numerical simulation of baffle-supported tube bundle vibration by the method of finite differences

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Numerical Simulation of Baffle-Supported Tube BundleVibration by the Method of Finite Differences

By Michael Fischer and Klaus Strohmeier*

Tube bundles in heat exchangers are often damaged due to flow-induced vibrations. To simulate the fluid-structure interac-tion between the tubes and the flow field, which seems to be the only reliable way to predict flow-induced vibration of tubebundles, a coupled numerical solver for the governing Navier-Stokes equations of the flow field as well as for the structuralresponse has to be used. Since commercial codes still require a large computational effort to calculate the fluid-structure in-teraction, one has to develop special program codes for certain technical problems. This paper deals with a numerical methodto calculate the structural response of baffle-supported heat exchanger tubes due to outer forces such as flow and contactforces. The equations of motion for a cylindrical tube including torsional movement are developed. They are discretized inspace by the means of Finite Differences and in time by Newmark's method of constant acceleration. Non-linear frictionalimpact forces due to the baffle supports are introduced as additional line loads and line moments. Flow forces at this stage ofdevelopment are modeled by simplified assumptions concerning vortex shedding. Numerical examples show the consistencyof the calculated solutions. The algorithm is stable and converges for parameters of practical importance. Only little mathe-matical effort is needed compared to the Finite Element technique or modal analysis. A Navier-Stokes flow solver alreadydeveloped can now be integrated into the vibration simulation code.

1 Introduction

Tube bundles are often used as a component in heat ex-changers for chemical engineering, industrial process engi-neering, or apparatus and plant design. To increase powerdensity and efficiency, flow velocities have to be increasedwhile tube wall thicknesses decrease. In the past, this led tonumerous unpredictable cases of loss due to flow-induced vi-brations where very large tube vibration amplitudes can oc-cur. Researchers have tried in the last 30 years to achievesome semi-empirical design rules by representing collectedexperimental data by simple algebraic formulae to avoidfurther damage (e.g., review by Price [1]). In that the tube vi-brations turned out to be driven by complex fluid-structureinteractions, the semiempirical statements led to non-satis-factory results with error of some 100%. Since computerpower has dramatically increased in the last 10 years,coupled numerical simulation of the time-dependent equa-tions of motion for the tube structure and the Navier-Stokesequations for the fluid became a new method to solve theproblem. In a project recently finished [2], the tubes whererepresented by single mass oscillators while the representa-tive flow field was calculated in two dimensions. Comparisonwith experimental data showed good agreement with numer-ical solutions. Since the computational effort becomes verylarge, there has still been no attempt to calculate the time-

dependent, fully-coupled problem three-dimensionally.Nevertheless, there is already a need to develop efficientcomputer codes which can simulate three-dimensional flowfields simultaneously with the structural response of baffle-supported tube bundles. This paper deals with the develop-ment of a fast and stable numerical method to calculate thethree-dimensional structural tube responses due to outerforces using the method of Finite Differences and New-mark's method [3].

2 Equations of Motion for the Tubes

For an infinitesimal beam element (Fig. 1) the equilibriumof moments, the dynamic equilibrium of forces, and the elas-tostatical equation1)

(1)

representing Hooke's law of linear elasticity can be com-bined to the well-known equation of motion for a beam

(2)

if axial forces and the moment of inertia of the elementare neglected. This equation is applicable to the y- as well asto the z-direction. For torsional movement there is the equa-tion of motion

Chem. Eng. Technol., 21 (1998) 5 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1998 0930-7516/98/0101-0431 $ 17.50+.50/0 431

±

[*] Dipl.-Ing. M. Fischer, Scientific Assistant at the Institute for Apparatusand Plant Design; Prof. Dr.-Ing. K. Strohmeier, Chair Institute for Ap-paratus and Plant Design, Technical University of Munich, Boltzmann-straûe 15, D-85747 Garching, Germany.

0930-7516/98/0505-0431 $ 17.50+.50/0

±

1) List of symbols at the end of the paper.

432 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1998 0930-7516/98/0101-0432 $ 17.50+.50/0 Chem. Eng. Technol., 21 (1998) 5

(3)

which can be derived from the dynamic equilibrium ofmoments for a torsional beam element; q and mz are lineloads due to flow and contact forces.

3 Discretization in Space

By replacing the fourth derivative used in Eq. (2) with a fi-nite difference scheme one yields the equation of motion

(4)

implicitly discretized in space where the upper index n meansthe time step and the lower index I the x-position. Introdu-cing the boundary conditions leads to the matrix equation

(5)

with the mass matrix

(6)

the damping matrix

(7)

and the stiffness matrix

(8)

Note that M, C, and K are symmetric matrices. If l equalsa value of 5, there is a hinge fitting; if l equals 7, the clamp-ing is fixed. By varying the two l values between 5 and 7 dif-ferent kinds of clamping can be modeled. For different de-signs of clamping, the corresponding l can be evaluated byvibration experiments as they are already known for the eva-luation of damping coefficients. Measured natural frequen-cies have to be compared with calculated natural frequenciesfor varying l to achieve an experimental relation betweenthe clamping design and the numerical clamping coefficient.As can be easily determined by numerical investigation, onlya small influence of any variation of l remains as l ap-proaches the value of 7. So the value of 7 might be a goodapproximation for any fixed clamping.

If there is a baffle support in contact with a tube at a nodes, the stiffness matrix for the two nodes representing the leftand right of the support has to be modified setting

(9)

These vectors are derived by introducing the two bound-ary conditions that the bending line has to have the same in-clination and the same curvature left and right of the bafflesupport. Additional line loads at a baffle support resultingfrom the frictional impact are treated in section 5.

4 Newmark's Method for Time Discretization

In Newmark's method [3] there are the assumptions

(10)

and

(11)

Figure 1. Infinitesimal beam element.

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for displacement and velocity. Combining Eqs. (10) and (11)with the equation of motion (5) yields a linear system ofequations for the displacement vector. For d = � the methodis proved to have no numerical damping, which is very im-portant due to the nonlinear contact forces being dependenton the amplitudes. If a is set to � the algorithm is proved tobe stable for all time steps, which is important for an every-day engineering tool. Using these values for a and d leads tothe linearized system

(12)

which is known to be the method of constant acceleration.Eq. (12) can be solved for the displacement vector if the lineloads for the following time step are known. The new velo-city vector and acceleration vector can be calculated by Eqs.(10) and (11).

Eq. (12) must be solved iteratively due to the fact that theadditional line loads at nodes of contact are nonlinear in na-ture. For all matrices that are symmetric and dominant inthe primary axis, it is easily possible to find the relaxedGauss-Seidel equation for the displacement in node I at timestep n + 1. As could be found, a number of iterations of fourper time step can be sufficient since the time step Dt has tobe rather small in the nonlinear case anyway. For the equa-tion of torsion the iterative equation can be found in thesame way. Frictional impact forces cause an additional linemoment at the nodes of contact which can be introducedinto the equation of torsional movement.

To prove the correctness of clamping modeling andboundary conditions in Eqs. (8) and (9), one can simulatetube motions in only one direction with the gapwidth zero.By the means of Fourier's transformation, the time seriescan be analyzed within the frequency space. Using constantdistances between the baffle supports and a random startcondition, the result after sufficient simulation time mustequal the natural frequencies of the vibrating tubes as it canbe easily calculated analytically by

(13)

with bi being the eigenvalues as shown in TEMA's catalogue[4].

Fig. 2 shows the time series of a fixed clamped test tube vi-brating for two seconds and Fourier's corresponding trans-formation. As can be seen from Table 1 up to the 3rd natural

frequency, the results of TEMA's catalogue Eq. (13) and ofthe numerical simulation differ only within the solubility of

Fourier's transformation, which here was 0.5 Hz due to thelength of the time series.

To prove the correctness of the torsional movement simu-lation one can use an analytic solution for the natural fre-quencies of a double fixed torsional beam and compare itwith a Fourier's transformation of a simulated torsional timeseries with no contact forces occurring. Good agreement be-tween simulation and analytic solution could also be foundhere.

5 Contact Forces and Chaotic Coupling ofMotions

Due to the fact that there are frictional impacts betweenthe tubes and the baffle supports, the motion in the y- and z-direction as well as the torsional motion become interdepen-dent. Fig. 3 shows a principal drawing of all occurring con-

tact forces. Because the contact forces are not parallel to thegeneral coordinate axes, normal and tangential loads have tobe transformed for being introduced into the equations ofmotion for the y- and z-direction. In the normal direction,the contact forces can be modeled by

(14)


Figure 2. Simulation time series and Fourier's transformation (EI = 1, rA = 1,m = 0, q = 0, L = 1).

Figure 3. Frictional impact and forces.

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434 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1998 0930-7516/98/0101-0434 $ 17.50+.50/0 Chem. Eng. Technol., 21 (1998) 5

which is a Hertzian law [5] in conjunction with an addi-tional damping term. Here k2 is the contact stiffness whichcan be dependent on the inclination angels between the tubesurface and the baffle support. At this stage of developmentof the code k2 is assumed to be constant as well as the powerof 3/2. The dependency on the inclination angels for k2 andalso the power 3/2 could be measured or simulated by themethod of finite elements to improve the structural model.

The tangential forces due to friction and normal loadshave been modeled including a distinction between stickingand slipping. While slipping, the frictional forces can bemodeled following Coulomb's law

(15)

where mt may also be dependent on the tangential velocity.If

(16)

then sticking occurs and sticking frictional forces can bemodeled by

(17)

which is a surface deformation law (see [6]) with a smalldamping term due to numerical reasons. The resulting addi-tional forces can be divided by the baffle support thicknessto get a line load which can be added to the line loads in theequation of motion, Eq. (12). The distinction algorithm wasdeveloped by Antunes et al. [7] and has been already suc-cessfully used, but within modal analysis in conjunction withlinear tangential forces.

In Fig. 4 as an example, the trajectory of the center of atube at the level of a baffle support is shown together withthe gap's center circle. As can be seen the way of movingchanges it's character completely if only one parameter, herethe slipping coefficient, is changed only slightly. This is abehavior as it is known from deterministic chaos. This meansthat any prediction of tube durability and wear work ratecan only be made as a statistical statement after a certainnumber of variations of governing parameters in some testcalculations which have to be long enough to create transfor-

mations of statistical significance. From the time series inFig. 5 it can be seen that the first natural frequency does notchange although the trajectories in Fig. 4 differ strongly. Thisis due to the fact that the natural frequency is a structuralparameter that does not change when changing the methodof measuring it. Another sign of chaotic behavior is the slope

of the Fourier's transformation which contains many moremaxima of significantly lower energy besides the main maxi-mum. The positions and levels of these maxima differ verymuch when changing the input parameter only slightly.These maxima do not represent any natural frequency.

6 Fluid Forces

The fluid was assumed to flow towards the tube bundle asit is shown in Fig. 6. With the main flow direction being y asa simplification at this stage of development

(18)

for the time dependent fluid forces was set constant overthe whole height of every chamber with fA being a vortexshedding frequency. Eq. (18) means that the drag forceschange with double the frequency of the lift forces which isknown to be true from vortex shedding experiments. Thesign of the qy flow forces in the three chambers of the as-Figure 4. Chaotic tube center trajectory.

Figure 5. Comparison of Fourier's transformation for different slipping coeffi-cients.

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sumed heat exchanger was fitted to the main flow directionof every chamber. Torsional damping in this test case resultsonly from structural damping and impact forces.

7 Conclusions and Future Developments

A nonlinear finite differencing method for the simulationof baffle-supported tube bundle vibrations was presented.Finite differences for space discretization and Newmark'smethod for time discretization were used to simulate themotion of heat exchanger tubes numerically including tor-sion and nonlinear baffle support frictional impacts. Neithermodal analysis nor finite element formulation were neces-sary to develop the discretized equation of motion. Manyimportant structural parameters can be varied. The solutionsproved to be stable, convergent, and exact. Fluid forces wereassumed to follow a simple vortex shedding law.

The structural code can now be used as a basis for a fully-coupled, fluid-structure interaction code where fluid forcesare calculated from the simulation of the flow field withinthe tube bundle. A fluid module is being currently devel-oped that will be integrated into the structural simulationcode. This opens the way to a fully-coupled, three-dimen-sional numerical simulation of flow-induced vibration oftube bundles in baffle-supported heat exchanger tube bun-dles.

Received: November 17, 1997 [CET 965]

Symbols Used

A tube cross sectionC damping matrixcD maximum additional transient drag

coefficientcD0 mean drag coefficientcL maximum transient lift coefficientD tube outer diameterE Young's modulusfA vortex shedding frequencyFn normal load

Ft tangential loadf natural frequencyG modulus of shearI moment of inertiaI index of space in direction of tube axisIT polar moment of inertiaJp moment of mass inertia per unit

lengthK tangential coefficient of dampingK stiffness matrix

modified stiffness vectork2 Hertzian stiffness of contactL tube lengthM mass matrixM momentmz moment per lengthn index of normal directionn index of timeDn lateral tube compression in normal

directionqy, qz line loads for y- and z-directiont timeDt time stepw bending linex tube axisDx discretization lengthy, z bending directionsa Newmark's coefficient of stabilitybi Eigenvalued Newmark's coefficient of numerical

dampinge tangential displacemente0 tangential displacement coefficientj torsional anglelL, lR numerical clamping coefficientsm bending damping coefficientmd normal contact damping coefficientmt Coulomb's coefficient of slippingmT torsional damping coefficientm0 Coulomb's coefficient of stickingr tube densityrF fluid density

References

[1] Price, S. J. Journal of Fluids and Structures 9 (1995) pp. 463±518.[2] Kassera, V. Doctoral Thesis, Institute for Apparatus and Plant Design,

Technical University Munich, Munich, Germany 1997.[3] Newmark, N. M. Journal of the Engineering Mechanics Division 85

(1959) ASCE, pp. 67±94.[4] TEMA Catalogue Standards of Tubular Exchange Manufacturers Asso-

ciation. Section 12, pp. 224±230.[5] Hertz, H. Journal für die reine und angewandte Mathematik 92 (1882)

pp. 156±171.[6] Oden, J. T.; Pires, E. B. Journal of Applied Mechanics 50 (1983) pp. 67±

76.[7] Antunes, J; Axisa, F.; Beaufils, B.; Guilbaud, D. Journal of Fluids a

Structures 4 (1990) pp. 287±304.


Figure 6. Assumed flow situation in a heat exchanger for test calculations.

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numerical simulation of baffle-supported tube bundle vibration by the method of finite differences

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