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2010,22(6):816-822 DOI: 10.1016/S1001-6058(09)60121-9

MATHEMATICAL ANALYSIS OF TRANSVERSE VIBRATION OF CONICAL SPIRAL TUBE BUNDLE WITH EXTERNAL FLUID FLOW*

YAN Ke, GE Pei-qi, SU Yan-cai, BI Wen-bo Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, Jinan 250061, China, E-mail: wssy@mail.sdu.edu.cn

(Received June 5, 2010, Revised August 9, 2010)

Abstract: The conical spiral tube bundle is a new type of heat transfer elements used to enhance heat transfer through flow-induced vibration. The effect of the external fluid flow on the transverse vibration of the conical spiral tube bundle is investigated with a mathematical method proposed in this article. Firstly, the natural vibration of the tube bundle is obtained by the hammering excitation method and the mode shapes of the transverse vibration are discussed. Then the effect of the external fluid flow on the transverse vibration of tube bundle is analyzed by a combination of experimental data, empirical correlations and FEM. The results show that in the frequency range from 0 Hz to 50 Hz, there exist six transverse vibrations. The external fluid flow has a significant effect on the frequency of the tube’s transverse vibration, which are decreased by about 18% to 24% when the external fluid flowspeed is 0.3 m/s.

Key words: conical spiral tube bundle, mathematical model, vibration characteristic, FEM

1. IntroductionQuite a number of theoretical studies of the

Flow-Induced Vibration (FIV) in heat exchangers were carried out. The flow-induced vibration is usually considered as a detrimental factor related with damages and is to be prevented[1,2]. However, the FIV also plays a positive role in enhancing heat transfer, as shown in Ref.[3,4] , where a novel approach is proposed to deal with the flow-induced vibration as a heat transfer enhancement in shell-and-tube heat exchangers. The heat transfer enhancement is achieved via the vibration of elastic tube bundles. The conical spiral tube bundle[5] is a new type of elastic tube bundles used in heat transfer enhancement, which consists of two spiral pipes connecting with a joint body. Because of its conical spiral structure, different bundles are fixed as a nesting structure in the shell-side of heat exchangers.

To enhance the heat transfer, the vibration of

* Project supported by the National Basic Research Program of China (973 Program, Grant Nos. 2007CB206900) Biography: YAN Ke (1984-), Male, Ph. D. Candidate Corresponding author: GE Pei-qi, E-mail: pqge@sdu.edu.cn

conical spiral tube bundles must be reasonably induced and controlled. Therefore, the investigation and the prediction of the vibration characteristics of elastic tube bundles are of importance[6]. The natural vibration of a conical spiral tube bundle was numerically studied by both sub-structure method and FEM[7]. The results show that the vibration modes of a conical spiral tube bundle are mainly longitudinal ones. In fact, the tube bundle is filled with fluid flow in heat exchangers in the real working conditions. The fluid flow has a great influence on the natural vibration of the tube bundles, as known as fluid-structure interaction. The influence of fluid flow inside the spiral tube bundle on its natural vibration was investigated by Ni[8] and Xu[9] with a modified Hamilton equation. Subsequently, the results of their work were used to predict the vibration of conical spiral tube bundles[10].

The fluid flow outside the tube bundle has a much larger influence than the internal fluid flow, therefore, the investigation of the vibration of conical spiral tube bundles with external fluid flow is highly desirable. But, related researches in this area, both for experimental and numerical simulations, have been

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very few due to the complex structure of conical spiral tube bundles. Most researches in this area were focused on the long straight cylinder with a circular cross section[11]. In experimental studies, the lift coefficient and the drag coefficient of a forced vibrating cylinder with external fluid flow were analyzed by Gopalkrishnan[12], and the results were used in many engineering designs and software simulations. In numerical simulations, some empirical correlations were proposed to predict the flow-induced vibration of cylinder with external fluid flow[13]. A novel approach was proposed by Pan[14] to determine the response characteristics of the flow-induced vibration of a deep-sea riser with data from forced vibration experiments. The vibration of the riser was simulated with a combination of empirical correlations and experimental data, and the prediction agrees very well with some experiments.

Due to the complex structure of the conical spiral tube bundle, it would be a formidable task to set up the simulation model in FEM software, and in addition, a numerical simulation of fluid-structure interaction of conical spiral tube bundles requires computers with performance of very high level and it is really a time-consuming work. In this article, the correlations of Pan[15] and the experimental data of Gopalkrishnan[12] are used to investigate the influence of external fluid flow on the transverse vibration of conical spiral tube bundles. The conical spiral tube bundles are divided into many small elements based on FEM. With respect to the external fluid flow, each pipe element is assumed to be a rigid cylinder acted by a periodical lift force induced by vortex shedding behind the tube bundle. Meanwhile, conjoint elements is acted by a daggle force from each other thus the vibration of a pipe element is assumed to be a forced Vortex-Induced Vibration (VIV). In this article, the transverse vibration modes of the conical spiral tube bundle were experimentally determined with the hammering excitation method. The flow force of each element was determined based on the above experimental data and the VIV model of a long straight riser. Finally, the additional mass of each element, induced by the external fluid flow, was calculated in iterations and the transverse vibration of the whole tube bundle with the external fluid flow was simulated by FEM.

2. The model of finite element analysis The structure of the conical spiral tube bundle is

shown in Fig.1. It consists of two pipes, I and II, which are conical spiral structures and connected via a joint body M, with an additional mass. The inlet and outlet of the fluid flow are supposed to be clamped. Because of its conical spiral structure, different bundles are fixed as a nesting structure in the

shell-and-tube heat exchangers. The transverse vibration of the conical spiral tube is in the direction vertical to the external fluid flow, as shown in Fig.1.

Fig.1 Structure of conical spiral tube bundle

2.1 The natural vibration of the tube bundle An experiment was conducted first to investigate

the natural vibration of the conical spiral tube bundle, as shown in Fig.2 and Fig.3. The inlet and outlet of the tube bundle were totally clamped with the substructure of a large size and weight to minimize the interference in the testing. The 192 testing points were uniformly distributed on the tube bundle, with a uniform central angle of 12o. The accelerometer was fixed on the 91st testing point, and the response signal was collected through a charge amplifier. The force sensor was connected with the exciting hammer. The hammerhead used in the experiment is plastic in view of the narrow range of the tube frequency in our consideration. The force hammer was used to impact the 192 measuring points in the transverse direction, and the response signals were collected by the DASP-V10 data acquisition system.

Fig.2 The hammering excitation method in experimental test

A FEM simulation was carried out before the experiment with APDL method and the results show that the difference between two contiguous natural frequencies is small. Therefore, in the experimental data analysis, the Eigen-system Realization Algorithm (ERA) was used to calculate the natural frequency of

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the tube bundle. Table 1 lists the structural parameters and material parameters of one type conical spiral tube bundle used in the experiment. The natural frequencies of the conical spiral tube bundle are shown in Table 2. The experimental data are compared to the simulation results, and the biggest deviation between the experiment and simulation results is 8.07%.

Fig.3 The experimental principle of hammering excitation method

Table 1 Structural and material parameters of the tube bundle

Parameter Value Bottom radius R (m) 0.4

Top radius r (m) 0.1Helical pitch winding H (m) 0.09

Sectional diameter D (m) 0.02Pipe thickness h (m) 0.0015

Space between two pipes e (m) 0.03Elastic modulus E (GPa) 127

Poisson ratio 0.33Joint body M (mm) 66×42×30

Table 2 Transverse vibration of tube bundle

Order Frequency (Hz)

2 4.5770 3 4.9360 4 6.1210 19 27.184 21 35.593 23 45.996

The Modal Assurance Criterion (MAC) was proposed to evaluate the independence of two different vibration modes. Theoretically, MAC being equal to zero indicates that two modes are totally independent, and MAC = 1 indicates that one mode can be derived by another. Taking the noise interference in the experiment into account, MAC > 0.9 means that modes are related, and MAC < 0.1 means independent modes. It can be seen from Fig.4 that in the relative matrix of vibration modes, MAC is equal to 1 in the diagonal of the

matrix, which means the transverse vibration modes are related to each other. MAC in the other parts of the matrix is much smaller that 1, which means that different physical modes are independent. Therefore, the experimental analysis of the transverse vibration of the conical spiral tube bundle is accurate and reasonable.

Fig.4 The relative matrix of transverse vibration modes

Fig.5 The transverse vibration of conical spiral tube bundle

The experiment shows that in the frequency range between 0 HZ and 50 Hz, there exist 6 modes of transverse vibration of the tube bundle, as listed in Table 2. In the investigation of vortex-induced

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vibration, the transverse vibration of a VIV cylinder is much larger than the longitudinal vibration. Therefore the transverse vibration of the tube bundle has a larger influence on the heat transfer enhancement. In this article, the transverse vibration of the tube bundle with external fluid flow was investigated. The mode shapes of the transverse vibration are shown in Fig.5, where the dashed line and the solid line represent the outline of the tube bundle deformation and the original position, respectively.

Figure 5 shows the deformation of the 2nd order transverse vibration. The two spiral pipes vibrate conformably in the x direction. The direction of the 3rd order deformation is normal to that of the 2nd order, which is along the y direction. The deformation of the 4th order transverse vibration is relatively small comparing to the other vibration modes. The vibration of the free end is along the xdirection, while the two pipes vibrate in opposite directions. The mode shape of the 19th order is similar to that of the 3rd order, with the two pipes vibrating conformably while different parts of the whole tube bundle vibrating independently. The vibration of the 21rt order is similar to that of the 4th mode shape, with much larger deformation. The two pipes vibrate independently in the 23rd order mode shape, with the vibration of the free end similar to the 2nd order. It should be pointed out that the listed vibrations of the conical spiral tube bundle in Table 2 are not strictly in the transverse direction due to its helical structure. Each transverse deformation contains some longitudinal component, which is rather small comparing to the transverse one and is negligible in the analysis of the transverse deformation.

Fig.6 Finite element mesh of conical spiral tube bundle

2.2 The finite element mesh of elastic tube bundle The finite element mesh of the conical spiral tube

bundle is shown in Fig.6. Pipe I is divided into 100 elements and Pipe II into 90 elements in order to be consistent with the experiment. The rigid joint M is simplified with the lumped mass method and its additional mass is distributed to the nodes of the conjoint elements. Each element is assumed as a VIV system with external fluid flow, and the lift force and the drag force are calculated in order to obtain the

additional mass of the whole tube bundle. 2.3 The VIV model of pipe element

The vibration equation of the circular cylinder with external fluid flow can be found in Pan[15]

excluding the effect of the axial force

2 2 2

2 2 2+ + =y y yEI x m x c xx x t t

,F x t (1)

where ( )EI x is the flexural rigidity of the cylinder, ( )m x is the mass linear density, ( )c x is the

structural damping, ( , )F x t is the transverse fluid force of the cylinder. For a stable transverse vibration mode of the tube bundle with natural frequency ,one single mode solution of Eq.(1) can be obtained as follows

, = siny x t A x t (2)

where ( )A x is the mode amplitude of the tube bundle. The transverse fluid flow force ( , )F x t can be decomposed into the inertia force component LaC ,in-phase with the pipe acceleration, and the drag force component LvC , in-phase with the pipe velocity

21, = cos2 f LvF x t U D C x t

sinLaC x t (3)

where f is the fluid density, D is the tube diameter, U is the external fluid flow speed. Rearranging the above three equations into sine part and cosine part, we have

2 22

2 2

AEI x m xx x

2

02

1 = 02

La fC x DUA

A (4)

20

1=2 f Lvc x A DU C x (5)

where 0A is the relative mode coordinate of the pipe element, which is not the real amplitude of the tube in

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Eq.(4). The effect of additional mass on the tube bundle with external fluid flow is shown in Eq.(4), which may be denoted as m

2

2

1=2

La fC x DUm

A (6)

For the whole tube bundle, the structural damping and the transverse fluid force should satisfy the energy conversation principle

2

0

1 cos cos d2

l

f LvDU C x t A t x

2

0cos d = 0

lc x A t x (7)

where the first part is the effect of the transverse fluid flow on the tube bundle, and the second part is the energy dissipation due to the structural damping. The amplitude of the tube bundle can be obtained by integrating Eq.(4) along the conical spiral tube bundle

2

0

1 d = 02

l

f LvDU C x c x A x (8)

where A is the real local amplitude of the tube bundle, as is different from what is in Eq.(4). 2.4 The iteration process

Combining Eq.(4) and Eq.(7), one obtains the natural frequency of the tube bundle and the additional mass of the fluid flow in an iterative process. Firstly, the natural frequency and the vibration mode shape of the conical spiral tube bundle are obtained assuming a constant additional mass, and based on the mode coordinate, the lift coefficient of each pipe element can be obtained according to experimental data, then the local amplitude of the tube bundle can be calculated with Eq.(7). Based on the local amplitude, one obtains the inertia force coefficient from experimental data, and the additional mass of each pipe element can be calculated with Eq.(6). Again the additional mass is used in Eq.(4) to obtain the natural frequency of the tube bundle and a second iteration begins. The iteration prcess ends if the difference of two iteration results is within a certain range. The flow chart of the calculation is shown in Fig.7.

There are two assumptions in the derivation of the above equation: (1) the vibration of the conical spiral tube bundle is strictly in one single mode, (2) the rigidity of the spiral tube is equivalent to a spiral spring.

The experimental data of the fluid flow force

coefficient can be found in Gopalkrishnan[12]. In the iteration process, while / 0.05 0.35f St , the lift and drag coefficients can be obtained from experimental data. It should be noted that the Strouhal number St is assumed to be 0.2 in the whole calculation and f denotes the frequency of the tube.

Fig.7 The flow chart of iteration process

If / 0.05f St , the empirical damping correlation of the low reduced frequency is used to calculate the fluid force coefficient[15]

= 2Lv vhAC CD

, = 0.6aC (9)

If / 0.35f St , the empirical damping correlation of the high reduced frequency is used to calculate the fluid force coefficient

332 3 28 2= 4Lv r sw r

w

A AC f k fD DRe

4 vl rAC fD

, = 1.0aC (10)

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where rf denotes the reduced frequency, aC the additional mass coefficient, and = 0.2vhC ,

= 0.18vlC , 2= /Re D v , = 0.25swk and all these empirical parameters are obtained from model tests.

3. Analysis and calculationThe external fluid flow velocity U is equal to

0.3 m/s in the following calculation. The parameter is thus selected for two reasons. Firstly, the fluid flow speed in the tube-side of the elastic tube bundle heat exchangers is between 0.01 m/s to 0.3 m/s. The critical speed = 0.3 m / sU used in the iteration process may represent the maximal influence of the external fluid flow on the transverse vibration of the tube bundle. Secondly, for the fluid flow speed

= 0.3 m / sU , the lift and the drag coefficients are mainly in the experimental range, which makes the calculation more accurate and convenient comparing to correlations by empirical data.

The effect of the external fluid flow on the transverse vibration of the conical spiral tube bundle is investigated in this article and the convergence process ( Q ) of the 2nd order mode ( 2ndf ) can be seen in Fig.8. It can be concluded that after three iterations, the results of the 2nd natural frequency of the conical spiral tube bundle tend to converge. A further calculation is carried out to verify the frequency and the convergence process curve is found to be smooth. In Fig.8, the last three iteration results of the 2nd natural frequency are 3.4772 Hz, 3.4668 Hz and 3.4792 Hz. The biggest difference between them is 0.36%.

Fig.8 The convergence process of calculation

The frequency of the transverse vibration of the conical spiral tube bundle in the condition of the external fluid flow is shown in Table 3. The six transverse vibration frequencies are compared to the natural frequencies of the tube bundle. It is concluded that the effect of the external flow on the vibration of the tube bundle is significant. The frequencies decrease by about 18% to 24% with the external flow

speed = 0.3 m / sU . The mode shape of the transverse vibration with the external fluid flow is not discussed in our analysis because the solution of the asymmetric dynamic matrix of the tube bundle is rather time-consuming, while the eigenvalues of the matrix or the frequencies of the tube bundle are easy to obtain.

Table 3 Effect of external fluid flow on transverse vibration Order Tube natural

frequencies(Hz)

Tubefrequencies

with external flow(Hz)

Ratio

2 4.5770 3.4792 23.9% 3 4.9360 3.4921 24.0% 4 6.1210 4.9770 18.7%

19 27.184 22.163 18.5% 21 35.593 26.379 34.9% 23 45.996 34.681 24.6%

4. ConclusionThe transverse vibration of the conical spiral tube

bundle is investigated in this article by the hammering excitation method and the ERA method is used to deal with the experimental data when the difference between two contiguous natural frequencies is small. The results show that in the frequency range between 0 Hz and 50 Hz, there exist six transverse vibrations. Each transverse deformation contains a small component of the longitudinal vibration, as compared to the transverse vibration, which is negligible in the analysis of the transverse deformation. The effect of the external fluid flow on the transverse vibration of tube bundle is studied with the combination of experimental data, empirical correlations and FEM. The frequencies of the transverse vibration are decreased by about 18% to 24% when the external fluid flow speed is 0.3 m/s. In elastic tube bundle heat exchangers, the heat transfer enhancement is achieved via the vibration of elastic tube bundles, therefore the FIV prediction for the tube bundle is of importance. The results of this article may be used in the FIV control of heat exchangers.

AcknwoledgementThis work was supported by the Independent

Innovation program of Shandong University (Grant No. 31360070613218)

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