Prediction of Flow-Induced Vibrations in Tubular Heat Exchangers—Part I: Numerical Modeling

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Y. A. Khulief1

e-mail: [email protected]

S. A. Al-Kaabie-mail: [email protected]

S. A. Saide-mail: [email protected]

M. Anise-mail: [email protected]

Department of Mechanical Engineering,King Fahd University of Petroleum & Minerals,

KFUPM Box 1767,Dhahran 31261, Saudi Arabia

Prediction of Flow-InducedVibrations in Tubular HeatExchangers—Part I: NumericalModelingFlow-induced vibrations due to crossflow in the shell side of heat exchangers pose aproblem of major interest to researchers and practicing engineers. Tube array vibrationsmay lead to tube failure due to fretting wear and fatigue. Such failures have resulted innumerous plant shutdowns, which are often very costly. The need for accurate predictionof vibration and wear of heat exchangers in service has placed greater emphasis on theimproved modeling of the associated phenomenon of flow-induced vibrations. In thisstudy, the elastodynamic model of the tube array is modeled using the finite elementapproach, wherein each tube is modeled by a set of finite tube elements. The interactionbetween tubes in the bundle is represented by fluidelastic coupling forces, which aredefined in terms of the multidegree-of-freedom elastodynamic behavior of each tube inthe bundle. Explicit expressions of the finite element coefficient matrices are derived. Themodel admits experimentally identified fluidelastic force coefficients to establish the finalform of equations of motion. The nonlinear complex eigenvalue problem is formulatedand solved to determine the onset of fluidelastic instability for a given set of operatingparameters. �DOI: 10.1115/1.3006950�

Keywords: flow-induced vibrations, heat exchangers, tube bundles, crossflow

IntroductionIt is well known that bundles of tubes used in heat exchangers

nd boilers in nuclear and chemical plants can be excited to vi-rate excessively when exposed to crossflow. Pressure fluctuationsround heat exchanger tubes result in fluid-structure dynamic cou-ling, which give rise to vibrations in each tube of the bundle.ccordingly, the fluid dynamic forces on one of the tubes in theundle are induced by the vibration of the tube itself and by theibration of the neighboring tubes. This is due to the fact thatressure fluctuations around the tubes result in fluid-structure dy-amic coupling. At sufficiently high flow velocities, a large am-litude vibration at the natural frequencies of the tubes may takelace. This phenomenon is known as fluidelastic instability and isf great concern to plant operators. The onset of large amplitudeibration is quite abrupt when the crossflow velocity is increasedbove the so-called “critical velocity.” These vibrations may ap-roach resonance, thus leading to failure of the tubes. Damagelso occurs when the tube vibration results in midspan collisionsetween adjacent tubes, as well as collisions between tubes andaffle holes.

To gain more insight into this phenomenon, researchers resortedo dynamic modeling of such systems. However, modeling theynamics of fluidelastic motions due to crossflow over a tubeundle is too complicated to be investigated only analytically. Atresent there is no such a reliable analytical model that accuratelyescribes the phenomena of flow-induced vibration over a bank ofubes. Accordingly, a reliable mathematical model requires someuning via experimental measurements.

The flow-induced vibration phenomena of tube bundle vibra-ions caused by shell-side flow in heat exchangers were addressed

1Corresponding author.Contributed by the Pressure Vessel and Piping Division of ASME for publication

n the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received January 4,007; final manuscript received April 27, 2008; published online November 10,
008. Review conducted by David Raj.
ournal of Pressure Vessel Technology Copyright © 20

oaded 28 Nov 2008 to 212.26.1.29. Redistribution subject to ASME

early in the literature �1,2�. Today, the literature in this area ofresearch has become very rich, with a large number of publica-tions addressing these problems and suggesting different methodsof predictions and solutions. These include approximate analyticalmodels, analytical models with purely structural emphasis, semi-analytical models where modeling simplifications were attributedto complementary experimental investigations, and pure experi-mental studies dedicated to the understanding and identification ofsuch excitation mechanisms. Progress in research activities relatedto this problem was reviewed by Paidoussis �3�, Price �4�, andWeaver et al. �5�. In addition, some books were dedicated to pre-senting a rather detailed account of this problem, e.g., the booksby Chen �6� and Katinas and Zukauskas �7�.

A number of theoretical investigations have been conducted onflow-induced vibrations in heat exchangers. Chen �8,9�, in hissequel papers, presented a general theoretical approach to charac-terizing the instability mechanisms of a group of tubes in cross-flow. It was concluded that no single stability criterion could beapplicable to all cases for all ranges of parameters. Instead, dif-ferent stability criteria have to be devised for different instabilitymechanisms and different parameter ranges.

Cai et al. �10� reported a theoretical investigation of the flu-idelastic instability of loosely supported tubes in nonuniformcrossflow, wherein the unsteady flow theory was employed. Theirinstability analysis, however, was restricted to the inactive phaseof the tube motion. Further investigations were suggested to studythe instability mechanism during the active mode of the tube mo-tion. Cai and Chen �11� extended the theoretical study of Cai et al.�10� to include chaotic vibrations associated with the fluidelasticinstability of nonlinearly supported tubes in crossflow. However,they indicated that chaotic analysis may differ based on the com-plexity of the model, and that results derived for a one-or two-degree-of-freedom systems may not be true for multidegree-of-freedom systems. Accordingly, further investigation of the
phenomenon was suggested. Eisinger et al. �12� applied a numeri-
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al model to simulate the fluidelastic vibration of a representativeube in a tube bundle based on the unsteady flow theory. Theumerical simulation was performed using the general purposeBAQUS-EPGEN finite element code using a special subroutine in-orporating fluidelastic forces. The results were validated againstome published data based on linear cases.

A generalization of the quasisteady theory was developed byranger and Paidoussis �13� and was applied to study the cross-ow induced vibration of tube arrays. This study was conductedn the basis of the continuity and Navier–Stokes equations, in thease of an impulsive movement of a body subjected to crossflow.umerical solutions showed that the quasi-unsteady model was a

lear improvement on the conventional quasisteady approach,eading to a more reasonable agreement with experimental resultsnd providing refined insights into the physical mechanisms re-ponsible for fluidelastic instability.

An analytical computational fluid dynamics technique was in-roduced by Ichioka et al. �14� to study the fluidelastic vibration ofube bundles in heat exchangers. The technique was based on the

oving mesh method, which was developed by the authors. Theechnique was examined for the case of two cylinders in an infi-itely arrayed cylinder row.

Kassera and Strohmeier �15� introduced a two-dimensionalimulation model for the flow-induced vibrations in tube bundles.he flow field equations including turbulence were solved using

he boundary element method; however the tubes were structur-lly treated as rigid cylinders supported by linear elastic strings.

The work-rate concept was also utilized by Yetisir et al. �16� tonvestigate the possibility of establishing a damage criterion forretting wear based on the flow-induced vibration characteristics.hey developed a simple criterion based on parameters such asibration frequency, midspan vibration amplitude, span length,ube mass, and an empirical wear coefficient. In Ref. �16�, thexcitation was due to turbulence only, wherein fluidelastic forcesere not modeled.Fischer and Strohmeier �17� introduced a coupled fluid-

tructure interaction model to evaluate the stability of tubeundles in crossflow. A three-dimensional transient model is de-eloped, which was augmented by a structural response modelased on beam theory and frictional impact. Reasonable resultsere obtained for the case of fixed-fixed tube in crossflow. Theodel, however, does not address the effect of the tube pitch ratio

nd tube arrangement in the bundle.Au-Yang �18,19�, in his sequel papers, reviewed the theoretical

evelopment of the acceptance integral method to estimate theandom vibration of structures subjected to turbulent flow. Thelosed-form and finite element solutions together with a standardommercial finite element structural-analysis computer programere employed to compute the joint and cross acceptances for

ubes and beams subjected to crossflow. The results were pre-ented in the form of charts, and steps were given to show how tose these charts together with standard commercial finite elementtructural-analysis programs to estimate the responses of singlend multispan tubes to crossflow turbulence-induced vibration.

It has become evident that the modeling of the complex dynam-cs of fluidelastic forces that give rise to vibrations of tube bundlesequires a great deal of experimental insight. Experimental inves-igations on the phenomenon of flow-induced vibrations in heatxchangers were recognized and pursued for the following twoeasons: �a� to gain more insight into the nature of such complexynamic behavior and �b� to reduce the complexity of the derivedathematical models in light of some experimental findings.rover and Weaver �20,21� presented a sequel of experimental

tudies of crossflow-induced vibration of the tube array, andointed out some observations over the range of their tests. Levernd Weaver �22� made use of the findings of such previous ex-erimental investigations and developed a theoretical model forhe fluidelastic instability in heat exchanger tube bundles. They
receded their theoretical model derivation by a series of experi-
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ments, thus paving the ground for considerable simplification ofthe model. Based on experimental results, they concluded thatonly a single elastic tube and the flow streams immediately adja-cent to either side of the tube are required to model the essentialfeatures of the fluidelastic system. The developed model was invery good agreement with the experimental results, and furthermodel improvement was recommended to include more sophisti-cated fluid mechanics, and to study the effect of tube pitch ratioand pattern on the predicted stability.

To account for some fluidelastic effects that could not be tack-led by the quasisteady flow theory, Tanaka and co-workers�23–25� introduced a method for calculating the critical flow ve-locity based on the unsteady flow theory. They presented a dy-namic model of a tube bundle in crossflow, wherein the unsteadyfluidelastic forces, for small tube vibrations, are represented by alinear equation with three terms representing the added mass,damping, and stiffness effects. The model addressed the couplingbetween adjacent cylinders in the bundle; however each tube isstructurally modeled by a single degree-of-freedom lumped-parameter system. Equations were presented in matrix form in-cluding coupling, and experimental measurements were utilized todetermine the fluid-induced force coefficients.

Granger et al. �26� used the same linear model of fluidelasticforces by Chen �8� to write the dynamic model for a bundle of Ntubes in crossflow. The 2N degree-of-freedom model was laterapproximated by a single degree-of-freedom system called theglobal system. Experimental measurements were performed on atube bundle with several instrumented tubes, and the measuredmodal parameters were then used to determine the global dampingand natural frequency, which in turn were used to determine thefluidelastic force coefficients. This approach, however, is depen-dent on indirect identification techniques and the associated digitalsignal processing method.

A methodology for modeling flow-induced vibrations of tubesin crossflow was presented by Chen and co-workers �27,28�. Theunsteady flow theory is utilized in establishing the fluidelasticforce coefficients. The fluid-induced dynamic forces are thenadded as exciting forces. Such forces were modeled by a similarlinear expression to that used by Tanaka et al. �24�. The fluid forcecoefficients were stated analytically and justified by experientialmeasurements. They concluded that fluidelastic coefficients de-pend on the tube arrangement, pitch, oscillation amplitude, re-duced flow velocities, and Reynolds number.

Although considerable progress has been made in the area offlow-induced vibrations since the early 1970, it remains necessaryto understand the flow-induced vibration mechanisms for all pos-sible flow situations. To date, there are no accurate criteria bywhich one could pinpoint the onset of fluidelastic instability inheat exchangers. The criteria set by TEMA �29� are relied on inindustry, though it is not adequate in predicting the onset of dam-aging flow-induced vibrations in many situations. Pettigrew et al.�30� presented a comprehensive account of the problem of flow-induced vibrations that still affects the performance and reliabilityof heat exchangers. They concluded that there are still some im-portant areas requiring further investigation, e.g., the prediction offretting wear and the understanding of damping mechanisms. Re-cently Pettigrew and Taylor �31,32� presented an excellent over-view of flow-induced vibrations in heat exchangers and outlinedsome design guidelines to avoid the onset of such damaging vi-brations.

Now, one must recognize that dynamic modeling in conjunctionwith experimental measurements is the most feasible route totackle such a problem. Accordingly, a reliable dynamic modelmust be tuned by the experimental evaluation of some fluid-induced force coefficients. However, most of the analytical mod-els presented used simple lumped-parameter models to representthe elastodynamic behavior of the tubes, wherein the fluidelastic
coupling is configured for a single degree-of-freedom tube. Few
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nvestigators utilized commercial finite element codes to modelube bundles, with no mention of whether the fluidelastic couplingas integrated into such models.In this paper, a fluidelastic dynamic model using the finite ele-ent approach is developed. In this context, each elastic tube isodeled by a set of finite tube elements comprising a set of nodal

egrees-of-freedom. The resulting nonlinear dynamic model ad-its the unsteady fluidelastic coupling in terms of experimentallyeasured force coefficients. The developed finite element formu-

ation allows the fluidelastic coupling to exist over the length ofhe entire tube, thus avoiding the assumption adopted by severalrevious investigations when the tube was treated as a singleumped mass suspended by an elastic support. The resulting non-inear eigenvalue problem is iteratively solved and the instabilityonditions are determined. The experimental investigation neededo measure the fluidelastic force coefficients is presented in a com-anion paper by the authors �33�.

The Elastodynamic ModelBased on the actual heat exchanger construction, all the tubes

f the tube bundle were normally made of the same material andad the same cross-sectional dimensions. The developed formula-ion, however, is written in a general form to admit different geo-

etrical and material properties for each elastic component.

2.1 Assumptions and Basic Coordinates. The following arehe basic assumptions underlying the elastodynamic modeling: �a�he material of the elastic tube is homogeneous and isotropic; �b�

he deflection of the tube is produced by the displacement ofoints of the centerline; and �c� the shear deformation for suchlender tube configuration is neglected. Now, let us consider anlement of the elastic tube, as shown in Fig. 1. The deformation ofgeneral point p in the cross section can be represented by the

osition vector rp, which can be expressed in the form

rp = ro + e �1�

here ro is the unreformed position of point p, and e is the elasticeformation vector, which can be written in the form e= �N��q�.ow, Eq. �1� becomes

rp = ro + �N��q� �2�

here �q� is the vector of nodal coordinates and �N� is the shapeunction matrix. The velocity vector is then obtained from Eq. �2�y differentiation with respect to time. Noting that the heat ex-hanger tube does not execute reference motion, the time deriva-ive of Eq. �2� can be expressed as

rp = �N��q� �3�

2.2 Kinetic Energy Expressions. In order to utilize theagrangean approach, one needs to derive expressions for theystem’s kinetic and potential energies. Using the velocity vector

Fig. 1 The generalized coordinate system

n Eq. �3�, the kinetic energy expression can be written as

ournal of Pressure Vessel Technology


KE =1

2�v

�s�rp�T�rp�dv �4�

Using Eq. �3� into Eq. �4�, one obtains

KE =1

2�s�

v

�q�T�N�T�N��q�dv =1

2�q�T�M��q� �5�

where �M�=��v�N�T�N�dv is the elastic mass matrix that accountsfor axial, bending and torsional deformations of the tube element,and �s is the mass density of the tube material; details of the massmatrix will be derived in the next section.

2.3 Strain Energy Expressions. There are six nodal coordi-nates, which are defined as follows: uaxial deformation alongthe x-direction, v translational deformation about the y-axis, w translational deformation about the z-axis, �y =�v�x , t� /�xnodal bending rotation about the y-axis, �z=�w�x , t� /�xnodal bending rotation about the z-axis, and �xnodal torsionalrotation about the x-axis Noting that the cross-sectional secondmoment of area Iz= Iy = I�x�, ��v /�x�=�y and ��w /�x�=�z, thestrain energy expression can be written in the form

SE =1

2�0

l EI�x�� y

�x 2

+ � ��z

�x 2� + EA� �u

�x 2�dx

+1

2�0

l

GJ� ��

�x 2

dx �6�

Using the standard finite element formulation, as explained in thenext section, Eq. �6� can be written in compact matrix form as

SE =1

2�q�T�K��q� �7�

where �K� is a 12�12 elemental stiffness matrix, which accountsfor bending, axial, and torsional stiffness, respectively.

Structural Dynamics FEM Formulation. In this formulation,a finite two-node tube element, with six degrees-of-freedom pernode, is considered. Accordingly, the elemental nodal coordinatevector, as shown in Fig. 2, can be written as

�q� = �u1,v1,w1,�x1,�y1

,�z1,u2,v2,w2,�x2

,�y2,�z2

�T �8�

Now, assuming a feasible displacement field, the axial and bend-

Fig. 2 Nodal coordinates of the tube element

ing deformations can be expressed in the form

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�u�x,t�v�x,t�w�x,t�

� = �Nu1 0 0 0 0 0 Nu2 0 0 0 0 0

0 Nv1 0 0 Nv2 0 0 Nv3 0 0 Nv4 0

0 0 Nv1 Nv2 0 0 0 0 Nv3 − Nv4 0 0��q�

= �Nu

Nv

Nw��q� = �Nt�x��q� �9�

��y

�z� = 0 N�1 0 0 N�2 0 0 N�3 0 0 N�4 0

0 0 − N�1 N�2 0 0 0 0 − N�3 N�4 0 0��q� = N�y

N�z��q� = �N��x��q� �10�

��x,t� = �0 0 0 0 0 N�1 0 0 0 0 0 N�2��q� = �N��q� �11�

here Nu, Nv and Nw are the translational shape functions, N�ynd N�z are the elastic rotational shape functions, and N� is theorsional deformation shape function. These shape functions wereerived by Perzemieniecki �34�, or Bazoune et al. �35�. The finitelement expressions of the coefficient matrices are given by

�K� =�0

l

�Be�TEI�Be�dx +�0

l

�Ba�TEA�Ba�dx +�0

l

�Ba�TEA�Ba�dx

�12�

nd

�M� =�0

l

�Nv�T�sA�Nv�dx +�0

l

�N��TID�N��dx +�0

l

�N��TIp�N��dx

�13�

here the three terms in Eq. �12� represent the bending, axial, andorsional stiffness matrices, respectively. The derivatives of thehape function matrices are given by �Be�=��N�� /�x, �Ba��Nu� /�x, and �B��=��N�� /�x. In Eq. �13�, the three terms rep-

esent the translational mass matrix, the rotational inertia massatrix, and the torsional mass matrix, respectively.

2.5 Equations of Motion. The dynamic equations of motionhat represent the elastodynamic behavior of the tube can be de-ived using the Lagrangean approach. Denoting �qi� as the vectorf nodal displacements of tube i, one can substitute theagrangean function in the variational form, carry out the associ-ted differentiations, and then perform standard finite element as-embly procedure to express the equation of motion of tube i inhe following final form:

�Mi��qi� + �Di��qi� + �Ki��qi� = �Qi� �14�

quation �14� represents the elastodynamic model of a tube ele-ent, where the vector �Qi� is the generalized force vector thatay contain all externally applied forces, and may also contain

he time-dependent fluidelastic coupling forces, if dynamic re-ponse analysis is required. In this context, some subscripts arentroduced to identify the constituent matrices of a given tube as

�Msi��qi� + �Ds

i��qi� + �Ksi��qi� = �Qi� �15�

or the rest of the derivation, the subscript s is introduced to refero the intrinsic structural properties, e.g., structural mass, damp-ng, and stiffness properties, while the subscript f refers to theorresponding fluidelastic terms. The fluid-structure interaction, as
anifested by the fluidelastic forces, can be represented to include
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the coupling between the adjacent tubes in the tube bundle. Letthe fluidelastic force that couples tube i and tube j be

Fij = CDijsij + CKij

sij �16�

where sij is a vector in the yz-plane, which represents the distancebetween tube i and tube j, and CDij

is the damping coefficient thatdepends on the tube diameter, fluid density, and the bundle gapvelocity. CKij

is the stiffness coefficient that depends on the fluiddensity and the bundle gap velocity. Using the virtual work ex-pression, one can write the work done by Fij as

�WFij= − Fij�sij = − Fij��sij/�qi��qi − Fij��sij/�qj��qj

= QFiT�qi + QF

jT�qj �17�

The vectors QFi and QF

j are the generalized forces associated withthe fluidelastic coupling between tube i and tube j. These forcesare to be added to the right hand side of the equation of motion.Now, Eq. �15� can be written for tubes i and j, including theadded-mass effect, as

Mi 0

0 Mj �� qi

qj� + Ds

i 0

0 Dsj �� qi

qj� + Ks

i 0

0 Ksj ��qi

qj� = �QF

i

QFj ��18�

where M is the mass matrix that includes both the structural iner-tia properties and the added-mass effects, i.e., Mi=Ms

i +Mfi . The

added-mass matrix Mfi is a function fluid density and the tube

dimensions. Equation �18� can be written in a general assembledform to represent all active tubes in the tube array. The added-mass effects, as well as the coupling fluidelastic forces, are deter-mined experimentally by estimating the fluidelastic coefficients.

3 FEM Formulation of the Fluidelastic ForcesIn this formulation, a crossflow across a �3�3� square tube

bundle of �three rows and three columns� is considered. Previousinvestigations have concluded that the fluidelastic coupling for agiven tube is dominated by the adjacent tubes, and that the effectof other distant rows may be insignificant �23�. Nevertheless, thepresented formulation is directly applicable to a tube bundle ofany size. Let us consider a tube arrangement, as shown in Fig. 3.In the case of multirow cylinders, the force on tube �O� is con-sidered to be predominantly influenced by the vibrations of theupper �U�, the lower �D�, the right �R�, and the left �L� tubes. Thefluid dynamic force acting on tube �O� is a function of the vibra-tional motions of the tube itself and the adjacent tubes �L, R, U,and D�, both in the x- and y-directions. The linearity assumptionof the augmented effect of the fluid dynamic forces has been
verified by some experimental investigators, e.g., Tanaka et al.


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25�. Consequently, the superposition principle can be applied toxpress the fluidelastic effect as a linear combination of the flu-delastic forces produced by different active tubes in the bundle asollows:

Fy =1

2� fV

2�k=1

5

�CykyYk + CykzZk� �19�

Fz =1

2� fV

2�k=1

5

�CzkyYk + CzkzZk� �20�

ere, k=1, . . . ,5 corresponds to the tubes �O, L, R, U, and H�,espectively. The subscripts y and z denote the vibration displace-ents in the y- and z-directions, respectively. Following Tanaka’s

otations, each fluid dynamic force coefficient is identified byhree subscripts. The first suffix is associated with the direction ofhe fluid force, the second with the position of the vibrating tube,nd third with the direction of the tube vibration. In the case of thequare array, the following relations are obtained from the geo-

Fig. 3 Instrumented tube array arrangement

etrical properties and symmetry considerations of the array:



CyRy = CyLy, CzRz = CzLz, CyRz = − CyLz, CzRy = − CzLy

CyOz = CzOy = CyUz = CzUy = CyDz = CzDy = 0

The following matrix explains the location of each tube with re-spect to its surrounding tubes:

Tube Position U L O R D

Tube Number

1

2

3

4

5

6

7

8

9

�0 0 1 2 4

0 1 2 3 5

0 2 3 0 6

1 0 4 5 7

2 4 5 6 8

3 5 6 0 9

4 0 7 8 0

5 7 8 9 0

6 8 9 0 0

� �21�

Using the standard finite element assembly procedure, Eq. �14�can be written for all tubes considered in the active bundle. Thestiffness, mass, and damping matrices due to fluid coupling areadded to the respective structural stiffness, mass, and dampingmatrices of the bundle to obtain the augmented equation of motionas

�M��q� + �D��q� + �K��q� = �Q�� 22�

where �M��= �Ms��+ �Mf��, �K��= �Ks��+ �Kf��, and �D��= �Ds��+ �Df�� are the assembled finite element coefficient matrices of thewhole bundle. In this context, the matrix �Ks�� is the structuralstiffness matrix of the bundle and �Kf�� is the corresponding addedstiffness matrix due to fluidelastic forces. The matrix �Ds�� is struc-tural damping matrix of the whole bundle, which represents thematerial damping and is commonly represented by a proportionaldamping model in terms of a linear combination of both the struc-tural stiffens and mass matrices. In this formulation, the spanwiseeffect on the fluidelastic forces is taken into consideration, andconsequently the model is capable of addressing the three-dimensional instabilities �36�. The finite element expressions ofthe fluid-induced mass, stiffness, and damping matrices �Mf��,�Kf��, and �Df�� are assembled from the constituent 9�9 couplingmatrices, which are established at each nodal plane of the bundle
as follows:
�M f� = ��M1/1� �M1/2� 0 �M1/4� 0 0 0 0 0

�M2/1� �M2/2� �M2/3� 0 �M2/5� 0 0 0 0

0 �M3/2� �M3/3� 0 0 �M3/6� 0 0 0

�M4/1� 0 0 �M4/4� �M4/5� 0 �M4/7� 0 0

0 �M5/2� 0 �M5/4� �M5/5� �M5/6� 0 �M5/8� 0

0 0 �M6/3� 0 �M6/5� �M6/6� 0 0 �M6/9�0 0 0 �M7/4� 0 0 �M7/7� �M7/8� 0

0 0 0 0 �M8/5� 0 �M8/7� �M8/8� �M8/9�0 0 0 0 0 �M9/6� 0 �M9/8� �M9/9�

� �23�

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�Kf� = ��K1/1� �K1/2� 0 �K1/4� 0 0 0 0 0

�K2/1� �K2/2� �K2/3� 0 �K2/5� 0 0 0 0

0 �K3/2� �K3/3� 0 0 �K3/6� 0 0 0

�K4/1� 0 0 �K4/4� �K4/5� 0 �K4/7� 0 0

0 �K5/2� 0 �K5/4� �K5/5� �K5/6� 0 �K5/8� 0

0 0 �K6/3� 0 �K6/5� �K6/6� 0 0 �K6/9�0 0 0 �K7/4� 0 0 �K7/7� �K7/8� 0

0 0 0 0 �K8/5� 0 �K8/7� �K8/8� �K8/9�0 0 0 0 0 �K9/6� 0 �K9/8� �K9/9�

� �24�

�Df� = ��D1/1� �D1/2� 0 �D1/4� 0 0 0 0 0

�D2/1� �D2/2� �D2/3� 0 �D2/5� 0 0 0 0

0 �D3/2� �D3/3� 0 0 �D3/6� 0 0 0

�D4/1� 0 0 �D4/4� �D4/5� 0 �D4/7� 0 0

0 �D5/2� 0 �D5/4� �D5/5� �D5/6� 0 �D5/8� 0

0 0 �D6/3� 0 �D6/5� �D6/6� 0 0 �D6/9�0 0 0 �D7/4� 0 0 �D7/7� �D7/8� 0

0 0 0 0 �D8/5� 0 �D8/7� �D8/8� �D8/9�0 0 0 0 0 �D9/6� 0 �D9/8� �D9/9�

� �25�

here

�Ki/j� = −1

2� fd

2�2��CO cos ��i/j�Vr2 − 2�3�Cm�i/j��/4�2 �26�

�Mi/j� = −�

4� fd

2�Cm�i/j� �27�

�Di/j� = −1

8�2� fd2�Vr

2��CO sin ��i/j� �28�

��CO cos ��i/j� = Block Diagonal��Ai/j� �0� �Ai/j� �0� �Ai/j� �0��29�

here �0� is a 2�2 null matrix, and �Ai/j� is given by

�Ai/j� = �CO cos ��YKY �CO cos ��YKZ

�CO cos ��ZKY �CO cos ��ZKZ� �30�

he subscript k refers to O, L, R, U, or H, depending on theosition of tube j with respect to tube i. The constituent couplingatrices in Eq. �23�–�25� are assembled into the associated coef-cient matrices according to the placement array of Eq. �21�. Forxample, if i=1 and j=2, then k stands for R because tube 2 isositioned to the right of tube 1, i.e.,

�A1/2� = �CO cos ��YRY �CO cos ��YRZ

�CO cos ��ZRY �CO cos ��ZRZ� �31�

ere, V is the crossflow velocity, � f is the fluid density, d is theuter diameter of the tube, and VR is the reduced velocity. Theoefficient submatrices �Mf�, �Kf�, and �Df� are the elementaldded-mass, stiffness, and damping matrices on tube i due to flu-delastic coupling with tube j. It is noted that when i= j, such

atrices account for the fluid-induced effects due to the vibration
f the tube itself.
11301-6 / Vol. 131, FEBRUARY 2009


4 The Eigenvalue and Stability AnalysisAssuming no externally applied forces, the fluidelastic forces

on the right side of Eq. �22� can be absorbed into the left side, thusleading to the representation of the final dynamic model in thefollowing compact form:

�M��q� + �D��q� + �K��q� = �0� �32�

where �M�, �D�, and �K� are the global mass, damping, and stiff-ness matrices that include both structural and fluidelastic coeffi-cients for the whole tube bundle. The eigenvalue problem associ-ated with this nonself adjoint form can be established by writingEq. �32� in the state space form as

d

dt�q

q� = �0� �I�

− �M�−1�K� − �M�−1�D��q

q� �33�

or simply as

�y� = �A��y� �34�

where �y�= ��q�T �q�T�T is the state vector, and �A�= �A�� is thestate coefficient matrix, which is a function of the natural fre-quency of the tube bundle. �A� is a general matrix, thus the result-ing eigenvalues are complex. The dependence of �A� on the natu-ral frequency results in a nonlinear eigenvalue problem. Neumaier�37� presented an inverse iteration scheme for the nonlinear eigen-value problem. A class of nonlinear eigenvalue problems encoun-tered in solid-structure problems has been addressed by Conca etal. �38�.

The scheme adopted in this paper employs an inverse iterationouter loop with the MATLAB complex eigenvalue solver as its innercore. The method first calculates structural stiffness, mass, anddamping matrices for the tube bundle. It also interpolates the fluidforce coefficients by curve fitting so that these coefficients can bedetermined at any iteration step of the reduced velocity. It thenupdates the stiffness, mass, and damping matrices with the currentvalue of fluidelastic effects. The modal characteristics of thewhole system �tube bundle� are then determined by solving the
associated complex eigenvalue problem. The critical reduced ve-


lTtoe�

5

tTr��p

tpoiiutT12vcbtl

a

idb

J

Downl

ocity is defined as the reduced velocity at the onset of instability.he iteration is indexed over the reduced velocity, and the itera-

ion is terminated once the critical reduced velocity is reached. Inther words, the critical reduced velocity is one at which the high-st real part of an eigenvalue changes its sign from negativestable region� to positive �unstable region�.

Numerical Results and ConclusionsIn order to test the developed numerical scheme, the data of the

est example reported by Tanaka and co-workers �23,25� are used.he case consists of a 3�4 square array with a pitch-to-diameter

atio of 1.33. Tubes 30 mm �1.18 in.� in diameter and 300 mm11.8 in.� in length were used. For this case study, Tanaka et al.25� tabulated the fluidelastic coefficients, which they have ex-erimentally determined using their experimental setup.

The developed scheme was invoked by utilizing the aforemen-ioned available test data. In this regard, the nonlinear eigenvalueroblem was generated and solved numerically to study the effectf different tube arrays and flow parameters on the onset of flow-nduced vibrations. Using the same mass ratio and reduced veloc-ty definitions of Tanaka, Table 1 displays the results obtainedsing the developed finite element model �FEM� formulation forhe case of having the same mass damping parameter of 0.19.able 2 shows the comparison of results for a fixed mass ratio of.852 and a range of damping values. As depicted in Tables 1 and, the results obtained by the developed finite element model areery comparable to those obtained by Tanaka; however they wereonsistently slightly lower. This may be anticipated, which coulde attributed to two reasons, namely, �a� the frequency calcula-ions and �b� the fluidelastic forces calculations. Firstly, the single

Table 1 Critical velocity estimates „�=0.19…a

Mass ratiob

��

Critical velocity

Ref. �24� Present method

2.22 2.024 1.9834.44 2.283 2.2048.88 2.394 2.31117.76 2.448 2.39835.52 2.474 2.42171.04 2.485 2.442

�=m� /� fd is the mass damping parameter, where � is the logarithmic decrement, ms the mass per unit length of the tube, � f is the fluid density, and d is the tube outeriameter.Mass ratio: =m /� fd

2.

Fig. 4 Effect of

umped-mass model assumed by Tanaka is known to underesti-



mate the calculated natural frequencies, while the consistent massFEM formulation overestimates calculated frequencies. Secondly,the fluidelastic coupling in Tanaka’s model is associated with asingle lumped-mass, while in the finite element it is a function ofa set of nodal degrees-of-freedom associated with the distributedmass over the whole tube length.

The developed numerical scheme is then used to study the ef-fect of the pitch-diameter-ratio �� on the onset of fluidelasticinstability, as marked by the recoded value of the critical velocity.Figure 4 shows the results for the case when =2 and =0.2,while varied in the range 1.2–2.5. The numerical prediction ofthe critical velocity showed that the value of critical velocity in-creased as increased. However, for values of above 1.75, theincrease in critical velocity is relatively negligible. A similar trendwas reported by Tanaka et al. �24�, in which they pointed out thatthe critical velocity did not vary for greater than 2.

As manifested by the aforementioned comparisons, the devel-oped finite element numerical scheme demonstrates a good degreeof accuracy in predicting the onset of instability associated withthe flow-induced vibrations in tubular heat exchangers. The devel-oped scheme has the advantage of representing the fluidelasticcoupling forces in terms of a set of degrees-of-freedom distributedover the entire tube length, thus providing a more accurate pre-diction of such forces.

AcknowledgmentThis research work is funded by Saudi Aramco, Project No.

ME2212. The authors greatly appreciate the support provided bySaudi Aramco and King Fahd University of Petroleum & Minerals

Table 2 Critical velocity estimates „�=1.852…

Logarithmicdecrement

��

Critical velocity

Ref. �25� Present method

0.0023 1.35 1.2910.0061 1.45 1.3870.0297 1.75 1.6960.033 1.80 1.7340.087 2.35 2.2210.166 2.70 2.586

h-diameter-ratio
pitc
during this research.

FEBRUARY 2009, Vol. 131 / 011301-7


N

R

0

Downl

omenclatureA � cross-sectional area of tube

Cijk � fluid dynamic force coefficientCm � added-mass coefficientCD � damping coefficientCK � stiffness coefficientC0 � constant amplitudeF � forceG � shear modulusd � tube diameter

�D� � damping matrix�K� � stiffness matrix�M� � mass matrix�N� � shape function matrix�q� � nodal coordinate vector�Q� � generalized force vector

rp � position vectorVr � reduced velocity�s � density of tube material� f � density of fluid� � torsional deformation� � phase difference� � frequency �rad/s�

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Prediction of Flow-Induced Vibrations in Tubular Heat Exchangers—Part I: Numerical Modeling

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