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Chapter 2 

FLOW-INDUCED VIBRATIONS IN HEAT EXCHANGERS 

2.1 Introduction

Heat exchanger’s tubes tend to oscillate under the effect of cross-flow velocities. The

heat exchange tubes usually get damaged, when the amplitude of vibration becomes

sufficiently large, due to following mechanisms; (a) thinning of mid-span due to

continuous collision, (b) impact and fretting wear at the baffle plate and tube interface,

and (c) corrosion due to significantly high wear rate. Heat exchanger failures are

extremely costly, they can cause plant shutdown. These issues are extremely serious in

heat exchangers used in the nuclear industry. It is important to ensure that the modern

shell and tube heat exchangers are safe from the influence of flow-induced vibrations.

Good thermal performance and low fouling generally require higher flow velocities,

while fewer baffle plates are desirable to minimize pressure drop. Higher flow velocities

and fewer baffles can lead to severe flow-induced vibration. It is essential to avoid costly

tube failures by a detailed flow-induced vibration analysis at the design stage. The flow-

induced vibration phenomenon and the underline physics responsible for these vibrations

have been studied rigorously over the past decades. This subject continues to gainsattention because of its significance especially in shell and tube type heat exchangers,

which contributes over 60% of operational heat exchangers in industry.

2.2  Basis of Flow-Induced Vibration

The shell side flow in the heat exchanger is the prime cause of excitation and vibration.

The tubes usually behave as slender elastic beams between the shell and tube heat

exchanger components. When dislocated from its mean position, it experience vibrations.

The tube motion due to cross-flow velocity is [22]:

1.  For low cross-flow velocities, the tubes vibration is at low amplitude with random

motion.

2.  At medium flow velocity, rattling of tubes occurs in the baffle holes.

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3.  At higher flow velocities above some threshold value, high amplitude motion

occurs which can lead to severe damage.

When the exciting frequency reaches the fundamental frequencies of the tubes, resonance

occurs. The relative motion between the tubes and the supports such as baffles and shell

vessel boundary can lead to cause impact and fretting wear of tubes.

2.3 Damages of Flow-Induced Vibration

Tube damages are due to [22]:

1.  Impact wear (Rattling of tubes and tube to baffle impact).

2.  Fretting wear at the tube/baffle interfaces due to impact and/or sliding motion.

3.  Combined impact and fretting wear.

Mechanical failure results either from fatigue and impact damage or welding failures

[23, 24];

   Mid-span collision: Collision with adjacent tubes takes place if the amplitude of

response at the mid-span is large enough. The resulting wear can damage the tube

wall under pressure.

  Wear damage: Heat exchangers tube bundles are generally designed with a

clearance between the tube and the baffle plates. Tubes that suffer lower

amplitude vibration close to baffle plates may fail by impact and fretting wear or

fatigue.

  Fretting wear: Oscillating motion is produced due to cyclic loads at tube to baffle

interface.

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   Excessive noise level:  Acoustic vibration will be induced within the tube bank

containment if shell-side medium is a gas, steam or air. The acoustic vibration is

characterized by pure-tone, low-frequency intense noise.

  Severe pressure drop: As the vibration of tube requires the energy from the shell-

side fluid, its pressure drop increases. If the vibration is severe, destructive

 pressure fluctuations can take place.

   Intensified stress corrosion: Intensive tensile stresses are induced on the tube

surface due to repeated impact with the baffle supports. Susceptible tube material

can fail due to the accelerated stress corrosion cracking in the shell-side medium.

However, corrosion due to flow-induced vibration is second to the tube failure in

corrosive environment.

2.4 Regions of Tube Damage

The high velocity regions are more vulnerable for flow-induced vibration such as:

  Largest unsupported mid-span between two baffles.

  Tubes located in the baffle window region at periphery of the tube bundle.

  Bend regions of U-tube bundle.

  Tubes located near the inlet nozzle.

  Tubes located in the tube bundle bypass area, next to pass partition lanes.

  Interfaces of tube and baffle support, tube and tube-sheet.

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2.5 Flow-Induced Vibration Mechanisms

The following excitation mechanisms are usually responsible for flow-induced

vibrations;

1. 

Vortex shedding2.  Turbulent buffeting

3.  Fluid elastic instability

4.  Acoustic resonance

Vortex shedding, turbulent buffeting and acoustic excitation are due to resonance

 phenomena. Fluid elastic instability sets in for tubes in a cross-flow at a critical flow

velocity or threshold velocity resulting in amplitude of tube response large enough to

collide with the adjacent tubes and cause failure. Below the critical velocity fluid elastic

instability (FEI) will not take place. Fluid elastic instability is a primary concern in all

flow mediums in heat exchangers. Table 2.1 shows a summary of the important Flow-

Induced Vibration mechanisms for different of the flow mediums in a tube bundle [25].

Table 2.1 Vibration Excitation Mechanisms Versus Flow Medium for Crossflow [25]

Flow Situation

(Cross-Flow) Fluid Elastic

Instability Vortex

Shedding Turbulent

Buffeting Acoustic

Resonance 

Single Cylinder

Liquid 

U 

I 

I 

U 

Gas U  P  P  U 

Two-Phase  U  U  I  U 

Tube Bundle

Liquid  I  P  P  U 

Gas I  U  P  I 

Two-Phase  I  U  I  U 

U-Unlikely, P-Possible, I-Most Important (to consider)

2.5.1 Vortex Shedding

 a) Single tube

 Now consider a circular cylinder subjected to cross-flow. As the flow passes the circle,

the wake behind the tube usually gets non-regular and distinct pattern of vortices can be

seen (Fig. 2.1). The regular shedding of the vortices from the sides of the body in a

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 pattern, give rise to fluctuating lift and drag forces, leading to periodic motion of the tube.

An example of such phenomenon is the Von-Korman Vortex Street.

Figure 2.1. Vortex shedding past a single cylinder.

Strouhal number  S  , and vortex shedding frequency s f   characterizes the Vortex

Induced Vibration (VIV) and are related as;

U 

 D f S  s   (2.1)

Here,  D  = Outside diameter of tube in meters and U  = Velocity of Upstream in m/s.

Whenever the vortex shedding frequency matches with, or closes to the natural frequency

of tube, resonance takes place. Resonance is characterized by large amplitudes of tubemotion with possible damage to the tube. The vortex shedding is also referred in

literature as periodic wake shedding, Strouhal periodicity or Strouhal excitation.

Since the vortex shedding drag force in the stream-wise direction (drag) occurs at twice

the vortex shedding frequency and the magnitude of drag force is lower than the

oscillating lift force, normally the analysis is carried out for lift forces only.

The vortex shedding phenomenon for a single cylinder with a peak response is well de-

fined and has been dealt with by various researchers. Information on the lift and drag

force coefficients and the Strouhal number for the Reynolds numbers of interest has been

reviewed and presented by Chen and Weber [27]. With reference to Figure 2.2, the

Strouhal number is around 0.2 for Reynolds numbers ranging from 300 to lower critical

value of 5102   for cross-flow over a tube. After this point the Strouhal number

Fluid

Movement

Movement

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increases due to wake contraction. But as the Reynolds number exceed the value of

5105.3     and when the supercritical range is reached, the wakes become completely

turbulent. No regular vortex shedding exists any more. This exceptional case lasts only to

a Reynolds number of  6105.3   . After exceeding this value, a Von Korman vortex can be

formed again. In the transcritical range the Strouhal number is about 0.27 [28].

Figure 2.2. Strouhal number for single cylinder [27].

 b) Vortex Shedding for Tube Bundles

It is usually assumed that for multiple tube arrays under cross-flow, vortex shedding

occurs in a pattern similar to a single isolated cylinder. But research shows that vortex

shedding results in a peak structural response that resembles to an isolated cylinder for

the tube rows in an array at the enterance, but a distinct resonance does not exit for most

 part of tube arrays.

Vortex shedding past a tube bank is shown in Fig. 2.3. Owen [29] argued the presence of

vortices deep inside a tube bank. The dominant frequency for lift and drag forces in

vortex shedding and turbulent buffeting coincides deep within the tube bank. For a

closely packed tube arrays, with pitch ratio less than 2, Blevins [30] and Zukauskas [31]

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Figure 2.3. Vortex shedding in a bank.

found that the vortex shedding degenerates into multiple turbulent eddies instead of a

single distinct frequency. Such mechanism is called turbulent buffeting [30, 31]. At the

same time vortex shedding is a design problem for the front tube rows of a tube bankespecially in liquid flows and can produce acoustic noise in gas flows [32]. Hence the

 possibility of the tube rows at the enterance excited by vortex shedding has to be

determined. Deep in the array, vortex shedding can be considered as a special case of

turbulent buffeting and should be calculated by the method of random vibration.

The expression for the Strouhal number for a tube bank is the same as equation 2.1 but

the velocity term U   should he replaced by the cross-flow velocity. Even though many

researchers used either pitch velocity or row velocity to account for cross-flow velocity

for their row models but for all these sections the velocity term is the cross-flow velocity

should be calculated by Tinker’s method [33] or Bell’s method [34] or through the stream

analysis method [35]. Liu [36] used measured vibration data to build a mathematical

method to find unknown mechanical parameters for inverse vibration problem. He chose

displacement data to categorize a time-dependent function of damping or stiffness.

In liquid cross-flow, vortex shedding frequency resonance may occur when there is

relatively uniform flow. Normally, this mechanism does not happen at the entrance of the

steam generator as the flow is non-uniform and turbulent [38]. The TEMA [39] approach

also uses Tinker’s method for calculation of shell-side (E-type) cross-flow velocity as

input for all excitation mechanism.

 D  X  L 

T  X    L

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2.5.2 Turbulence Buffeting

High flow rates generally produce high turbulence in the fluid, which enhances heat

transfer, but at the same time it is an additional cause of structural excitation. Tubes

respond in a random fashion to the flow turbulence. Turbulence in the flow can influencethe existence and strength of other excitation mechanisms.

Turbulent buffeting phenomenon, sometimes referred to as ‘subcritical vibration’, is the

low amplitude response of the bundle before the critical velocity, usually away from

lock-in region.

The turbulent flow has been described by random velocity perturbations. The turbulent

eddies in turbulent flow have a wide range of frequencies populated around a central

dominant frequency. As this dominant frequency coincides with the tube’s lowest natural

frequency, resonance state occurs due to large amount of energy transfer which leads to

high-amplitude tube vibration. Turbulent buffeting can cause fretting wear and fatigue

failure even in the absence of resonance.

With a design objective of 40 years code life for nuclear power plant steam generators

and heat exchangers, even relatively small tube wear rates cannot be acceptable [32,40].

Hence, turbulence excitation has importance in the design of reliable heat exchangers.

Turbulent Buffeting Frequency

Owen [29] correlated an empirical expression relating turbulent buffeting frequency tb f   as

 

  

  28.0

1105.3

2

t t l

tb X  X  DX 

U  f    (2.2)

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Where;

l X  = longitudinal pitch ratio =  D L p /

t  X   = transverse pitch ratio =  DT  p /

 p

 L  = longitudinal pitch

 pT   = transverse pitch

U = cross-flow velocity

Weaver and Grover [41] suggested that Owen’s approach is suitable for prediction of

 peak frequency in the turbulence, given that minimum gap velocity is used in the

expression. The preceding correlation is only applicable for a tube bank with transverse

 pitch ratio more than 1.25.

2.5.3  Fluid Elastic Instability

Fluid elastic vibration occurs in tube arrays at some flow velocities and sometimes leads

to large amplitudes if the flow velocity is further increased. Common examples of such

vibration are aircraft wing flutter, transmission line galloping, and vibration of tube

arrays.

The tube bank experiences instability when the energy input to the tube mass-damping

system is more than the energy dissipated by the system. The tube failure occurs in a

relatively short period of time due to fluid elastic instability and this can be avoided by

controlling the cross-flow velocity [42].

Some tube responses cannot be avoided and tube can damage due to fretting wear in

turbulent buffeting or vortex shedding [43]. If vortex shedding resonances are predicted

at velocities above the fluid elastic critical velocity, then vortex shedding would not be aconcern and no need to predict the associated amplitudes of vibration.

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 2.5.3.1 Fluid Elastic Forces

The fluid elastic force induced on tube in a cross-flow falls into the three major

categories [26, 44]:

1.  Forces proportional to the displacement of the tube. This instability mechanism is

called displacement mechanism.

2.  Fluid forces dependent on velocity of fluid.Forces such as fluid inertia, fluid

damping, and fluid stiffness forces. This instability mechanism is called velocity

mechanism.

3. 

A combination of both displacement and velocity dependent forces.

 2.5.3.2 General Characteristics of Instability

The characteristics of unstable tube vibrations include:

1. 

High vibration amplitude of tube.

2.  Synchronization between adjacent tubes of the bundle.

3.  Coupling of fluid and structure.

Typical trajectories of unstable tube motion of a tube array are shown in Fig.2.4.

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Figure 2.4. Tube motion during instability. [45]

 2.5.3.3 Fluid Elastic Instability Analysis

The work on FEI was initiated by Connors [46] in 1970. He studied tube rows in a

wind channel using a quasi-static model. The schematic model is shown in Fig. 2.5.

Figure 2.5. Connors single row FEI model. [46]

A quasi-static model uses fluid dynamic coefficients, obtained from tests on a stationary

 body, to determine the fluid dynamic forces acting on a vibrating body. According to

Connors, when the total amount of energy input to the tubes in a cross-flow exceeds the

energy that is dissipated through damping, FEI occurs. This can lead to tube vibration

intensified such that clashing occurs with adjacent tubes. Quasi-steady model force

Upper wing tube (tube 2)

moved in a trajectory

Lower wing tube (tube 2)

moved in a trajectory

Central tube (tube 1)held fixed

Air flow

1.41D

D

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coefficients developed a semi-empirical stability criterion for estimating the start of fluid

elastic instability of tube arrays. The criterion relates the critical flow velocity ‘ cr U  ’ and

the properties of the fluid and structures given by,

a

sn

cr 

 D

mK 

 D f 

U  

  

 

2  

   (2.3)

Here

cr U  = critical velocity

    = logarithmic decrement of damping (    2 )

    = damping ratio,

n f    = tube natural frequency (Hz)

s     = density of shell-side fluid,

m   = mass/unit length of tube= sca mmm    

am   = fluid added mass/unit length

cm   = fluid mass/unit length

sm   = structural mass/unit length

K and a are fluid elastic instability constants

For his single-row experimental model with p/D = 1.41, the value of K  =9.9 and a = 0.5.

(Here p is the tube pitch and D is the diameter of tube).

Accordingly, the expression for instability is given by

5.0

29.9

 

  

 

 D

m

 D f 

U 

sn

cr 

  

   (2.4)

In this expression, the two main parameters are

 

  

 

 D f 

U 

n

cr   is the reduced velocity and

 

 

 

 2

 D

m

s  

  is the mass damping parameter.

The Connors vibration mechanism was later referred to as a displacement mechanism and

the model is known as a quasi-static model.

The value of K  =9.9 in tube bundles does no hold good. So, several experiments have

 been conducted to form a more appropriate value of K . Several new models; the

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2.5.4 Acoustic Resonance

The cross-flow of gas/steam causes acoustic resonance. Standing waves arise which are

transverse to both tube axis and flow direction as given in Figure 2.6. The resonating

oscillation of the standing waves surrounding the tubes is known as acoustic resonance oracoustic vibration.which is characterized by intense, low-frequency, pure-tone noise with

the frequency of oscillation closer to the standing wave frequency. If the standing wave

frequency coincides with the natural frequency of structural members (such as supports,

shell, tubes, etc.) this may become structurally harmful [58]. This can reduce the life of

the heat exchanger by increasing the pressure drop. Acoustic resonance is seen in

staggered/in-line tube banks, single rows of tubes, superheaters, ducts of

rectangular/circular shells, and others [59, 60].

Figure 2.6. Fluctuating transverse velocity (solid lines) and fluctuating pressure

(dashed lines) related to standing waves with 1, 2 and 3 half waves [61].

This section represents the

 pressure node and

displacement antinode

WALL WALL

Pressure antinodedisplacement node

n=1

n=2

n=3

La 

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Fluid elastic

instability

Response to flow

“periodicity” Nonresonant buffetingresponse

   T  u   b  e   D   i  s  p

   l  a  c  e  m  e  n   t

Velocity, U  Ucr  

Vorticity

Turbulence

 Natural

Frequencies

5 10 15 20

Tube Response Spectra (Hz)

0.45

0.30

0.15

0.00

   M  a  g  n   i   t  u   d  e

(a) (b)

2.6 Tube Response

The response of tube to flow-induced vibration in a tube bundle due to any of the

excitation mechanisms are shown in Fig. 2.7. Each one is seen only for a range of flow

velocities except turbulent buffeting which is seen over the entire velocity range.

Figure 2.7. Response spectrum of a tube (a) Ideal diagram, (b) 1.5 pitch ratio

rotated square array (Aluminum tubes) in water having cross-flow upstream

velocity of 0.117 m/s [62].

2.7 Tube Bundle Dynamic Behavior

The dynamic behavior of an array of tube splits into three regimes depending on cross-

flow velocity (U ) [62]:

1.  For low flow velocities the structure responds mainly to turbulent buffeting and

its amplitude is proportional to 2U  .

2. 

At medium range flow velocities resonance may occurs through vortex shedding,

turbulent buffeting and acoustical oscillation.

3. 

At higher flow velocities, fluid elastic instability is the prime source with the

amplitudes increase with the flow velocity.

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2.8  Hydrodynamic Forces

The hydrodynamic forces that play part in the flow-induced vibration are generally

characterized into following groups:

1.  Those which arise from the turbulent fluctuations of the flow.

2.  Those resulting from vortex shedding from structures, tubes and its wakes.

3.  Those forces which arise due to motion of tube from their equilibrium position

while interacting with the flow.

2.9  Flow-Induced Vibration (FIV) Analysis

Flow-induced vibration in shell & tube heat exchangers are usually studied using

following methods:

1.  F.E. modeling: 

Finite element model is used to simulate time dependent motion of heat exchanger

tube while in presence of tube/baffle hole clearances. This approach is normally used

in heat exchangers under extremely critical services. Using modal analysis resonant

frequency and shape of tubes can be found [68].

2.   Limiting amplitude of vibration:

This method simplifies the structural members by assuming tubes as classical beams.

The support plates are usually modeled as simple support. The model predicts the

amplitude of vibration and instability thresholds based on an reliable criterion.

2.10 Empirical Nature of Flow-Induced Vibration Analysis

Flow-induced vibration is very complex phenomenon in shell & tube heat exchanger. Itcannot be understood using empirical correlations only[25, 31]. The behavior of shell &

tube heat exchangers is affected by the following parameters [22]:

1.  Tube bundle dynamics is extensively a multi-body dynamic with the tubes

supported at multiple baffles with holes slightly larger in diameters than the tube.

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2.  The interaction between the tube and its support is usually characterized by

impacting as well as sliding motion.

3. 

The tubes and its surrounding fluid are coupled resulting in motion dependent

forces using added mass, coupled modes and damping.

4.  The flow is generally non-uniform and time dependant [63].

5.  Mechanical tolerances, fit-ups and manufacturing process may add complexities

in appropriate boundary conditions [64].

Therefore, most of the methods in the analysis of tube bank dynamics are semi-empiricalin nature. To render the problem amenable for most analytical studies and experimental

investigations, the flow conditions are idealized as:

1.  The flow is uniform and steady.

2.  The incidence of the flow is either axial or normal to the tubes.

3. 

The tube motion is linearized and it is assumed that the frequencies are well

defined.

4. 

The baffle supports provide a simply supported condition.

2.11 Hydrodynamic Mass, Fundamental Frequency and Damping

Investigation of flow-induced vibration in heat exchangers requires the estimation of

hydrodynamic mass, frequency of vibration and related damping of tube bundle.

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2.11.1  Hydrodynamic Mass

The vibrating tubes shift the shell-side fluid during flow-induced vibration. When the

fluids involved are liquids or very dense gases, the natural frequency of tubes is

influenced by the inertia of the fluid. Hence, while finding the natural frequency, the

effect of the displaced fluid is considered using added mass. The added mass is defined as

the displaced fluid mass times an added mass coefficient mC  . Since the mass of the

vibrating tube augmented, the natural frequency of the tube will be reduced as compared

to tube vibration in vacuum or gas.

 Added Mass Coefficient, mC   for Single-Phase Flow

The added mass coefficient is estimated either by the analytical method of Blevins [30] or

from the experimental database of Moretti et al. [66].

   Blevins Correlation

Blevins [30] determined added mass coefficient mC for a single flexible tube surrounded

 by rigid tubes (see Fig. 2.8 a) using the following analytical formula.

1/ 2  D DC  em   (2.5)Where  D p D p D De /)/5.01(/    

and e D  = tube array equivalent diameter.

This is a reasonable approximation for more complex flexible cylinders.

   Experimental Data of Moretti et al. [66]

Moretti et. al. [66] performed experiments using a flexible tube in neighbour of rigid

tubes having a triangular/square pitch (see Fig. 2.8 b) and a P/D ratio from 1.25 to 1.50.

His experimental results of mC   are shown in Fig. 2.9 and same are tabulated in Table 2.2

for different pitch to diameter ratios. TEMA standard included this figure (Fig. 2.9) to

find added mass coefficient [39].

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Figure 2.8. Tube patterns to find mC  (a) Blevins model [30] & (b) Moretti model [66].

Figure 2.9. Added mass coefficient. [66]

Table 2.2. Added Mass Coefficient.  mC   

 pitch

 pitch

dia

Triangular pitchexperiments

Square pitchexperiments

1.2 1.3 1.4 1.5 1.6 1.7 1.8

Pitch to diameter ratio

2.5

2.0

1.5

1.0

      C  m

(a)

(b)

 p

 pitch

dia

dia

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2.11.2 Natural Frequencies of Tube Bundles

The tubes in a heat exchanger bundle are slender beams and the most flexible members.

They vibrate at discrete frequencies when they are excited. The natural frequency

depends on their geometry, material properties like Young’s modulus, density andmaterial damping, tube-to-support interactions, etc. The lowest frequency at which the

tubes vibrate is known as the natural frequency. If the exciting frequency coincides with

the natural frequency of the tube resonance occurs. The conventional shell and tube heat

exchangers consist of either straight tubes or U-tubes with baffle supports at intermediate

 points and fixed at their ends.

2.11.3 Damping

The stability of the tube array depends on damping. It limits the tube response when the

tube is excited by any one of the excitation mechanisms. Greater the damping, the lower

the tube response will be. Damping will also ascertain the critical flow velocity ( cr U  ) for

fluid elastic instability (FEI). The critical velocity increases with increasing damping.

Damping in a vibrating system is due to several possible energy dissipation mechanisms.

This phenomenon has been explained in detail in next chapter.