induced Vibration in Heat ExChangers

0263–8762/02/$10.00+0.00# Institution of Chemical Engineers

Trans IChemE, Vol 80, Part A, April 2002

FLOW-INDUCED VIBRATION IN HEAT EXCHANGERSH. G. D. GOYDER

Cran�eld University at RMCS, Shrivenham, Swindon, UK.

H eat exchanger tube bundles may fail due to excessive vibration or noise. The mainfailure mechanisms are generated by the shellside � uid that passes around andbetween the tubes. This � uid may be a liquid, gas or multi-phase mixture. The

most severe vibration mechanism is a � uidelastic instability, which may cause tube damageafter only a few hours of operation. Clearly, such extreme causes of vibration must always beavoided. In contrast buffeting due to � ow-turbulence causes very little vibration. However, aftermany years of service such remorseless low level vibration will produce tube wall thinning, dueto fretting, which may be unacceptable in a high-integrity heat exchanger. Consequently, theissues of life cycle and integrity must frequently be included in heat exchanger speci� cation.This paper reviews the various mechanisms that cause vibration and noise. Particular attentionis given to methods for achieving good tube support arrangements that minimize vibrationdamage. References to the most recent sources of data are given and good working practice forthe design and operation of standard and high-integrity heat exchangers is discussed.

Keywords: heat exchangers; vibration; noise; � uidelastic; � ow-induced vibration; tubes.

INTRODUCTION

In principle, heat exchangers are simple engineering deviceswith no moving components. However, anyone who workswith heat exchangers will agree that for a successful designall the skills of almost every engineering discipline arerequired. The aim of this paper is to provide the processengineer with a review of the mechanical engineering issuesassociated with � ow-induced vibration and noise in heatexchanger tube bundles. This should enable the processengineer to feel forewarned and prepared when tackling avibration issue.

Heat exchangers are not mass-produced to a few � xedcon� gurations. They are individually designed and craftedfor a speci� c process-plant duty. This is an exception tomost modern engineering where mass-production andtesting provides an opportunity to eliminate problemsobserved in prototypes. If a mistake is made in a heatexchanger design then the exchanger may fail (possiblydramatically) and a replacement exchanger must be suppliedwith an improved design. It can be seen that the conse-quences of failure are signi� cant.

A particular complicating issue with � ow-induced vibra-tion in tube bundles is that past experience of design andoperation is not necessarily a guide for future performance.Small changes in � ow rates, or mechanical design, canlead to conditions that result in the dramatic failure of aunit. This suggests that designers and operators need a goodunderstanding of the engineering issues associated with� ow-induced vibration. This paper aims to provide thatunderstanding.

The paper is divided into three parts. The � rst partreviews the various mechanisms causing tube vibration

while the second part considers the practical issues arisingfrom the mechanical design of the tube support system. Thethird part of the paper presents solutions, problem issues,and good working practice.

The literature on � ow-induced vibration in tube bundlesis extensive. The ASME1 symposium in 1997 is a goodstarting point together with the books by Blevins2 andBlake3.

MECHANISMS CAUSING VIBRATION

The following list of � ow-induced vibration mechanismsis used by the author as a checklist when investigating themechanical integrity of a heat exchanger. The list is orga-nized in decreasing order of importance:

(1) � uidelastic instability;(2) vortex shedding;(3) multi-phase buffeting;(4) acoustic resonance;(5) turbulence buffeting;(6) hydraulic transients;(7) environmental excitation;(8) transmitted mechanical vibration.

The mechanisms of � uidelastic instability, vortex shed-ding, buffeting (single and multi-phase) and acousticresonance, all arise because of the variety of forces thatact on a tube due to the shell-side cross � ow. Thesemechanisms will be dealt with � rst. The remaining mechan-isms will be considered later in the paper.

Figure 1 shows an example of three mechanisms (buffet-ing, vortex shedding and � uidelastic instability) which

226

induce vibration on a laboratory tube bundle. The largeamplitudes of vibration due to vortex shedding and � uid-elastic instability should be noted. In this study4, the� uid was water and the tubes were simple cantilevers.The behaviour shown in this � gure will be discussedsubsequently.

Figure 2 shows the typical con� guration of a tube in ashell-and-tube heat exchanger. The shell-side � ow is madeto zigzag over the tubes with the � ow direction approxi-mately perpendicular to the tube axis. The � ow is directedby baf� es, which also provide a method for supporting thetubes. The tubes in the windows are often the most vulner-able tubes because they have fewer supports than the tubesin the interior of the bundle but have the full cross � owvelocity.

One of the main issues when assessing a tube bundle is totake data from experiments of the type shown in Figure 1,which is a single span tube bundle, and use this data forrealistic tube bundles (see Figure 2), which have many spansand a variety of � ow velocities.

The � ow inside the tubes does not cause vibration andonly adds mass to the tube. It is the cross-� ow outside the

tubes that gives rise to the various vibration phenomena. The� ow outside the tubes is complex and it is perhaps worthnoting that the forces acting on the tubes due to this cross-� ow can give rise to every known form of � ow-inducedvibration.

The shell side � ow passes across and between the tubes ina heat exchanger leaving a low velocity, stagnant, region as awake behind each tube. The wake is a low pressure regionrelative to the upstream side of the tube. The wake is nota static region but oscillates in two distinct manners. It isthese two different types of oscillations that lead to themain causes of � ow-induced vibration, namely � uidelasticinstability and vortex shedding.

Fluidelastic Instability

This is the most severe form of � ow-induced vibration.The forces on the tube are generated by the wake moving inresponse to the tube displacement. As the tube moves the� ow redistributes and the force on the tube associated withthe redistribution enhances the tube motion. This is anexample of positive feedback.

Figure 1. Tube vibrational response to increasing � ow velocity.

Figure 2. A heat exchanger tube bundle showing some window and interior tubes.


FLOW-INDUCED VIBRATION IN HEAT EXCHANGERS 227

Referring to Figure 1 it may be seen that for small � owvelocities there is relatively little vibration compared to thesigni� cant vibration that starts at a � ow velocity of about0.21 m s¡1. The � ow rate of 0.21 m s¡1 marks a de� nitethreshold and beyond this threshold the � uidelastic mechan-ism is acting. This threshold is often called the critical � owvelocity, and the relationship between this critical � owvelocity and the � uid and tube properties is of key impor-tance in predicting the onset of instability in tube bundles.

Figure 3 shows a typical plot that gives the thresholdvelocity for the onset of � uidelastic instability in single spantube bundles. The two axes are nondimensional propertiesof the � uid and the tubes. The vertical axis is a nondimen-sional � ow velocity U=f D where U is the gap velocity, fthe tube natural frequency and D the tube diameter. Theproperty plotted on the horizontal axis, 2pzm=rD2, is amixture of damping and mass ratios. Here m is the tubemass per unit length, z is the mechanical damping ratio (seebelow) and r the � uid density. This plot is for a rotatedtriangular tube layout. Similar plots for other tube layoutsmay be found in Weaver and Fitzpatrick4. These plots arebased on compilations of experimental data, such as thatshown in Figure 1, from which critical � ow velocities havebeen extracted and nondimensionalized.

The plot in Figure 3 is divided into a stable and unstableregion. Increasing the � ow velocity corresponds to movingvertically up a straight line. When the threshold line iscrossed vibration is predicted. Examination of the para-meters on this chart show that the designer has very littleroom for manoeuvre. In order to avoid vibration, withoutchanging the tube diameter, � uid properties, and � ow rate,the only option is to increase the tube natural frequency, f ,or to increase the damping ratio, z. Methods for managingboth these tube properties are given later in the paper.

The horizontal axis in Figure 3 may be divided into tworanges. Small values of the nondimensional mass-dampingparameter usually correspond to liquids while large valuescorrespond to gases. Multi-phase � ows are in-between.The horizontal form of the threshold line for small mass-damping values shows that damping is not important forliquids. In contrast, for gas or multi-phase � ows the damp-

ing is the main parameter controlling the threshold velocityfor � uidelastic instability.

The plot in Figure 3 is suitable for single span tubes witha uniform cross � ow velocity. The extension of this plot toheat exchangers with a range of � ow velocities and a rangeof span lengths is an important part of the heat exchangerassessment methodology. The method is given in Goyder5

and involves integrating the � uid force acting on a tube overthe whole length of the heat exchanger. The integrationincludes the effect of tube supports by using a weightingbased on the vibration energy at each location along thetube. The result of the integration is a value for the dampingratio needed to maintain stability. The following assessmentequation emerges:

z >D2

16p2

„L

0

U (x)fD

³ ´2

r(x)C(x)f2(x)dx (1)

Here, z is the actual damping ratio for the tube and x is thedistance along the tube which is of total length L. Most ofthe key parameters are allowed to vary along the length ofthe tube in order to model the varying � ow conditions atdifferent points in the heat exchanger, thus the density, r(x),the gap � ow velocity, U (x), the tube energy, f2(x), and a liftcoef� cient, C(x), are all functions of location x. The valuesof the lift coef� cient, C(x), are deduced from single spantube bundles and results of compilations of data such asthose presented in Figure 3. An example of the functionf(x) is shown in Figure 4.

The use of this assessment equation will be discussedfurther later in the paper. The general approach is tocalculate the terms on the right hand side and then checkto see if the inequality is satis� ed. It can be seen that acomputer-based method is usually necessary to undertakethis calculation. In particular, it is not easy to determine theenergy distribution, f2(x), from hand calculations. Handcalculations are usually restricted to cases where the vari-ables are not functions of the location within the heatexchanger.

The most dif� cult case to estimate, even with experience,is where there is a small � ow velocity on a span with a large

Figure 3. Fluidelastic stability chart for a rotated triangular geometry.


228 GOYDER

f2(x) and a large � ow velocity on a span with a small f2(x).A computer calculation in this case is essential.

Vortex Shedding

All wakes behind all tubes oscillate. The oscillationfrequency of a wake is proportional to the � ow velocityand gives rise to an oscillatory force on the tube. Theoscillatory force is in the direction perpendicular to boththe � ow and the axis of the tube. The wake oscillation isassociated with the shedding of vorticity from the stagnantregion in the wake behind a tube. This vorticity is shedalternately from one side of the tube and then the other, thus,giving the oscillatory effect.

When the wake oscillation frequency equals a tubenatural frequency then resonance may occur and largeamplitude tube vibration, perpendicular to the � ow direc-tion, is possible. A tube resonance induced by vortexshedding in a liquid can be seen in Figure 1 at � ow ratesof about 0.11 m s¡1.

The coincidence between tube natural frequency and thefrequency of vortex shedding does not have to be exact6. Ifthe vortex shedding frequency is close to the naturalfrequency, then the tube motion that is generated willcause the vortex shedding frequency to change to the naturalfrequency, and the wake oscillation to correlate with the tubemotion. This effect is called lock-in or lock-on. The condi-tions for lock-on are usually estimated from:

0:8f < f < 1:2f with f = StUD

(2; 3)

where f is the tube natural frequency and f is the frequencyof vortex shedding. The frequency of vortex shedding isgiven by the second equation in which St is the Strouhalnumber. Values for Strouhal numbers for a range of pitchand patterns may be found in Weaver7 et al., Ploak andWeaver8, Ziada and Oengoren9, and Ziada and Oengoren10.These recent papers constitute a signi� cant advance.Previous data collections were often very complex and

should now be regarded as highly suspect in the light ofthe detailed modern studies.

The amplitude of vibration due to vortex shedding isdif� cult to predict. The maximum amplitude is typically onetube diameter, but this value is not usually reached becausethe tube is not excited with the same � ow velocity along itswhole length, and because damping can act against largeamplitudes.

Vortex induced tube vibration can often not be avoided ingases, however, damping in gas � ows is generally suf� cientto prevent large amplitudes of vibration. Vortex inducedvibration is, thus, of most concern in liquid � ows or for gas� ows where tubes have very small damping.

Acoustic Resonance

The cause of the very loud noise sometimes produced byheat exchangers with gas on the shellside is an acousticresonance set up by the oscillation of the wakes behindtubes. The mechanism is very similar to that described in thelast section, but, in this case, it is not the tube that is made tovibrate but the gas in the shell. The gas in the shell hasacoustic resonance frequencies that depend on the speed ofsound and the shell geometry. A lock-in effect occursbetween the acoustic natural frequency and the wake oscil-lation frequency. This lock-in effect, as in the case of tubevibration, causes the wake oscillation frequency to change tothe acoustic natural frequency and the wake oscillations tobe correlated everywhere with the acoustic particle veloci-ties. The whole system is thus subject to an acoustic feed-back, which is the cause of the often-surprising intensity ofthe noise.

An acoustic resonance may be suppressed by insertingacoustic baf� es into the shell. These baf� es change theacoustic natural frequency and mode shape thus preventinglock-in. Such baf� es are dif� cult to � t after the constructionof the heat exchanger and may interfere with the shellside� ow. It is therefore important to determine if an acousticresonance could present a problem at the design stage.Investigations into compromises with baf� e geometry andshellside � ow may be found in Goyder11.

Figure 4. A typical mode shape for a heat exchanger tube.



Buffeting

All heat exchanger tubes are in constant motion due torandom buffeting. Although the vibration amplitudes aresmall, often only a fraction of the tube wall thickness, therandom vibration is remorseless and goes on throughout thelife of the heat exchanger. Some wear is therefore inevitable.If all other vibration mechanisms are avoided this cause ofvibration will eventually produce tube damage. Buffetingcomes in two forms; single phase buffeting and multiphasebuffeting. In single phase buffeting the random excitation ofthe tubes is due to turbulence. In a multi-phase � ow thebuffeting is due to the changing density of the � uidimpacting on the tube. Wear due to buffeting, particularlyin multi-phase systems, is of considerable importance if thetube bundle will be in service for many years. Buffeting inmulti-phase � ows has been considered by Pettigrew et al.12

and Axisa et al.13.

OTHER DAMAGE MECHANISMS

It is sometimes necessary to determine the in� uence of awater hammer in a heat exchanger. A possible cause ofa water hammer could be the rupturing of a tube containinghigh pressure gas. If this occurs, a rapidly growing gasbubble forms in the heat exchanger and sets up waterhammer waves in the shellside liquid. If this type of incidentis possible it is usual to include a pressure relief system inthe shell side circuit. The assessment problem is to deter-mine if the shell will rupture before the pressure reliefsystem can vent the high pressure in the shell. This problemis described in an Institute of Petroleum14 publication.

A heat exchanger may be damaged by vibration due tosources outside the vessel. For example the wind may act ona large shell and cause vibration due to vortex shedding.Alternatively, vibrating pipework or machinery may transmitmechanical vibration. If the frequency of transmitted vibra-tion is equal to a tube natural frequency then resonance mayoccur and tubes can be damaged.

TUBE NATURAL FREQUENCY AND DAMPING

Heat exchanger tubes are supported by the tube sheet andthe baf� es. When the tubes vibrate the amplitude of vibra-tion is unlimited in the span between baf� es, but at thebaf� es it is constrained to the tube hole clearance. The loosenature of the tube support at baf� es adds an importantcomplication to tube vibration analysis. These complicationswill be considered in more detail below. As a � rst approx-imation it is often assumed that the tube-to-baf� e clearanceis zero so that the tube can rock like a seesaw but the point atthe baf� e does not otherwise move. A typical shape taken upby a vibrating tube is shown in Figure 4. In this case the tubehas seven spans and eight supports, shown as triangles. Thesupports at the ends are tube sheets while the interiorsupports are provided by baf� es with zero clearance. Thisshape is called a mode shape and one mode shape isassociated with each natural frequency of vibration. Thefunction f2(x) in equation (1) is the mode shape functionsquared. A tube has a sequence of natural frequencies withthe lowest being the one that is usually involved in vibration.

The damping of a heat exchanger tube is of primaryimportance when the shellside � uid is a gas and the tubes

are being assessed for � uidelastic instability. The dampingof a tube is a measure of its ability to absorb energy.Damping is usually expressed in terms of a nondimensionaldamping ratio. A value much less than one is normal; thesmaller the value, the less energy absorbed in each cycle. (Ifthe damping ratio is greater than 1.0 the damping is so largethat a tube will not oscillate but just decay exponentially ifdisturbed.) Experiments have been undertaken on heatexchanger tubes5 to determine their damping ratio. Figure5 shows the results of these experiments presented as astatistical distribution. It can be seen that there is a peak atabout z = 0:015 and that there is a � nite probability of avery small damping value. The existence of small values ofdamping that occur by chance is problematic. Some tubes ina heat exchanger have a probability of small damping,and thus, have a probability of suffering from � uidelasticvibration.

The clearance between a tube and a baf� e has twoimportant in� uences. Firstly, the natural frequencies maybe modi� ed because the tube vibration is less constrainedand secondly, the interactions between a tube and a baf� eprovide friction sites that dominate the tube damping. Thevibration of a tube with loose supports is very complex andhas not yet been fully understood. It is usual to assume thatthe vibration is a perturbation from the case where there iseither no clearance or alternatively a very large clearance.

If a tube passes through a baf� e without making contact itwill have a natural frequency that is much smaller than ifthere were no clearance (the natural frequency is inverselyproportional to span length squared so a double spandecreases the frequency by a factor of 4). The small naturalfrequency may make the tube vulnerable to � uidelasticinstability. The tube will thus vibrate, and consequently, itsvibration amplitude will grow. The growth in amplitude willcause the tube to strike the baf� e through which it passes.The impact of the tube against the baf� e will cause thenatural frequency to increase and may add damping. Thus,the tube vibration will be limited to an amplitude similar tothe tube-to-baf� e clearance. The point to note is that thisvibration will continue throughout the life of the heatexchanger. If the forces generated by this persistent vibrationare large then the wear may be considerable.

An alternative to being loose is for the tube to be pressedagainst a support due to tube lack-of-straightness or baf� e

Figure 5. Experimental values for the damping ratio of heat exchangertubes.


230 GOYDER

misalignment. In this case, the support may be effective andthe tube may not vibrate. In practice, this is probably adesirable con� guration because the small sliding motions atthe baf� es generate friction forces that contribute to thedamping. Some wear will, however, be inevitable.

HIGH INTEGRITY AND STANDARD DESIGNS

When assessing the structural integrity of a heat exchan-ger it is necessary to consider the duty it is undertaking. If astandard heat exchanger is being designed then it may bepermissible to plug one or more tubes after the heatexchanger has been used for three or four years. This typeof heat exchanger will be called a standard heat exchanger.For such an exchanger it is only necessary to ensure that the� uidelastic mechanism and possibly vortex sheddingmechanism (in liquid � ows) is not occurring.

If the tube life must be longer than a few years, or ifmixing of the tubeside and shellside � uid due to a tube leakis unacceptable, then a high integrity design must beconsidered.

For a standard assessment it is convenient to determinethe damping required to prevent � uidelastic instability usingequation (1) and then to compare this value to valuesmeasured for actual heat exchangers in Figure 5. Theprobability of a tube being damaged can then be determinedand the technical risk accepted or rejected. For example anassessment may proceed as follows. A heat exchanger withgas on the shell side and 1000 tubes has 50 window tubeswhich may be vulnerable to vibration. The assessmentequation, equation (1), indicates that a damping ratio of0.003 is required to prevent � uidelastic vibration. Thecumulative probability for damping indicates that 15% ofvulnerable tubes will have a damping smaller than this valueso approximately 8 tubes may suffer from � uidelasticinstability. This risk is considered too large and conse-quently the heat exchanger is redesigned to have moresupports for the window tubes. It can be seen that the useof a required damping value together with a statisticaldistribution of actual damping values provides a rationalbasis for decision making.

Vibration investigations sometimes reveal that signi� cantvibration is almost inevitable in the heat exchanger beingconsidered. In this case, it may be necessary to change to ano-tubes-in-the-window design. This removes all doublespan tubes with a low natural frequency from the exchanger.If the remaining tubes still suffer from vibration, thenintermediate support plates can be installed to raise thenatural frequency until there is no signi� cant vibration.Intermediate support plates only occupy the bundle cross-section and do not extend into the window regions. Conse-quently, intermediate support plates do not interfere with the� ow, and thus, can be added with the sole purpose ofcontrolling vibration.

A common special case is where just one part of the heatexchanger is vulnerable to vibration. This is usually the caseclose to the inlet and outlet nozzles where high � owvelocities are to be found. Two solutions are possible.Redistribute the � ow using impingement plates so that thetubes do not see the full nozzle � ow velocity. Perforatedimpingement plates allow for detailed velocity adjustments.Alternatively, add additional support plates for the tubes.

These support plates may have to contain windows, to allowthe � ow to enter the bundle.

If a high integrity heat exchanger is required then a moredetailed study must be undertaken in addition to the � uid-elastic calculation shown above. The detailed study shouldinvestigate the following issues:

(1) The possibility of double span behaviour due to loosesupports. This can be investigated by seeing if the tubeswill suffer from � uidelastic instability if any baf� e isremoved. If this is possible then extra baf� es will berequired. In very high integrity, work treble span beha-viour should be investigated.

(2) The magnitude of the forces acting at effective supportsshould be compared. It may emerge that some supportsare more important than others and that when there is asmall level of vibration most of the vibration is resistedby one support. This con� guration would result in wearbeing concentrated at one location. A more favourablecon� guration would be for the forces at supports to beapproximately equal so that wear progresses evenly atall support locations.

(3) Arrangements can be made to ensure that the tubes arepressed against the supports. This can be achieved bydeliberately misaligning baf� es or by using bent tubes.In particular the tubes should not be perfectly straight.Similarly, the tube-to-baf� e clearances can be reduced.The probability of tubes being loose has been estimatedby Goyder15.

(4) Baf� e thickness and tube wall thickness can be madelarge. During sliding and impacting the volume ofmaterial removed per unit time tends to remain constant.However, if the surfaces are thick the depth wear ratewill be proportionally small. Also, if the tube is thickthen it has a reserve of material for wear before damageis signi� cant. A simple wear method is proposed byYetsir et al.16.

(5) The U-bend region of a heat exchanger is of particularconcern and usually requires special attention. Thenatural frequency of a U-tube is often much lowerthan that of a straight tube. Arranging for supports inthe U-bend may be dif� cult because of the largeclearances needed for assembly, poor tolerances inconstruction and the lack of roundness of bent tubes.Finally, the U-bend may have a complex � ow � eldwhich is dif� cult to predict. Good detailed design istherefore essential.

The above list of points for high integrity design can alsobe considered as suggestions for good working practice forstandard designs.

CONCLUSIONS

The following conclusions may be drawn:

(1) From the start, a heat exchanger should be identi� ed aseither standard or high integrity. A standard heatexchanger may require some tubes to be plugged afteronly a few years of service. A high integrity heatexchanger will require much detailed design to ensureacceptable tube wear rates after many years of service.

(2) All heat exchangers should be assessed for � uidelasticinstability. This will require a computer calculation if



there are mixtures of large � ow velocities on short spansand small � ow velocities on long spans.

(3) If a heat exchanger has gas on the shellside, it isnecessary to rely on the damping of the tubes toremove energy from � uidelastic vibration. The dampingof a heat exchanger tube has a wide statistical spreadand thus any failure assessment is probabilistic.

(4) Standard heat exchangers should be checked for vortexshedding. For exchangers with liquid on the shell sidevortex shedding can lead to damaging vibration due towear after a few years in service. For exchangers withgas on the shell side vortex shedding often leads tounacceptable noise.

(5) The usual method for preventing � uidelastic or vortexinduced tube vibration is to decrease the baf� e pitch,thus raising the tube natural frequency. A reliablemethod for managing vibration is to have no-tubes-in-the-window possibly with additional intermediate tubesupport plates.

(6) In high integrity designs, consideration must be given totubes which are loose in their supports and the in� uenceof buffeting due to turbulence or multi-phase � ows.

(7) To minimize tube wear and for high integrity designs, itis necessary to consider the effect of tube-to-baf� eclearance, the forces between tubes and baf� es, thebaf� e thickness and the degree to which wear isshared between supports or concentrated at one tubesupport location.

NOMENCLATURE

C lift coef� cientD tube diameterf tube natural frequencyf frequency of vortex sheddingSt Strouhal numberU � ow velocity

Greek symbolsf mode shaper � uid densityz damping ratio

REFERENCES

1. ASME, 1997, 4th Inter Symp Fluid-Structure Interactions, Aeroelasti-city, Flow-Induced Vibration and Noise (Dallas, USA), Vols 53-1, 53-2and 53-3.

2. Blevins, R. D., 1990, Flow-Induced Vibration (Van Nostrand Co, NewYork, USA).

3. Blake, W. K., 1986, Mechanics of Flow-Induced Sound and Vibration(Academic Press, UK).

4. Weaver, D. S. and Fitzpatrick, J. A., 1988, A Review of Cross-FlowInduced Vibration in Heat Exchanger Tube Arrays, J Fluids andStructures 2: 73–93.

5. Goyder, H. G. D., 1992, A practical method for assessing tubevibrationsin heat exchangers, ASME Winter Annual Meeting Vol 230=NE-Vol 9.

6. Zdravkovich, M. M., 1997, Flow around circular cylinders: A compre-hensive guide through � ow phenomena, experiments, applications,mathematical models and computer simulations (Oxford UniversityPress, UK).

7. Weaver, D. S., Lian, H. Y. and Huang, X. Y., 1993, Vortex shedding inrotated square tube arrays, J Fluids and Struct, 7: 107–121.

8. Ploak, D. R. and Weaver, D. S., 1995, Vortex shedding in normaltriangle tube arrays, J Fluids and Struct, 9: 1–17.

9. Ziada, S. and Oengoren, A., 1992, Vorticity shedding and acousticresonance in an in-line tube bundle. Part I: Vorticity shedding, J Fluidand Struct, 6: 271–292.

10. Ziada, S. and Oengoren, A., 2000, Flow periodicity and acousticresonance in parallel triangle tube bundles, J Fluid and Struct, 14:197–219.

11. Goyder, H. G. D., 2001, Acoustic modes in heat exchangers, ASMEPVP, Vol. 420-2: 107: 114.

12. Pettigrew, M. J., Tromp, J. H. and Mastorakos, J., 1985, Vibration oftube bundles subjected to two-phase cross-� ow, J Press Vess Tech, 111:446–477.

13. Axisa, F., Boheas, M. A. and Villard, B., 1985, Vibration of tubebundles subjected to steam water cross-� ow: A comprehensive study ofsquare and triangular arrays, 8th Inter Conf on Structural Mechanics inReactor Techn (Brussels, Belgium).

14. Guidelines for the Design and Safe Operation of Shell and Tube HeatExchangers to Withstand the Impact of Tube Failure, Institute ofPetroleum, 2000.

ADDRESS

Correspondence concerning this paper should be addressed to Dr H. G.D. Goyder, Engineering Systems Department, Cran� eld University, RMCS,Shrivenham, Swindon, SNG 8LA, UK.E-mail: [email protected]� eld.ac.uk

The paper was received 1 November 2001 and accepted for publicationafter revision 18 December 2001.


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Flow Induced Vibration in Heat ExChangers