Experimental study on cross-flow induced vibrations in heat exchanger tube bundle

Vibration in heat exchangers is one of the main problems that the industry has faced over last few decades. Vibration phenomenon in heat exchangers is of major concern for designers and process engineers since it can lead to the tube damage, tube leakage, baffle damage, tube collision damage, fatigue, creep etc. In the present study, vibration response is analyzed on single tube located in the centre of the tube bundle having parallel triangular arrangement (60°) with P/D ratio of 1.44. The experiment is performed for two different flow conditions. This kind of experiment has not been reported in the literature. Under the first condition, the tube vibration response is analyzed when there is no internal flow in the tube and under the second condition, the response is analyzed when the internal tube flow is maintained at a constant value of 0.1 m/s. The free stream shell side velocity ranges from 0.8 m/s to 1.3 m/s, the reduced gap velocity varies from 1.80 to 2.66 and the Reynolds number varies from 44500 to 66000. It is observed that the internal tube flow results in larger vibration amplitudes for the tube than that without internal tube flow. It is also established that over the current range of shell side flow velocity, the turbulence is the dominant excitation mechanism for producing vibration in the tube since the amplitude varies directly with the increase in the shell side velocity. Damping has no significant effect on the vibration behavior of the tube for the current velocity range.


Introduction
Flow-induced vibration (FIV) in heat exchanger tube bundle is one of the major concerns while designing heat exchangers. Fluid-elastic instability, turbulence, vortex induced instability and the acoustic resonance are the FIV mechanisms that can cause vibrations in the tube bundle in heat exchangers (Weaver and Fitzpatrick, 1988). But for single phase liquid flow, the acoustic resonance is not applicable since this excitation mechanism is prominent in two-phase and the gas flows. Owing to flow-induced vibrations in heat exchanger tube bundles, the tube bundles are subjected to a chance of failure. Thus, flow-induced vibration is of deep concern for designers and engineers. Each tube in a tube bundle is supported by tube sheets and baffles in a multi span tube bundle. Natural frequency of the tube is the central concern for designers, since it indicates the level of vibrations in tube bundle. Many researches have been carried out in the field of flow-induced vibrations driven by the importance of the field and its applications in the industry.
Fluid-elastic instability is considered to be the most im-portant excitation phenomena since it is the main cause of tube failure in heat exchangers. According to Price (1995), the turbulence in the flow induces small displacement in the tube. This displacement alters the flow pattern and fluid forces on the tube. This alteration induces more displacement in the tubes. If the displacement goes on increasing, then fluid-elastic instability occurs. Chen (1984) presented the overall review of different instability models and stability maps developed based on analytical models and published experimental data. Roberts (1966) discovered for the first time that a tube displacement mechanism can account for the dynamic instability of the tube row in cross-flow. He found experimentally that the motion of the alternating tubes produce changes in the wake pairing downstream of the tube row. This results in the net work done by the fluid on the moving tubes over every cycle. Therefore, he established that the lack of space between tubes in the tube bundle results in "jet switch phenomena" in tightly packed tubes bundles. Researchers have performed the numerical simulation of fluid elastic instability of multiple cylinders subjected to cross-flow (Lin and Yu, 2005). The tubes are arranged in a rotated triangular bundle. The results showed that the fluid elastic vibrations with high amplitude occur in multiple cylinders as the free stream velocity goes beyond the critical value. Also beyond the critical value, the cylinders start vibrating in the elliptical orbits. Connors (1978) analyzed different models to predict instability but found that the most reliable model is the elliptical motion model. He obtained energy balances in both inflow and cross-flow directions which satisfies Eq. (1).
where K is the Connors coefficient, f n is the natural frequency of tube, m is the mass ratio, δ is the logarithmic decrement, ρ is the density of shell side fluid, D is the outer diameter of the tube, and V pc is called the pitch velocity calculated by the relation presented in Eq.
where V is the free stream fluid velocity, and P is the pitch of the tube bundle. Later on, Blevins (1979) modified Connors model to account for the flow dependent fluid damping in his equation presented in Eq. ( where ζ x is the damping factor in x-direction and ζ y is the damping factor in y-direction. Blevins (1979) concluded that the damping factors also play a vital role in the prediction of fluid elastic instability in tube bundle. Forced vibration occurs when the fluid forces are the only function of time. These forced vibrations are called Turbulence Buffeting. Turbulent flow is characterized by the random changes in the fluid velocity and the flow pattern around the tubes (Blevins, 1977). Researchers have studied the turbulence in triangular tube array by hot wire measurements in an aerodynamic channel and flow visualization in water channel (de Paula et al., 2012). The tube bank has P/D=1.26 and Reynolds number ranges from 7.5×10 3 to 4.4×10 4 . Visualization shows that the flow arises from the gap between two tube forms coalescent jets. In some cases, changing flow direction occurs. This mechanism is termed as metastable. The turbulence features past three, four and five rows seem to be similar to the predicted results.
Vortex excited vibration is also the major excitation mechanism in the tube bundle subjected to cross-flow. When the flow passes over the tube, it produces a series of vortices past the tube. These series of vortices produce alternative forces results in relatively large tube vibrations. For a single tube subjected to cross-flow, the frequency of vortex shedding is given by the relation presented in Eq. (4) (Blevins, 1977).
where f vs is the vortex shedding frequency, U is the fluid velocity, St is the Strouhal number, and d o is the outer diameter of cylinder. The vortex shedding phenomena has been investigated by the number of researchers over the past few decades. Grotz and Arnold (1956) investigated the phenomena of vortex shedding. They presented a relationship between vortex shedding frequencies and the tube spacing ratios. For tube banks with vortex shedding, it is established that Strouhal number is a function of arrangements of tube in a tube bundle (Lienhard, 1966). Research has been carried out to predict the procedures and to recommend design guidelines for periodic wave shedding in addition to other flow induced vibration considerations for shell and tube heat exchangers (Pettigrew and Taylor, 2003). The conclusion shows that the fluctuating forces due to periodic wave shedding depend on the number of considerations like geometric configuration of tube bundles, its location, Reynolds number, turbulence, density of fluid, and pitch to diameter ratios. Table 1 presents the ranges of the reduced gap velocity for different excitation mechanisms in a tube bundle subjected to cross-flow (Iqbal and Khushnood, 2009).

Mathematical model
Heat exchanger tubes in a bundle are modeled as a hollow cylinder acting like a beam. Single length of the tube consists of unequal lengths. In the present study, the tube span consists of two boundaries i.e., at the tube sheet and at the baffle plate. The following assumptions have been taken while developing model for the tube vibration.
•Each tube is considered as a separate independent beam.
•Only the flow (drag) direction vibration is considered.
•Tube material is isotropic.
•Loose baffles are considered as a damped boundary and the tube sheet is considered as a fixed boundary.
•Only forced vibration caused by the cross flow is considered.
The governing differential equation of the forced vibrational motion of the tube is given in Eq. (5).
where , y is tube displacement in the flow (drag) direction and is the harmonic force magnitude of the fluid in cross flow. The harmonic steady state solution may be written as.
is the tube vibration response in the flow (drag) direction due to cross flow. Applying damped boundary conditions, we obtain the final solution of the form. y and C 1 , C 2 , C 3 , and C 4 are constants. Eq. (7) is the final solution of the vibration response of the tube in flow (drag) direction due to cross flow.

Experimental setup
A low speed water closed test loop is designed to couple with the shell and tube heat exchanger made of glass as presented in Fig. 1. The two water storage tanks each with capacity of 150 gallons are installed such that one tank is used for the closed loop shell side flow and the other is used for the close loop tube side flow. The variable water pump (750 W) is installed to generate a flow velocity ranging from 0.8 m/s to 1.3 m/s in the shell side and the constant speed water pump (300 W) is installed to generate constant velocity of 0.1 m/s in the tube side. At the entrance of water in tube bundle, the doppler flow meter is installed to monitor the inlet flow velocity and the flow rate of water entering in tube bundle. The reduced velocity ranges from 1.80 to 2.66 and Reynolds number varies from 44500 to 66000. Flow straighteners such as honey comb are generally used at the inlet of shell side flow in order to provide uniform flow to the tube bundle. Since, in current experiment, the flow straighteners have not been used, the flow entering the tube bundle is not uniform. The tube response may slightly differ from the uniform flow, but the dominant flow regime remains the same, which in the current case is turbulence excitation because of the reduced velocity range (1.80 to 2.66) ( Table 1). Table 2 lists the specifications of the heat exchanger tube bundle used in the current experiment. Fig. 1 shows the complete experimental test rig on which the experiment has been performed. Fig. 2b shows the cross-sectional view of the tube bundle with black circle indicating the instrumented tube in the bundle. The tube is instrumented with strain gages arranged on the tube in a full wheatstone bridge configuration with two gages on the upper side and two gages on the lower side as indicated in Fig. 2a. These gages are along the direction of flow, so it can only measure the vibrations response of the tube in the flow (drag) or y-direction (Fig. 2a).
The target tube is located in the center of the bundle as presented in Fig. 2. The tube has a tube sheet at one end and the baffle at the other end. So, the tube has a fixed-damped    boundary. Fig. 3 presents the diagram of the longitudinal layout of the target tube with its supports. The strain gages are mounted on the tube in full Wheatstone bridge configuration connected to SG-Link wireless sensing node originally developed by Microstrain Corporation, USA. This node converts the measured strain into displacement, and gives the output signal in the form of timedisplacement graph. It is worth mentioning here that the wireless sensors are used to acquire data from the test rig for the first time. The advantage of using wireless sensors instead of wired sensors is that the data is more reliable with less noise and distortion and also accounts for the signal distortion due to the resistance and the geometry of the wires. The complete data acquisition loop is presented in Fig. 4.
As presented in Fig. 4, The SG-Link wireless sensing node is connected to WSDA wireless base station through wireless signals. The base station is directly connected to the PC through the Node Commander software. The software displays the live streaming of the signal. The data sampling rate for data acquisition is 256 samples/sec. For the spectrum analysis of the signals, SIGVIEW software is used originally developed by Signal Lab Company.

Vibration amplitude response
Vibration amplitude response is one of the key concerns in the analysis of flow induced vibrations in tube bundle especially in cross flows (Connors, 1978). In the present study, vibration measurement is carried out on the target tube under two different flow conditions. Under the first condition, there is no flow inside the tube, and under the second condition, there is constant flow inside the tube having a velocity of 0.1 m/s. But in both conditions, the shell side velocity ranges from 0.8 m/s to 1.3 m/s. The amplitude response comparison for both conditions is presented in Fig. 5.
As shown in Fig. 5, the vibration response pattern for both flow conditions varies directly with the increase in the shell side velocity. The difference between the two flow conditions is very distinctive since the tube with internal tube flow vibrates with relatively high amplitudes as compared with the other tube without internal flow. The reason may lie in the fact that the internal tube flow increases the vibrating mass of the tube, which results in the high amplitude of vibration when excited by the shell side flow (Weaver and Yeung, 1984). The current study is very critical and more reliable since the tubes vibrate with a constant   internal tube flow as in most heat exchangers during operation. Table 3 presents the comparison of the current experiments with the published results of Lin and Yu (2005) .   Fig. 6 presents the graphical plot of the vibration response of the current experiment compared with that of Lin and Yu's experiment (Lin and Yu, 2005).
It is worth mentioning that Lin and Yu's experimental data is in a good agreement with the current experimental data for no internal flow condition because Lin and Yu's experimental data only accounts for tube vibration without internal flow (Fig. 6). Also, the vibration amplitude for internal tube flow conditions is larger than that without internal tube flow condition (Fig. 6).

Damping
Damping is a mechanism by which energy is being dissipated from a vibrating body. System damping has a strong influence on the amplitude of the vibration. The damping extracts energy from the vibrating body, thereby decreasing the amplitude of vibration. Most heat exchangers have very low value of damping (Blevins, 1977). In flow-induced vibrations, if the energy input to the vibrating tubes cannot be dissipated in damping, the amplitude of the vibration increases dramatically, leading to the structure failure. In the current experiment, the damping factor is estimated using the bode plot (Fig. 7) (Khushnood, 2005).
And the mathematical relationship used to find logarithmic decrement is presented in Eqs. (8) and (9).
where ζ is the damping ratio, δ is the logarithmic decrement, and ω 1 , ω 2 and ω n are the frequencies determined from the bode plot. The logarithmic value indicates how fast the amplitude is decaying in the vibrating tubes. The faster the decay of amplitude of vibration, the higher will be the damping in the structure (Blevins, 1977). In the present experiment, the logarithmic values range from 0.0005 to 0.007. Fig. 8 presents the plot of logarithmic decrement (damping) of the vibrating tube for both flow conditions as a function of free stream shell side fluid velocity. The plot shows that the value of logarithmic decrement (damping) under two different flow conditions varies slightly around a constant trend with the increase in the shell side fluid velocity. Generally, the tube bundle in heat exchanger is under the damped system. So, the value of logarithmic decrement (damping) generally lies in the range of 0.0001 to 0.1 (Pettigrew and Taylor, 2004). It is observed that the damping has no significant effect on the vibration amplitude of the tube for both flow conditions in the current velocity range and is independent of the shell side fluid velocity but highly depends on the fluid forces and distribution of fluid around the tube.

Spectrum analysis
When the inlet velocities are below the critical value, the spectrum of the signal exhibits specific peaks indicating the excitation mechanism of the monitored tube. Fig. 9 presents the actual signal and the Fast Fourier Transform (FFT) of the signal.   As presented in Fig. 9, there is a significant peak observed in the Fourier transform of the actual signal. This frequency peak is observed around 50 Hz (Fig. 9b). Fig. 10 presents the behavior of this observed frequency for two different flow conditions.
As presented in Fig. 10, the observed excitation frequency over the current velocity range seems to remain the same. Actually, the shell side fluid velocity has direct relation with the excitation frequencies of the tube (Lin and Yu, 2005). The reason for the consistent frequency behavior for both flow conditions is that the velocity range for the current experiment (0.8 m/s to 1.3 m/s) is low, so the turbulence is the dominant excitation mechanism in the tube which is also evident from the reduced velocity range (1.80 to 2.66) for the current experiment (Table 1).
In the analysis on cross flow induced vibrations in heat exchanger tube bundle, one of the important frequencies that should be always in consideration is the natural frequency of the tube. In the current experiment, the fundamental natural frequency is observed both in Fourier transform of the signal (Fig. 9b) and calculated by using the relation presented in Eq. (10) (Jones, 1970). (10) where f n is the natural frequency of tube, λ n is the frequency factor (which in this case is 1.875 for the fixed-damped boundary), l s is the target span length, E is the elastic modules of the tube, I is the moment of inertia, and m is the mass per unit length of the tube (Table 3). By putting the values in Eq. (10), we obtain the fundamental natural frequency of 101 Hz. There is a peak observed in Fig. 9b near 100 Hz. It is concluded that the frequency observed in Fig. 9b is the natural frequency of the tube since it is found in a good agreement with the value calculated from Eq. (10).

Conclusions
The following conclusions can be drawn from the current study.
(1) The tube with internal tube flow vibrates with larger amplitude than the tube without internal tube flow. This is because the internal tube flow increases the mass of vibrating tube resulting in large vibration amplitudes.
(2) The vibration amplitude for both flow conditions varies directly with the increasing shell side velocity indicating that the turbulence excitation is dominant for producing vibration in the tube.
(3) Logarithmic decrement (damping) shows a consistent trend with the increase in the shell side velocity for both flow conditions indicating that the damping has no obvious effect on the vibration of the tube.