Abstract

The amplitudes induced by random excitation forces on the tubes bring continuous friction between the tube and supports, which results in gradual failure of the tubes due to fretting wear. Therefore, it is very important to determine the envelope lines of the random excitation force spectrum for the coil tube. To the authors' knowledge, there are no published studies on the normalized force spectrum of coil tubes. In this paper, a simplified three-layer experimental model was established. The robustness of the numerical method was demonstrated by comparing the experimental and simulated results, including the vibration response and the fluid excitation force spectrum. Then, a semi-empirical equation for predicting the dominant frequency of turbulent buffeting was constructed by employing the threshold envelope method. Through the observation of time-history and root-mean-square (RMS) data, it was found that the pitch diameter ratio between adjacent tube layers, a, had the greatest influence on the force coefficients. The smaller a is, the larger the force coefficients are. The pitch diameter ratio in the same layer, b, and helix angle, α, had little effect on the force coefficients. With the increase of α, the flow instability in the shell-side flow enhanced and the fluctuation of force coefficients became larger. Finally, the mechanisms of the tube position, Reynolds number (Re), and bundle structure on the normalized force spectrum were studied. The normalized envelope force spectrum for coil tubes was proposed as the guidelines to predict and evaluate the random excitation force acting on the tubes.

1 Introduction

The coil tube bundle is superior to the straight tube and U-tube bundles because of its compactness and greater heat transfer efficiency [13]. Therefore, in recent years, coil-wound heat exchangers (CWHE) have been used widely in pharmaceutical [4], chemical industries [5], and especially in the nuclear power industry. It is utilized extensively in the fourth generation reactors, such as high-temperature gas-cooled reactors [6], lead-cooled fast reactors [7], and small modular reactors [8].

The coil tube bundle is subject to the same cross flow impacted in the shell side as other structural bundles, resulting in flow-induced vibration. There are three main mechanisms that can lead to tube failure for single-phase flow: vorticity shedding excitation, turbulent buffeting, and fluid-elastic instability [9]. Among the mechanisms cited above, both vorticity shedding excitation and fluid-elastic instability can lead to large-degree vibration and destructive tube failure [10]. However, turbulent buffeting does cause continuous small-degree vibration. Thereby appear fretting wear between the tube and support. It can produce progressive damage in terms of tubes [11,12]. Due to the high maintenance and downtime costs of nuclear power plants, it is important to be able to assess the effects of random excitation to ensure the maximum safe operating life of the steam generator. At the same time, in order to design a safer and longer life steam generator, it is also necessary to accurately evaluate the random excitation force exerted by the fluid on the tube during the design process. To this end, it is meaningful to carry out the research and making contributions to the design guidelines of the coil tube bundle.

Owen obtained the semi-empirical formula of the main frequency of turbulent buffeting by using the experiments of gas-phase flow subject to the bundle and determined the relevant parameters by using the experimental data [13]. Then, previous experimental results were used to verify the formula, which proved that it had a certain reliability in engineering design [14]. As a result, Owen's method can be used for reference to explore the prediction method of the main frequency of turbulent buffeting with respect to the coil tube, as presented in the later part of this paper.

Since heat exchanger with straight tube bundle has been developed for a long time, the random excitation forces acting on straight tubes have been studied quite a lot. In order to study the fluid excitation force of vortex shedding on a bluff cylinder, Blevins and Burton established a semi-empirical dynamic model using random vibration theory, which included both the spanwise correlation effects and the amplitude dependence of the related vortex forces [15]. The model parameters were determined according to the experimental data. Finally, the model was applied to assess the force acting on the elastic cylinder, and the results were in good agreement with the experimental data [16]. This method provides a new research idea, and the random excitation of the coil tube can also be evaluated by using such a semi-empirical model. Chen and Jendrzejczyk introduced the experimental results of a square tube array with a pitch diameter ratio of 1.75 under turbulence. And the assumption that the normal component of velocity was taken as the effective velocity was also proposed for the bundle which was inconsistent with the normal direction of velocity [17]. Taniguchi and Miyakoshi conducted an experimental study on the flow around the cylinder close to a horizontal plane [18], which does not exist in the coil tube bundle. Only the flow in the bundle is considered for coil tube, and there is no direct impact of the constant flow acting on the tube. It will be discussed in this article. Taylor et al. directly measured the fluctuating forces of bundles with pitch diameter ratios of 1.5 and 3 and finally gave power spectral density (PSD) and excitation force coefficients of these structural parameters [19]. Axisa et al. systematically studied the random excitation mechanisms of tube bundles subject to single-phase cross flow and included them in the reduced force spectrum versus reduced frequency [20]. This plays an active role in predicting the excitation forces on the tube accurately in engineering design. It also provides an idea for the establishment of random excitation mechanisms in terms of the coil tube. Paїdoussis and Price revealed the physical mechanisms of flow-induced instability caused by cross flow. Under the framework of quasi-steady fluid dynamics theory, fluctuation excitation was discussed [21]. Norberg shed light on root-mean-square (RMS) lift and drag factors over time in the range of Re= 1 × 103–104 [22,23]. The experimental results of excitation force of normal triangular bundle were given [24]. It found that there are significant forces in both directions of drag and lift. In addition to random excitations, constant frequency and quasi-periodic fluid forces were also found. These forces depend on the position of the tube within the bundle. This finding was confirmed by the experimental results of Liu et al. within our project team [25]. Sun et al. applied the improved Van der Pol model to study the vortex-induced vibration, which has a positive effect on the study of the forced vibration with respect to the tube under the elastic boundary conditions [26].

At present, there are few researches on random excitation forces of coil tubes under cross-flow impact. In Blevins's test, the tube response corresponding to the reduced velocity of 1–50 was studied. And the dimensionless force spectrum was derived, which was used to predict the tube response with respect to a single mode [27]. And then, Xu et al. proposed and implemented an RMS response estimation method for multisupported coil tubes based on the ansysapdl command object [28]. Choosing this method, Yuan et al. [29] co-developed multiphysics simulation toolkit sharp to explore random vibrations for coil tubes. It was worth noting that the simulation resulted in a higher peak frequency than the experiment. It is conservative for engineering design and evaluation.

Our research team has done some work on the flow-induced vibration mechanisms for coil tube bundles. The natural vibration characteristics and vortex shedding for coil tube bundles were explored. The details are presented in Refs. [3032]. In this paper, a simplified three-layer experimental model was established to verify the reliability of the numerical method. In terms of the simulation data, the semi-empirical equation of the dominant buffeting frequency was obtained. The mechanism of structural parameters on the force coefficients was discussed. Finally, the normalized force spectrum for the coil tube was proposed, which provided the guidelines for the prediction and evaluation of random excitation.

2 Experimental Setup

Coil-wound heat exchangers could be divided into curved tube region and coil tube region. The fluid entered by inlet must flow through the curved tube region to enter the coil tube region, as shown in Fig. 1. Therefore, there was no uniform flow impact on the coil tube region directly, only the tube vibration inside the region should be considered. The bundle was simplified reasonably. Three layers were determined, and the adjacent layers were wound in opposite directions. The radial directions were left-handed, right-handed, and left-handed from inside to outside. Three supports were uniformly arranged in the circumferential direction to constrain the coil tube structure. The outer diameter of the coil tube, d, was 0.008 m. The pitch diameter ratio between adjacent tube layers, a, and the pitch diameter ratio in the same layer, b, were quantified as
a=Td;b=Ld
(1)
Fig. 1
Simplification of the experimental section
Fig. 1
Simplification of the experimental section
Close modal

In the test section, a was 1.5, b was 2.6, and the helix angle, α, was set to 15 deg. The tube bundle height, Hc, and the straight tube section height, Hs, constituted the overall height of the section, which was 0.760 m and 0.100 m, respectively. In addition, Din denoted the internal diameter of the shell, which was 0.426 m.

The experimental system is indicated by Fig. 2. Pump was used to realize the water circulation. A throttle valve was used to regulate the flow, and a flowmeter was applied to monitor the flow. In this test, in order to ensure that the sensor was completely attached to the surface of the tube, resistive strain gauges with the size of 2.2 mm × 1.5 mm were selected, and the half-bridge connection was adopted as presented in Fig. 2. The sensitivity coefficient was 1.91±1%, and resistance was 120.1±0.1 Ω. In addition, support bars of the two sides that constrained the same tube were connected by welding bridges. Figure 3 shows the testing site.

Fig. 2
Schematic of experimental system
Fig. 2
Schematic of experimental system
Close modal
Fig. 3

3 Test Method

Tested coil tubes were in the middle layer of the span next to the inlet. A total of ten coil tubes were monitored, the 14th–23rd tubes, which were sequenced based on flow direction. Note that the model used in the simulation analysis below was the extraction of 25 tubes per layer of the tested span (orange tubes). If the tested tubes were ranked according to the simulation model, there were 9th–18th tubes. In order to facilitate the comparison between the test and simulated data, the tubes were numbered according to the simulation, which was T9–T18, respectively, as shown in Fig. 4. The sensors were placed in the middle of the coil tube in the single-span and monitored the time-history amplitude of the tubes in the drag and lift directions. The sampling frequency was set at 1 × 104 Hz, and the collection was for a total of 30 s. The test was carried out three times based on the same Reynolds number (Re).

Fig. 4
The position of the tested tube in the test
Fig. 4
The position of the tested tube in the test
Close modal
Fluctuating forces generated at the tube wall are caused by the turbulent flow. And the forces along the tubes can be described as random functions in terms of time and space. Starting from the classical linear random vibration theory, convenient formulas can be developed to describe the excitation field and calculate the tube response. Provided that F(s, t) is the fluctuating force per unit tube length at location s along the tube. And such function is assumed stationary Gaussian. As a result, it can be represented by its cross-correlation spectrum, which can be defined as the Fourier transform
ψF(s1,s2,f)=+RFe2iπftdτ
(2)
where RF denotes function of the cross-correlation spectrum between location s1 and s2. When uniform mass tube is subjected to an uneven cross flow, fluctuation is conveyed along the tube. ψF remains a real-valued function, if the flow is perpendicular to the tube axis. Then, such function is
ψF(s1,s2,f)=[ϕF(s1,f)ϕF(s2,f)]12γ(s1,s2)
(3)
where ϕ denotes the autocorrelation spectrum of the force per unit tube length. The coherence function, γ, describes the degree of correction of forces along the tube, approximated by
γ=exp{|s1s2|λc}
(4)

where λC denotes the correlation length. When λC is no more than a few tube diameters and even smaller, its exact value is not acquired. The λC/l is needed, where l is tube length, for practical applications [16,20].

A dimensionless formulation is desirable in order to summarize the data obtained from various experiments. The gap flow velocity, Ugap, can be approximately scaled by the uniform cross-flow velocity, UH. The detail is found in the previous work of Wang et al. [32]
Ugap=UH1.37εA0.37εV
(5)
where εA and εV are the fluid area and fluid volume porosity fractions for the shell-side. Excitation force per unit tube length is characterized by the permanent force density
F=12ρUgap2dlCf
(6)

where d represents the tube outer diameter.

In addition, the aim is to normalize the experimental data and apply the reduced frequency to represent the frequency
fr=fdUgap
(7)
In that way, the local dimensionless spectrum is
ϕ˜F[s,fr(s)]=[12ρUgap2d]2UgapdϕF(s,f)
(8)
Taking Eq. (9) back into Eq. (3) to obtain
ψF(s1,s2,f)=(12ρUgap2d)2dUgapexp{|s1s2|λc}[ϕ˜(fr)F]
(9)
The tube response to excitation force induced by cross-flow turbulence is indicated as a Gaussian random function [17]. And the autocorrelation spectrum represents
ϕR(s,f)=[12ρUgap2d]2dUgap[ϕ˜F(fr)]116π4n=1δn2(s)Lcn2mn2[(fn2f2)2+4ζn2fn2f2]
(10)
where δn(s) denotes the modal displacement at s, mn is the total mass including the added mass, fn is the natural frequency, and ζn is the damping ratio of the nth order. Here, Lcn denotes the modal joint acceptance. Referring to the generalized correlation length in the case of uniform cross flow, it is shown that
(Lcn/l)2=anλc/l
(11)

Provided that λc/l1, such as less than 0.1. It is worth noting that an values depend on the mode.

The mean square tube displacement is
A2(s)=0ϕR(s,f)df=n=1An2(s)
(12)

where σn is the contribution of the nth order RMS displacement.

Therefore, one obtains
An(s)=12ρUgap2dδn(s)Lcn8π3/2mnfn2{frnζn[ϕ˜F(frn)]}1/2
(13)
Axisa et al. present a PSD, ϕ˜F(fr)e, defined as
ϕ˜F(fr)e=λcLeϕ˜F(fr)
(14)
where An summarizes all parameters
A¯n2=(12ρUgap2d)2δn2(s)an64π3mn2fn3ζndUgap
(15)
Collecting random excitation data in the experiments, generally, includes the two methods. One is random excitation determined from reaction force. These experiments apply to force transducers to measure the reaction forces. The other is random excitation determined from the tube response. Strain gauges are utilized to test tube displacement. And then use tube displacement to calculate excitation force PSDs. The latter is chosen and obtains
ϕ˜F(fr)e=64A¯n2π3fn3m2ζnδn2an1(12ρUgap2d)2Ugapd
(16)

where mode shape displacement constant, (δ1,max)2, and the modal factor, a1, for first order are shown in Table 1 [20,33].

Table 1

(δ1,max)2 and a1 for first order

ParameterRigid tubeClamped–clampedClamped–pinnedClamped–freePinned–pinned
(δ1,max)220.80.90.51.1
a1Translation2.5222.2784.02.0
ParameterRigid tubeClamped–clampedClamped–pinnedClamped–freePinned–pinned
(δ1,max)220.80.90.51.1
a1Translation2.5222.2784.02.0

4 Numerical Details

4.1 Computational Domains and Boundary Conditions.

Several computational simulation cases were carried out to obtain random excitation forces of fluid and tube vibration response of analyzing fluid–structure interaction mechanisms.

The adjacent layers are wound in reverse, and the influence of α on the flow cannot be ignored. In a word, the effect of coil tube bundle structure on the interaction mechanism cannot be determined. Therefore, the three-dimensional transient simulation in terms of the flow impinging on the bundle was carried out. The coil tube bundle is constrained by circumferential uniformly distributed support bars, and every four bars are placed in the same position of the bundle. Two bars clamp the tube with grooves and smooth edges to act as constraints. In addition, the adjacent bars are bound by welding bridges at multiple points along the axial direction as shown in zoomed figure of Fig. 2. It makes the flow between adjacent spans weak or even negligible. Therefore, the numerical model only chose the span where the tested tubes were located, that is, the span close to the inlet. In order to reduce the calculation quantities, 25 tubes in the middle of each layer of the bundle were extracted to form the coil tube bundle, and then the impinging bundle model was established by adding fluid domain, as described in Fig. 5.

Fig. 5
Extracted computational domain and boundary conditions
Fig. 5
Extracted computational domain and boundary conditions
Close modal

Six bundles with different structures were set up in order to comprehensively consider the effect of various structural parameters on buffeting excitation, as listed in Table 2. Figure 5 also shows the boundary conditions applied to the domain. The inlet boundary was set at the uniform velocity inlet. The outlet was considered the outflow condition, where static pressure with poutlet = 0 Pa was set. The inlet condition is set to high turbulence intensity (10%) to meet wall boundary layer requirements [34]. No-slip condition was used in the walls, which consisted of the coil tube wall, inner wall, the walls of bars, etc., in addition, to reduce computations, the domain was divided into three parts: the upper, middle, and lower domains, where the coil tube bundle was in the middle domain, and the connected surfaces of each two domains were defined as interfaces to ensure that the mesh shared nodes. The medium in the shell-side is water with density, ρ = 999.87 kg/m3 and the dynamic viscosity, μ = 1.01 × 10−3 Pa·s. The middle 1/3 part of the detection tube was taken as the monitoring surface to ensure that the random excitation of the fluid acting on the tube was not affected by the model simplification. Monitoring tubes were named T9–T18 along the flow direction, as shown in Fig. 6.

Fig. 6
The position of the tested tube in the numerical model
Fig. 6
The position of the tested tube in the numerical model
Close modal
Table 2

Detailed structural parameters of the coil tube bundles

Case123456
a1.11.21.51.51.52.0
b1.11.51.51.52.61.5
α (deg)15157151515
Case123456
a1.11.21.51.51.52.0
b1.11.51.51.52.61.5
α (deg)15157151515

4.2 Computational Method and Meshing Strategy.

Note that the coil tube was regarded as a rigid body, which did not consider deformation, but only displacement. The motion of the tube midpoint was monitored as the tube displacement in a simplified manner [35]. Fan et al. in our research team studied the flow-induced vibration with respect to a flexible cylinder and discussed the instabilities of aerodynamics forces affected by vortices by using this method [34,36,37]. In view of coil tube constraints, both ends of a single-span tube could be regarded as simply support [38,39]. And then, the elongation direction of the tube was not prone to vibration compared with the other two directions. Therefore, only the radial and axial directions were considered as lift and drag directions, respectively [31]. As indicated in Fig. 7, the motion of the tube is presented by the two-dimensional second-order oscillator equation [36]
x¨+4πξUrx˙+(2πUr)2x=0
(17)

where x, x˙, and x¨ denote the displacement, velocity, and acceleration of the center of the coil tubes, respectively. Here, the damping and stiffness are not equal in both directions. Before the water tunnel test, the modal test was carried out by direct percussion. Because T9 was exactly the same as the restraint and structural feature of the others, one only measured vibration characteristics of T9 for three times. Figures 8 and 9 show the results of one of the times. In the end, the average values were used as the input value. The natural frequencies in the lift and drag directions are 456.21 Hz and 172.81 Hz, respectively, and the damping ratios are 0.0462 and 0.0201.

Fig. 7
The simplified motion model of the coil tube
Fig. 7
The simplified motion model of the coil tube
Close modal
Fig. 8
(a) Time-history curve and (b) Fourier transformed data in the lift direction
Fig. 8
(a) Time-history curve and (b) Fourier transformed data in the lift direction
Close modal
Fig. 9
(a) Time-history curve and (b) Fourier transformed data in the drag direction
Fig. 9
(a) Time-history curve and (b) Fourier transformed data in the drag direction
Close modal
In the fluid domain, scale-adaptive numerical simulations applying the double-equation shear stress transport model can be dynamically tuned to produce large eddy simulation-like behavior in the unstable region of the domain and provide standard RANS functionality in the stable domain [4042]. The momentum transport equation is written as
ρkt+(ρUk)=P˜kβρkω+[(μ+σkμt)k]
(18)
ρωt+(ρUω)=ανtP˜kβρω2+[(μ+σkμt)ω]+(1F1)2ρσω2ωkω+QSAS
(19)

where k is defined as the turbulent kinetic energy, QSAS is set to the additional source term, and F1 is set to the blending function. More details are indicated in Refs. [42] and [43].

The ansyscfx solver was used to analyze the fluid-structure interaction problem in terms of finite volume technique. In order to reduce computation, the whole computational domain was divided into three parts. The whole fluid domain was performed in three-dimensional and nonstructured grids were applied to the region close to the bundle, and structured grids were applied to the rest, as shown in Fig. 10. Note that boundary layers were added close to the tube wall to accurately capture the tube motion. A total of 12 boundary layers are set up. Table 3 lists the grid independent results of the domain for the bundle for a = 1.5, b = 1.5, and α = 15 deg at Re=2426. It illustrates that y+ ≈ 1 and the accuracy of the results could be guaranteed when the thickness of the first boundary layer is set as 1/155d. In addition, the residual is set to 1 × 10−5. The time-step is set to 1 × 10−4. Furthermore, to ensure that the calculation results are not affected by the initial conditions of the cases, a stable state calculation with a certain amount of computation was carried out first, and then the transient calculation was implemented. This ensures that time-averaging of the solution for calculating the mean flow quantities can be started.

Fig. 10
The computational domain and mesh
Fig. 10
The computational domain and mesh
Close modal
Table 3

The grid independence of the domain in regard to a =1.5, b =1.5, and α = 15 deg

Mesh sizeTotal nodes/×107y+RMS A/d (%)Error (%)
1/135d2.761.130.015710.191
1/145d2.921.050.015690.064
1/155d3.090.980.01568
1/165d3.210.920.01568
Mesh sizeTotal nodes/×107y+RMS A/d (%)Error (%)
1/135d2.761.130.015710.191
1/145d2.921.050.015690.064
1/155d3.090.980.01568
1/165d3.210.920.01568
Table 4

The gap velocity, Ugap, for UH = 3 m·s−1

Case123456
Ugap7.076.136.705.035.014.39
Case123456
Ugap7.076.136.705.035.014.39

4.3 Validation Cases.

Water tunnel experiments with respect to coil tube bundle with a =1.5, b =2.6, and α = 15 deg were carried out to verify the reliability of the numerical model for the study. Figure 11 shows amplitude of T9 versus sampling time at Re=7905 in the lift and drag directions. Here, the amplitude is dimensionless. Simulated and experimental data fluctuate in a similar range in both directions. The amplitude in the lift direction is asymmetric with respect to the axis of x = 0, which is different from the response of the straight tube [44,45]. It also illustrates that the asymmetry structure in the plane affects the fluid forces acting on the coil tube. In addition, the data line in the lift direction is denser than those in the drag direction, indicating indirectly that the natural frequency in the lift direction is greater than that in the other. RMS amplitude versus Re of T9 and T12 in the drag direction are delineated in Fig. 12. The amplitude of test and simulation increases with the increase of Re, and the error is low, less than 10%, indicating that the numerical model has a certain reliability.

Fig. 11
Comparison of simulated and experimental vibration responses of T9 in the (a) lift and (b) drag directions
Fig. 11
Comparison of simulated and experimental vibration responses of T9 in the (a) lift and (b) drag directions
Close modal
Fig. 12
RMS amplitudes versus Re of T9 and T12 in the drag direction
Fig. 12
RMS amplitudes versus Re of T9 and T12 in the drag direction
Close modal

What's more, the normalized force PSD (obtained by Eq. (17) refer to a reference length, Le, of 1 m) in both directions was checked (Fig. 13). The normalized force PSD for random excitation forces of the three tubes at Re=7905 was obtained. It is found that the order of magnitude of the test PSD has a high degree of fitting with the simulation, which proves that the fluid excitation force data obtained by the simulation is applicable. Besides, PSD in the lift direction is about one decade larger than that in the drag direction. In short, the robustness of the numerical methodology is proved by comparing with the experimental data.

Fig. 13
Normalized PSD versus the reduced frequencies of all the tubes in both directions
Fig. 13
Normalized PSD versus the reduced frequencies of all the tubes in both directions
Close modal

5 Results and Discussion

5.1 The Dominant Buffeting Frequency.

Turbulent energy is generated at the front of the bundle and convects into it at an approximate rate ΔCfρUH3T. ΔCf(1/2)ρUH2 is the pressure drop through a row of tubes embedded in the bundle. If the turbulent kinetic energy of the fluid per unit mass is 3q2/2 (provided that the mean square component of a velocity component is q2) and it mainly exists in eddies at scale l, the dissipation rate is about 3q2/2l. Thereby, the rate of turbulence attenuation in the shell-side is 3ρq3TL/2l.

The tendency of turbulence to decay results from the breaking up of its energy-containing vortices and the transfer of this energy to smaller vortices that can be affected by viscous dissipation. According to the energy balance assumption
ΔCf12ρU3=3ρq3L/2l
(20)
Adopting a suggestion of Owen [13], ΔCf can be obtained by assuming that the drag coefficient, Cd, of each tube is dependent on the dimensions of the tube array, based on the gap velocity, and independent of the transverse tube spacing
ΔCf=CD(d/T)(1d/T)2
(21)
If eddies of scale l flow through the bundle at an average velocity, U, they pass through the plane containing the downstream row of tubes, and the frequency for the excitation force on these tubes is
ft=U/l
(22)
which, combining Eqs. (21) and (22), become
ftLUgapTd=13CD(Ugapq)3(1dT)2
(23)
In fact, turbulence is not induced impulsively at a row of tubes, nor does it exhibit a uniform energy density until it reaches the next. Rather, it is partly the result of a redistribution of the average velocity between successive rows, in which neither q nor l is constant. In addition, the above equation used for the turbulence decay rate can be accepted as merely suggestive, because the time scale for the breakdown of the high-energy eddy is similar to the time required for the fluid to flow from one row of tubes to another, so that the average flowrate changes markedly. It follows that it is not at all worth considering improvements such as the dependence of CD and Ugap/q on the geometry of the tube array [13]. Then, the above equation can be written as
ftLUgapTdFb(1dT)2
(24)

where Fb, which is a constant, is fit factor for various spaced in a high-Reynolds-number flow. It has been demonstrated in studies of straight tube bundles [14,46]. Resonance occurs when the dominant buffeting frequency, ft, fits the natural frequency for the tube. Once such a resonance is established, the tube amplitude will be greatly increased, possibly greater than the amplitude threshold of 0.02d. ft can be used to predict whether resonance occurs according to the relation proposed in the above equation, that is, when 1/2 f1ft, resonance is prone to occur [47].

Although one cannot determine the force direction, it can be noted that even if it is entirely in the drag direction, corresponding to a pure drag fluctuation, the vibration in the lift direction would be excited in the bundle due to the lack of consistency in the phase distribution along the direction of the tube elongation: similar in its effect to that of a nonuniform piston executing vibration in the drag direction within the shell-side. The PSD of eight tubes of all monitoring tubes for various structural bundles is carried out to obtain the main peak value of the spectrum of random excitation forces acting on coil tubes to determine the constant Fb, as demonstrated in Fig. 14. Here, Re = ρUHd/μ = 2.371 × 104 corresponding to UH = 3 m·s−1. Particularly, maximum three peaks of all the tubes are extracted from the figure, and if two of the peaks are equal, only two peaks are marked.

Fig. 14
PSD of various structural bundles for UH = 3 m·s−1
Fig. 14
PSD of various structural bundles for UH = 3 m·s−1
Close modal
Table 4 lists the gap velocity, Ugap, calculated according to the calculation correlation of Ugap and UH obtained in our own works [32], which is substituted into Eq. (25). The results of spectral analysis of various structures are plotted in the form of (FL/Ugap) × (T/d) to [1 − (d/T)]2 in Fig. 15. Considering the safety of engineering design and evaluation, a straight line is drawn above all data points to envelope all points, so as to ensure that no resonance occur when the dominant buffeting frequency calculated according to the following formula is not within 1/2 f1ft:
ftLUgapTd=1.89(1dT)2+0.39
(25)
Fig. 15
The theoretical relation for the dominant frequency taken in conjunction with the gap velocity and tube arrangement
Fig. 15
The theoretical relation for the dominant frequency taken in conjunction with the gap velocity and tube arrangement
Close modal
Note that this equation is satisfied with relatively small d/T. In the structure for a =1.1, b =1.1, and α = 15 deg, d/T =0.9 does not meet this condition and does not conform to Eq. (22). However, it can be found that the frequency envelope of the bundle with a =1.1, b =1.1, and α = 15 deg calculated by this equation, namely, the point in the circle, is below the line by observing Fig. 15. Although the structure does not meet the establishment condition of the equation, it can still be used for conservative evaluation. For brevity, it should be applied to engineering. Equation (25) can also be written as
ft=UgapdLT[1.89(1dT)2+0.39]
(26)

5.2 The Excitation Force Coefficient for Different Variables.

The random excitation forces acting on the tube are dimensionless by using the following formula:
Cf,l,d=Ff,l,d12ρdlUgap2
(27)

where C denotes coefficient. The subscripts f, l, and d are related to the excitation force, lift, and drag.

Figure 16 shows the time-varying force coefficient of T9 within the tube bundle of various structures. The motion of fluid in the shell-side was still unstable for the first about 0.3 s, after that the flow developed fully. The closer the structure is, the larger the Cf is, as shown in Fig. 16(a). This phenomenon is especially evident when a increases from 1.1 to 1.2 and then to 1.5. When a =1.5, there is little difference between Cf of three different structures. Compared with the other two types, the shear flow on the tube surface of b =1.5 and α = 15 deg (case 4) is more significant, resulting in a larger fluctuation of Cf. This is because the increase of helical angle enhances the partial motion of the flow along the tube to generate a more uneven distribution of forces along the tube and a large fluctuation of excited force acting on the tube. The increase of the pitch diameter ratio between the tubes of the same layer leads to the formation of eddies behind the tube, and the interaction of eddies on both sides of the tube wall makes the weakening of excitation force fluctuation. However, the excitation force fluctuation of a =1.2, b =1.5, and α = 15 deg (case 2) is greater than that of a =1.1, b =1.1, and α = 15 deg (case 1). Because the tube clearance of case 1 is too low, the shear flow on the tube wall is weak and jet switching is obvious. Although the tube clearance of case 2 increases and jet switching weakens, the shear flow increases and the two modes of case 2 interact with each other, making the excitation force coefficient fluctuates significantly.

Fig. 16
Force coefficients for T9 over time at Re = 2.371 × 104
Fig. 16
Force coefficients for T9 over time at Re = 2.371 × 104
Close modal

Figure 16(b) presents the variations of lift coefficient (Cl) of T9 over time. Trends for Cl of different structures are similar to Cf of them. The difference is that Cl of case 2 fluctuates more obviously and shows obvious periodicity. There are periodic vortex shedding in the lift direction between the tube clearance. In addition, Cl is asymmetric with respect to x =0. It is caused by the asymmetry of the structures for the tube bundle and is exacerbated by the presence of α. Except for cases 1 and 2, Cl of other structures fluctuates around 0. The gap between adjacent layers increases, and a flow channel is formed in the gap. It makes the flow pass through the bundle rather than going around it.

The variations of drag coefficient (Cd) of T9 over time are shown in Fig. 16(c). The law of Cd over time is similar to the other two force coefficients. It is found that the smaller the gap between tubes, the greater the random excitation forces of the fluid on the tube. All in all, the influence of fluid acting on the tube is formed by the coupling of turbulence, vortex shedding, and jet switching. The tube clearance is small, the flow goes around the tube, and the streamlines are constantly changing. This phenomenon is mainly caused by jet switching. With the increase of tube clearance, the shear flow on the tube wall becomes more obvious, forming eddies behind both sides of the tube and moving backward. It enhances the fluctuation of random excitation forces. With further increase of tube clearance, the eddies formed behind the tube may interact with each other and then cancel out, so that the effect on the back tube is weakened. In this phenomenon, the vortex shedding plays an important role, while jet switching becomes weaker gradually.

Figure 17 shows the broken lines of the RMS force coefficients' distribution of all the tubes for the tube bundle with a =1.5, b =2.6, and α = 15 deg (case 5) at various Re. It can be seen from Fig. 17(a) that the farther back the tube is, the greater Cf is in terms of the overall trend. As the fluid continues to pass through the tube bundle, the flow disturbance is intensified, and the flow instability is further enhanced on account of the reverse wound of the tubes and the asymmetry of the structure. The excitation force on the tube not only increases the fluctuation but also the RMS force. And the Cf of at Re = 2.371 × 104 is greater than that of other Re numbers, which indicates that when the Re number is greater than when a certain value is reached after Re = 2.371 × 104, the flow state of the fluid in the basin or the vibration response of the bundle changes suddenly, which may be because the flowrate between the tubes is greater than the fluid elastic instability threshold, the bundle vibrates violently, and the coupling between the bundle and the fluid also changes greatly. However, the laws are not seen in Fig. 17(b). The reason is that the spacing between adjacent layers for case 5 is large, and obviously, flow channels are formed between the tube layers, which has been observed in previous studies, detailed in Ref. [32]. The fluid can easily pass through the tube bundle along the channels. Therefore, there is no regularity for Cl in this figure. The flow develops obvious shear flow on both sides of the tube, and eddies are formed in the gap between the tubes of the same layer, which enhanced vortex shedding in the shell-side. As the flow goes deeper into the bundle, Cd increases, as illustrated in Fig. 17(c). In addition, because of the complex and variable structure of the gaps of the bundle, the uncertainty of the flow is enhanced. The force on a certain tube may not obey the overall trend. For example, Cf of T13 is greater than them of T12 and T14 in Fig. 17(a).

Fig. 17
RMS force coefficients of all the tubes with various Re (a = 1.5, b = 2.6, and α = 15 deg, case 5)
Fig. 17
RMS force coefficients of all the tubes with various Re (a = 1.5, b = 2.6, and α = 15 deg, case 5)
Close modal

Root-mean-square force coefficients of all the tubes for various structural bundles at Re=2.371 × 104 are shown in Fig. 18, where RMS Cf of case 1 is the largest, and that of a =2.0, b =1.5, and α = 15 deg (case 6) is generally the smallest. And Cf of all the tubes is uniform. There is no prominent change. In general, a has a great influence on RMS Cf (the smaller a is, the larger Cf is), but b and α have little influence on it. When α decreases, RMS Cf increases slightly. With the increase of b, the RMS Cf does not decrease obviously. The more compact the bundle, the greater the role of jet switching. As the gaps between the adjacent layers widen, the effect of vortex shedding becomes more and more obvious. And turbulence works all the time. Figure 18(b) shows the distribution of Cl of all the tubes. The existence of α strengthens the flow in the lift direction and makes the flow more unstable. The smaller a is, the greater Cl is. The RMS Cd of all the tubes is shown in Fig. 18(c), and its law is the same as that of Cf, which will not be repeated here.

Fig. 18
RMS force coefficients of all the tubes for various structural bundles at Re = 2.371 × 104
Fig. 18
RMS force coefficients of all the tubes for various structural bundles at Re = 2.371 × 104
Close modal

5.3 The Envelope Spectrum for Random Excitation Forces.

The vibration amplitude caused by random excitation forces is relatively small and will not lead to short-term failure. However, these vibrations do cause constant friction between the tubes and its supports, which makes the tube gradually break down due to fretting wear. As a result, it is very necessary to determine the guidelines of random excitation force for coil bundle. As discussed earlier in Eqs. (9), (15), and (17), it is necessary to refer to a single excitation tube length, Le, in order to properly compare the spectrum obtained by the coil tube bundle with different geometric parameters. This paper uses 1 m as the reference length. The flow into CWHE first passes through curved tube or straight tube regions (Fig. 1) and then enters the coil tube region, so there is no direct fluid impact on the tube. Therefore, only the coil tube interior the bundle is considered. In this paper, the effects of tube position, Re, and bundle structure on the normalized PSD are analyzed in order to obtain the normalized spectrum of excitation force for the bundle.

Figure 19 gives normalized PSD with respect to Le = 1 m for all the tubes of case 5 at Re=2.371 × 104. It is observed that PSD varies from tube to tube within the bundle. T18 is the most backward of the monitoring tubes, and its PSD is located above the rest. It can be conjectured that the more coil tubes the flow passes through, the stronger the flow instability and the larger the value of PSD. In future studies, the number of tubes for per layer can be increased to get closer to actual number of tubes. To simplify the spectrum, the narrow peaks in the PSD are ignored. The broken line that envelops PSD is plotted, and the envelope 1 is proposed as
{[ϕ˜F]e=3.6×104fr0.39fr0.25[ϕ˜F]e=8.2×108fr6.4fr0.25
(28)
Fig. 19
Influence of tube position on the normalized PSD (Le = 1m)
Fig. 19
Influence of tube position on the normalized PSD (Le = 1m)
Close modal
The influence of Re on normalized PSD is shown in Fig. 20, where the structural parameters of the bundle are a =1.5, b =2.6, and α = 15 deg. In addition, PSD of T10 and T18 under three representatives of Re is considered. The narrow peaks in PSD may be the vortex shedding frequency or the dominant buffeting frequency of flow-induced vibration. Only the universal existence of random excitation force on the tube is studied, so the narrow peaks are ignored. Draw the envelope 2 as
{[ϕ˜F]e=0.0065fr0.65[ϕ˜F]e=5.1×105fr11.2fr0.65
(29)
Fig. 20
Influence of Re on the normalized PSD (Le = 1 m)
Fig. 20
Influence of Re on the normalized PSD (Le = 1 m)
Close modal
Fig. 21
Influence of bundle structure on the normalized PSD (Le = 1 m)
Fig. 21
Influence of bundle structure on the normalized PSD (Le = 1 m)
Close modal
Figure 21presents the influence of the bundle structure on the normalized PSD in regard to Le = 1 m. Three structural parameters, a, b, and α, are used to determine the structure of the coil tube bundle. The variation of each structural parameter affects the flow characteristics in the shell-side, thus changing the random excitation force on the tube. The selected bundle structure includes compact arrangement and transition arrangement, which are common types of bundle arrangement [48]. The helix angle, α, is within the usual range. PSD of T11 and T18 is selected to determine the envelope 3, delineated as
{[ϕ˜F]e=0.00727fr0.50[ϕ˜F]e=4.06×105fr7.5fr0.50
(30)
Gathering the above three normalized PSD envelope lines, an upper bound above all broken lines is drawn as shown in Fig. 22. Then, the final envelope for the random excitation force of coil tube bundle can be proposed
{[ϕ˜F]e=0.00727fr0.64[ϕ˜F]e=1.82×104fr8.3fr0.64
(31)
Fig. 22
Proposed normalized PSD in regard to excitation force for coil tube bundle
Fig. 22
Proposed normalized PSD in regard to excitation force for coil tube bundle
Close modal

And the final envelope of the coil tube is compared with that of the straight tube. Note that the three envelope lines are representative. Axisa et al. took into account a variety of factors to draw the envelope [20]. Taylor and Pettigrew summarized various previous research results and proposed two other envelope lines [33]. It is found that the broken line for interior the straight tube bundle intersects with the coil tube. The broken line of the straight tube bundle close to the inlet has crossing points with the coil tube at the sharp position, but most of it is enveloped. However, when fr = 0.01, the difference in values is 1 order of magnitude, indicating a large error. Therefore, it is necessary to analyze the normalized envelope spectrum independently to predict and evaluate the force on the coil tube.

6 Conclusion

The coil tube bundle in the CWHE was simplified, and the test equipment was established. Flow-induced vibration of the coil tube bundle with a =1.5, b =2.6, and α = 15 deg was carried out through water tunnel tests. And the numerical models with respect to various structural parameters were studied in order to investigate turbulent buffeting mechanism and the influence of Re and bundle structures on the force coefficients. The main conclusions of this paper are the following.

  1. The results show that the vibration response error is within 10%, and the order of magnitude corresponding to the normalized force spectrums and the trends are a little different for the experimental and simulated data, which proves the robustness of the numerical method.

  2. The dominant frequencies of turbulent buffeting corresponding to various structures were obtained by monitoring the fluid excitation of the tubes. Using the method of threshold envelope, the semi-empirical equation was set up as
    ft=UgapdLT[1.89(1dT)2+0.39]
  3. The smaller the clearance within the tube bundle, the more obvious the jet switching, resulting in the increase of the force coefficients. As the pitch diameter ratio between adjacent tube layers (a) increases, the shear flow on both sides of the tube enhances, and eddies are formed behind the tube. Lift on the tube has no obvious change law. Vortex shedding plays a major role at this moment. a has a great influence on the force coefficients, while the pitch diameter ratio in the same layer (b) and the helix angle (α) have little influence on it. In addition, force coefficients in the bundle a = 1.2, b = 1.5, and α = 15 deg (case 2) fluctuate greatly and the flow instability is obvious, which is the result of the interaction of jet switching, vortex shedding, and turbulence.

  4. The vibration amplitude caused by random excitation forces is relatively small. However, these vibrations do cause constant friction between the tube and its supports, which causes the tube to gradually break down due to fretting wear. Therefore, it is important to determine the random excitation force of the coil bundle. The mechanisms of tube position, Re, and bundle structure on the normalized force spectrum were investigated. And the envelope of each case is determined respectively. Eventually, the normalized envelope spectrum of random excitation acting on the coil tube was proposed.

Acknowledgment

The authors would like to thank Heat Exchanger Committee, National Technical Committee for Standardization of Boiler and Pressure Vessels for technical support. Thanks also to Ami Tektronix Industrial Equipment Co., Ltd. for its help in the design and implementation of experiment.

Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

a =

pitch diameter ratio between adjacent tube layers

A =

tube displacement, m

b =

pitch diameter ratio in the same layer

C =

coefficient

d =

tube diameter, m

f =

frequency, Hz

F =

function

l =

tube length

L =

length

m =

tube mass per unit length, kg/m

Q =

term

U =

flow velocity

Greek Symbols
α =

helix angle, deg

γ =

the degree of correction of forces along the tube

δ =

mode shape displacement constant

ε =

porosity

ζ =

damping ratio

λ =

the correlation length

ρ =

fluid density in the shell side, kg/m3

ϕ =

the autocorrelation spectrum of the force per unit tube length

ψ =

the fluctuating force per unit tube length

Subscripts
A =

area fraction

d =

drag

e =

equivalent

f =

force

gap =

gap

H =

uniform cross flow

l =

lift

n =

the nth order

r =

reduced

SAS =

the double-equation shear stress transport model

V =

volume fraction

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