Vibration of tubes in heat exchangers is an important limiting factor in heat exchanger operation. The vibration is caused by nonstationary fluid dynamic processes occurring in the flow. These are turbulent pressure pulsations (see Turbulent flow), vortex initiation and separation from tubes in crossflow (see Crossflow), hydroelastic interaction of heat transmitting element (tubes) assemblies with the flow, and acoustic phenomena. The greatest effect of nonstationary hydrodynamic forces is observed in tubes in flows with separation. Flow separation from tube surfaces occurs where there is a transverse velocity component of the flow and mainly affects the vibration strength of the tubes in heat exchangers. The dynamic effect of the flow on a vibrating tube depends on the flow velocity and the vibration characteristics of the tube. With a separated transverse flow over a tube bank, the reference velocity is assumed to be the flow velocity in the narrowest section of the bank in the tube plane (Figure 1) and is calculated by the formula

where u_{0} is the velocity in the absence of the tubes, b the tube pitch, D the tube diameter and where β is the angle of slope of tubes to the flow direction. For transverse flow β = 90° and for longitudinal flow β = 0°. Generalization of the data for calculation of the hydrodynamic forces and tube vibrations excited by them makes it possible to use the reference velocity, calculated by Eq. (1), with a relatively low error for β in the range from 90° to 15°. Decreasing β increases the error. It is assumed that all kinds of tube vibration come into play simultaneously with the onset of fluid flow. However, each type of vibration dominates over a certain range of flow velocity, this range depending on the vibrational parameters of tubes, the fluid properties, and the conditions of the flow. This is obvious from the amplitude and velocity characteristics (Figure 1) of tube vibration in the first and fifth rows of the staggered tube bank with transverse to longitudinal pitch ratios 1.61 × 1.38 with a natural tube frequency f_{n} = 99 Hz. High amplitude ratios A/D are observed in excitation by vortex separation (region 2 in Figure 1 which shows relative root-mean-square values of Ā_{y}, the amplitude of tube vibrations in the transverse direction relative to the free stream) and in hydroelastic instability (region 3). In region 1 low amplitude vibrations are brought about by turbulent pressure pulsations. In the case of longitudinal flow, the disturbance of tube assembly stability is determined only by excitation by turbulent pressure pulsations.

In calculating *tube vibration*, it is important to find the *natural frequency* of vibration of the tubes. For a tube with pivoted ends vibration may occur according to mode shapes 1,2, and 3 as shown in Figure 2. The natural frequency of vibrations depends on both the mode shape and the physical characteristics of the tube, and the way its ends are fixed; it can be calculated by the formula

where E is the modulus of elasticity of the tube material, I the area moment of inertia (= π(D^{4} - D_{i}^{4})/64 where D_{i} is the tube internal diameter), m is the overall tube mass per unit of its length (including the mass of the tube itself, the mass of the tube side fluid and the mass of the shell side fluid displaced in the vibration), 1 is the tube length, B_{n} is a constant depending on vibration shape and the manner of tube fixation in the heat exchanger. The constant B_{n} used for determining the frequency harmonic of natural vibrations in a quiescent fluid in the absence of axial forces is derived from tabulated data. For shell-and-tube heat exchanger with more than 4 baffles, and where the end spaces between the tube sheets and the nearest baffles does not exceed the baffle spacing by more than 20%, a value of B_{n} = 10 may be taken [Chenoweth (1983)]. Alternatively, the expression B_{n} =
, where λ_{n} is calculated from the expression given in Table 1, may be used.

*Vortex excitation of tubes* depends on periodic hydrodynamic forces originating in vortex formation and separation from tubes. The tube (Figure 3) is subjected to the periodic hydrodynamic forces which are cabable of rocking an elastically mounted tube. Vortices are separated in turn, from one side, then the other side of the tube. Therefore, the transverse hydrodynamic force continuously changes direction and is a source of energy for excitation of tube vibrations. The hydrodynamic force brought about by separation of vortices varies sinusoidally. It is obvious from Figure 3 that, in the longitudinal direction, its frequency is twice as high as in the transverse one. This means that the component of the nonstationary hydrodynamic force acting across the flow varies both in magnitude and direction. At the same time, the component of this force along the flow changes only in magnitude, while its direction remains the same. Nonstationary hydrodynamic forces, arising due to vortex separation, can excite high-amplitude vibrations of tubes if the natural frequencies of their vibrations coincide with the frequency of vortex separation or are twice as high.

The root-mean-square value of amplitude of tube vibrations, in the transverse direction relative to the flow, which are excited by separation of vortices, is calculated from the equation

The frequency of vortex separation from tube surface fs is determined by the characteristic value of the Strouhal number Sr = D f_{s}/
, i.e.,

In a practically important range of working velocity variation in the flow in heat exchangers of power plants, which is characterized by the variation of Re numbers from 10^{3} to 2 × 10^{5}, the Strouhal number, determining the frequency of vortex separation from the tubes, is calculated as follows:

Sr 0.2 for a single tube,

for a single cross row,

for a staggered tube bank at b_{1}/D ≥ 1.15, where

and

for an in-line tube bank at b_{1}/D ≥ 1.15.

Hydroelastic (or "fluid-elastic") vibrations of tubes in banks prevail at high flow velocities. They arise as a result of hydrodynamic forces which originate as a result of the vibration itself. The larger the amplitude of the vibration, the larger the force and, hence, a rapid increase in vibration amplitude with velocity occurs in the region. The self-amplified characteristic is often given the name "fluid-elastic instability." In transverse flow over tubes the flow velocity which equalizes hydrodynamic exciting and damping forces and gives rise to hydroelastic vibrations is known as the critical velocity. A slight increase of velocity above this value sharply increases vibration amplitudes and leads to tube failure. For banks with equilateral triangular and square arrangement of tubes this velocity is calculated as

where S is the longitudinal tube pitch and δ a damping factor (logarthemic decrement). Tube damping limits the vibration amplitude. It consists of hydrodynamic damping due to tube material and damping for structural reasons. Hydrodynamic damping is attributed to viscous forces appearing during interaction of the tube with the flow. When the tubes vibrate the energy also dissipates in the surrounding fluid since its particles move. Damping for structural reasons is attributed to friction forces appearing in tube constant and rotation in the holes of the supports.

In real *heat exchangers* an overall *damping* of tubes in assemblies is determined by the logarithmic decrement δ = 0.15−0.3. Commonly hydrodynamic damping constitutes about 50% of the overall damping and depends on the bank configuration and the pitch ratios. The lower the pitch ratio, the stronger the damping. Hydrodynamic damping of a single tube in a stagnant water on the average is determined by the logarithmic decrement δ
0.05−0.1. In the bank with pitch ratios b_{1}/D × b_{2}/D = 1.15 × 0.98 hydrodynamic damping is nearly three times that in the case of a single tube and in the 2.0 × 1.77 bank it is approximately identical to that for a single tube.

In heat exchangers with gas flow, high amplitudes of tube vibration or noise may arise if the natural frequency of transverse vibrations of the gas column coincides with the frequency of vortex separation and with natural frequencies of tube vibrations. This is referred to as *acoustic vibration*. Natural frequencies of transverse vibrations of gas column are calculated by the formulas

f_{n} = nu_{sound}/2h for rectangular channels, where h is the channel width, n = 1, 2, .. ., u_{sound} the sound velocity in a coolant,

f_{n} = α_{n}u_{sound}/d_{1} for circular channels, where dl is the inner diameter of the channel,

α_{n} = 0.59, 0.97, 1.34, 1.69, 2.04, ... .

To avoid acoustic vibrations, the natural vibration frequency of the transverse gas column must be separated from the frequency of vortex separation by no less than 20%. A similar tuning of the vortex separation frequency away from the tube's natural frequency is needed to avoid high vibration amplitudes occurring due to vortex excitation.

The highest vibration amplitudes are caused by fundamental natural harmonic vibrations. The tubes are damaged mechanically due to collision of vibrating tubes, attrition against hole edges in the interspacing tube plates, tightness faults in stiff joints, and fatigue failure as a result of cyclic load.

In general, vibration diminishing of streamlined heat-transmitting elements is achieved by (e.g., decreasing the flow velocity) clearing of faults in damaged tubes, rearrangement of tubes to make passages for a coolant, replacement of damaged tubes by bars, mounting of regulating baffles at the coolant inlet in the heat exchanger, and appliances reducing damping and tube stiffness. Bearing in mind that the strongest vibrations are observed in the first rows of tube assemblies, tube clips in midspan, stiffness-girdling rings, and other appliances reducing vibration amplitude can be mounted.

When choosing the method of prevention of tube vibration one should take into account advantages and disadvantages of each. For instance, reducing the coolant velocity, and making passages for the coolant in tube assemblies both result in a decrease in heat transfer coefficient. Replacing damaged tubes by new ones in practice is not advisable because vibration level does not change and the tubes rapidly fail again. In this case it is better to substitute bars or higher stiffness tubes, (e.g., steel tubes for brass ones) for the damaged tubes.

#### REFERENCES

Chenoweth, J. M. (1983) Flow-induced vibration, *Heat Exchanger Design Handbook*, 4, Hemisphere, New York.

Zukauskas, A., Ulinskas, R., and Katinas, V. (1988) *Fluid Dynamics and Vibration of Tube Banks in Fluid Flows*, Hemisphere, New York.

Blevins, R. (1977) *Flow-Induced Vibration*, Van Nostrand Reinhold Comp., New York.