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Nuclear Engineering and Design 284 (2015) 19–26

Contents lists available at ScienceDirect

Nuclear Engineering and Design

jou rn al hom ep age: www.elsev ier .com/ locate /nucengdes

perational modal analysis of flow-induced vibration of nuclear fuelods in a turbulent axial flow

. De Pauwa,b,c,∗, W. Weijtjensb, S. Vanlanduitb, K. Van Tichelenc, F. Berghmansa

Vrije Universiteit Brussel (VUB), Brussels Photonics Team (B-Phot), Brussels, BelgiumVrije Universiteit Brussel (VUB), Department of Mechanical Engineering (AVRG), Brussels, BelgiumBelgian Nuclear Research Centre (SCK•CEN), Boeretang 200, Mol, Belgium

i g h l i g h t s

We describe an analysis technique to evaluate nuclear fuel pins.We test a single fuel pin mockup subjected to turbulent axial flow.Our analysis is based on operational modal analysis (OMA).The accuracy and precision of our method is higher compared to traditional methods.We demonstrate the possible onset of a fluid-elastic instability.

r t i c l e i n f o

rticle history:eceived 19 August 2014eceived in revised form9 November 2014ccepted 30 November 2014

a b s t r a c t

Flow-induced vibration of nuclear reactor fuel pins can result in mechanical noise and lead to failure ofthe reactor’s fuel assembly. This problem can be exacerbated in the new generation of liquid heavy metalfast reactors that use a much denser and more viscous coolant in the reactor core. An investigation ofthe flow-induced vibration in these particular conditions is therefore essential. In this paper, we describean analysis technique to evaluate flow-induced vibration of nuclear reactor fuel pins subjected to a tur-bulent axial flow of heavy metal. We deal with a single fuel pin mockup designed for the lead–bismuth

eutectic (LBE) cooled MYRRHA reactor which is subjected to similar flow conditions as in the reactor core.Our analysis is based on operational modal analysis (OMA) techniques. We show that the accuracy andprecision of our OMA technique is higher compared to traditional methods and that it allows evaluatingthe evolution of modal parameters in operational conditions. We also demonstrate the possible onset ofa fluid-elastic instability by tracking the modal parameters with increasing flow velocity.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Investigating flow-induced vibration of reactor componentsuch as fuel rods is essential in view of ensuring safe operation ofuclear plants (Pettigrew et al., 1998). In order to understand ando assess the possible safety hazards due to flow-induced vibra-ion of the fuel rods in realistic operating conditions of the reactor,e propose a method based on operational modal analysis (OMA)

echniques to analyze those vibrations (Fu and He, 2001; Guillaumet al., 2003; Ewins, 2000). To illustrate the potential of our methode have measured the response of a single fuel rod in a dedicated

∗ Corresponding author at: Vrije Universiteit Brussel, Brussels Photonics TeamB-Phot), Pleinlaan 2, 1050 Elsene, Belgium.

E-mail address: bdepauw@vub.ac.be (B. De Pauw).

ttp://dx.doi.org/10.1016/j.nucengdes.2014.11.040029-5493/© 2014 Elsevier B.V. All rights reserved.

experimental setup and we show that our technique yields higherprecision and accuracy than traditional direct or spectrum basedmethods. By increasing the flow velocity we were able to cre-ate similar turbulence conditions as in the actual lead–bismutheutectic (LBE) cooled MYRRHA reactor (De Bruyn et al., 2011; AïtAbderrahim et al., 2012). We estimated the modal parameters foreach flow condition and we compared those in order to assess theactual effect on the fuel pin. The high precision and accuracy of ourmethod allows for a reliable and unbiased modal identification ofthe MYRRHA fuel assembly. This identification can then be com-pared with the design specifications in order to draw conclusionsabout safety.

Since our method can be applied in operational environments,i.e. without need for any other excitation, features can be revealedthat would go undetected otherwise such as the onset of an insta-bility as discussed in Section 4.3.

20 B. De Pauw et al. / Nuclear Engineerin

Nomenclature

d channel diameterf0 fundamental frequency� damping ratio mode shapeD fuel pin diameterRe Reynolds numberU flow speedU/Df0 reduced velocitydh hydraulic diameterCm dynamic amplification factor

itTatdfyrpsit

2

c(rrw

mf mass of displaced fluid

The remainder of this paper is structured as follows. We proceedn the next section (Section 2) with a discussion of the experimen-al facility and of the used materials and measurement techniques.he section explains the experimental setup as well as the supportsnd construction of the fuel rod. Section 3 then briefly introduceshe theoretical models and vibration analysis techniques. We thenevote a section to the vibrational analysis of the fuel rod using datarom a laser Doppler vibrometer (LDV) system. The vibrational anal-sis includes four major parts. The first part is an estimation of theeduced amplitude as a function of the flow velocity to situate theroblem in the field of flow-induced vibration, followed by a discus-ion of the operational modal analysis techniques used to identifyndividual vibrational modes of the rod. The (critical) damping andhe added mass of the fuel rod are the subjects of the last two parts.

. Materials and measurement techniques

We have manufactured fuel rod mockups from stainless steelylindrical tubes with a diameter of 6 mm and a length of 1400 mm

Fig. 2). We chose these dimensions to mimic those of the actualods that will be used during operation of MYRRHA, the nucleareactor specifically targeted in our research. The cylindrical tubesere filled with a piece of solid lead–bismuth (5) supported by

Fig. 1. Concept drawing

Fig. 2. Design of construct

g and Design 284 (2015) 19–26

hollow PVC spacers (3) (Fig. 2). A spring (6) was inserted to simulatepressure applied to the lead–bismuth. Once filled the hollow tubewas sealed with dedicated tips. These tips fix the rods in the fuelassembly with a slit and key mechanism and hence they determinethe boundary conditions of the system.

To produce similar flow conditions as in the reactor we haveconstructed a dedicated water loop as shown in Fig. 1. The fuelpins were installed in that experimental set-up and their vibrationswere monitored using two LDV systems. We have shown in DePauw et al. (2013, 2012) that the LDV technique yields a superiormeasurement signal-to-noise ratio for this application.

3. Theoretical background and analysis methods

3.1. Flow-induced vibration correlation

The fuel rods in the MYRRHA reactor will experience a nom-inal flow of approximately 2 m/s of LBE at 300◦C along the axialdirection from bottom to top (i.e. left to right in Fig. 2). Such flowconditions yield a turbulent regime and can excite flow-inducedvibrations (FIV) of the fuel rods and of the entire sub-assembly asa whole. Since we have axial flow and turbulence as the main exci-tation mechanism we can use the simulation criterion establishedby Burgreen (Prakash et al., 2011). This criterion allows calculatingthe required flow speed and temperature of any fluid, e.g. water, tosimulate the same flow conditions (and thus vibration amplitudes)as those that one would obtain using LBE. In terms of the velocityratio, that criterion becomes,

UwULBE

= �0.5LBE �−0.33

LBE E−0.33warm M0.165

warm

�0.5w �−0.33

w E−0.33exp M0.165

exp(1)

where Uw is the water flow speed and ULBE is the LBE flow speed.At a given temperature, �LBE and �w are the respective densitieswhile �LBE and �w are the dynamic viscosities. Finally Ewarm/Eexp

and Mwarm/Mexp are the ratios at reactor operation (‘warm’) andtest temperature (‘exp’) of the Young moduli and the mass per unitlength, respectively. The values for the Burgreen ratios that applyto our tests are summarized in Fig. 3.

of the test facility.

ed fuel pin mock-up.

B. De Pauw et al. / Nuclear Engineerin

2

TrTRSdlap

R

wf

3

sLgwrao(aplstffiaod

3

biAedivmtTmt

f = 12�

√ks

m + Cmmf(6)

Fig. 3. The experimental flow speed as a function of the Burgreen ratio.

Our test temperature was fixed around room temperature at5 ± 2 ◦C and therefore

UwULBE

= 2.6 (2)

he resulting flow speed of water at 25 ◦C to simulate similar fuelod excitation conditions for an LBE flow of 2 m/s at 300 ◦C is 5.2 m/s.his flow speed yields a Reynolds number (Re) of 2.4 × 105. Theeynolds number for an LBE flow at 2 m/s at 300 ◦C is 4.6 × 105.ince we are working in a fully developed turbulent flow region, thisifference is nevertheless of little importance as the (local) turbu-

ence intensity is very similar. The experiments were conducted at reduced velocity in the range of 80–220 and at nearly atmosphericressure. This reduced velocity is dimensionless and defined as,

educed velocity ≡ U

Df0(3)

here U is the mean fluid flow, D the diameter of the fuel rod and0 the fundamental natural frequency of the fuel pin.

.2. Reduced amplitude estimation

The vibration velocities in the transverse direction were mea-ured at 12 individual positions along the fuel pin using the twoDV systems. The displacements were estimated using direct inte-ration and noise filtering. The vibration amplitudes of the fuel pinere derived from the displacements by calculating the average

ms values of a sliding window subset of the data. These vibrationmplitudes are typically normalized using the characteristic lengthf the structure under test, in this case the diameter of the fuel pins6 mm). Absolute values of the vibration amplitude at a certain timere not meaningful since the excitation mechanism is a randomhenomenon. Time-averaged values of the amplitude are nonethe-

ess useful but they still do not describe the entire random vibrationignal. Given the random nature of the turbulent forces and the cen-ral limit theorem, we expect that the average displacements of theuel rod adhere to a normal or Gaussian distribution. We thereforet a Gaussian distribution through the histogram of the vibrationmplitude. To obtain the reduced amplitude the standard deviationf the resulting distribution should again be normalized with theiameter of the fuel pins (6 mm).

.3. Modal parameter estimation

Identifying the individual vibration modes of the fuel rod cane informative in order to study the nature of the turbulence

nduced vibrations under the prevailing experimental conditions.s demonstrated in the previous section, it should be possible toxtract information about the flow-induced vibrations from theisplacement data obtained by direct integration. However direct

ntegration does not provide information about the individualibration modes or about the response between the measure-ent points. Furthermore and whilst traditional direct or FFT based

echniques for nuclear fuel pins (and assemblies thereof) (IAEA-WGFPT, 2005; Prabhakaran et al., 2012) – such as a peak detectionethod combined with modal bandwidth or modal magnifica-

ion factor method – can be used to estimate modal parameters

g and Design 284 (2015) 19–26 21

(Akishita et al., 2005), the accuracy of such techniques remainsdebatable. To cope with these shortcomings we used an adaptedoperational modal analysis technique (see also Fu and He, 2001;Guillaume et al., 2003; Ewins, 2000) in order to identify the systemwith high accuracy and precision under real operational conditions.We describe this technique below.

In modal analysis it is assumed that the force F(s) and displace-ment X(s) are related by:

X = H ∗ F (4)

with

H(s) =Nm∑m=1

m Tms − �m

+ ∗m

∗Tm

s − �∗m

(5)

and where m is called the ‘mode shape’ of mode m and �m thepoles of the system. This model holds for linear and time-invariantsystems. The estimation procedure follows the four steps describedbelow, with the first including preprocessing of the data and thethree following steps pertaining to the actual modal parameterestimation.

PreprocessingWe obtain the experimental frequency response function

H(ωk) within the selected bandwidth from the averaged Fouriertransforms of the correlation functions (so-called corrolelogramapproach) after segmenting the data. An exponential window isapplied to the correlation functions (readily obtained from theexperimental data) with a decay chosen so that the window is at1% of the amplitude at the end of the segment.

LSCF1 estimation of poles and reference vectors (Guillaumeet al., 2003)

We fit the spectra with a least-squares algorithm in the complexZ domain. This process is repeated for every selected model order(=number of poles in modal model).

Estimation of mode shapes for all model ordersWe use a singular value decomposition (SVD) of the residues of

the solution to evaluate the mode shapes and participation fac-tors. If the system under test is excited by ambient forces only(so called operational condition) the added value of obtaining theparticipation factors is nevertheless not that high.

Processing of estimated resultsFinally, we remove all unstable poles from the solution set and

we group the stable poles of different model order with correspond-ing modevectors in a cluster. The averaged values of the clustersresult in the final value of the modal parameter estimation for thefuel pin given the flow conditions.

3.4. Added mass coefficient calculation

Calculating the added mass and dynamic amplification factorunder different flow conditions is a traditional method to evaluatethe effect of changing the flow conditions on the fuel pin vibrations.Calculations of the added mass rely on accurate estimations of themodal parameters. The idea of added mass is especially interest-ing when fuel rods are combined in assemblies. We calculated theadded mass coefficient and the dynamic amplification factor usingthe ratio of the oscillating frequencies in air, stagnant water andfluid flow. This method is described in Someya et al. (2010). Theoscillating frequency of the fuel rod f in fluid flow is:

1 ‘Least-squares complex frequency’ estimator based on the commercially knownLMS PolyMax

2 eering and Design 284 (2015) 19–26

wfaatci

T

m

Tof

a

C

4

4

icnwebtpwa

ruuoae

Fw

2 B. De Pauw et al. / Nuclear Engin

here ks and m are the structural stiffness and the mass of theuel, rod respectively. Cmmf is the added mass with Cm the dynamicmplification factor. This factor equals 1 when the fluid is stagnantnd the oscillation amplitude is small. The added mass correspondso the mass of the displaced fluid mf, which can be estimated byomparing the oscillating frequencies of the fuel rod in air fair andn stagnant fluid fu=0:

f 2air

f 2u=0

= m + mfm

(7)

he mass of the displaced fluid becomes:

f =(f 2air

f 2u=0

− 1

)m (8)

he added mass coefficient can now be estimated using the ratiof the natural frequency in stagnant fluid fv=0 with the oscillatingrequency in fluid flow f,

f 2u=0

f 2= m + Cmmf

m + mf(9)

nd thus the amplification factor becomes

M =[(

f 2u=0

f 2− 1

)m + mfmf

]+ 1 (10)

. Results and discussion

.1. Reduced amplitude estimation

Two excitation mechanisms can be distinguished in the stud-ed application. The first mechanism, known as near-field noise,omes from the drag force which forms a turbulent boundary layerear the fuel pin surface. The second mechanism arises from theall pressure fluctuations and far-field noise (the latter includes

.g. bends in the loop, acoustic wave, etc.). These larger scale tur-ulences occur in the fluid flow in random time intervals. Shouldhese time intervals coincide with the eigen-frequencies of the fuelin, resonant behavior would occur and the probability for failureould increase. However, this is not the case in nominal conditions

nd the main excitation mechanism remains the near-field noise.Fig. 4 shows the variation of the vibration amplitude with the

educed velocity. The vibration amplitude has been normalizedsing the outer diameter of the fuel rods D equal to 6 mm. This fig-

re shows that the averaged amplitudes increase with flow speedwing to the increase in turbulence intensity (i.e. turbulent bound-ry layer pressure fluctuations or near-field noise). Solving thequation of motion for this system is not straight forward. We

ig. 5. Left: A histogram based analysis of the displacement values for different flow condiith the standard deviation from the histograms.

Fig. 4. Experimental reduced amplitude as a function of flow speed.

therefore follow the proposed model in Chen (1985) and Paidoussis(2004). In this model an approximation to the pressure fields isused to derive the relation between the (reduced) vibration ampli-tude and the flow velocity. For a Strouhal number smaller than0.2 (like in this paper), it is shown that the vibration amplitude isproportional to U1.5. This model only takes near-field noise or theturbulent boundary layer into account. The agreement between thetheory and the experiments is good (R2 values higher than 97% withrms errors smaller than 0.004), which indicates the adequacy of themodel. Below a reduced velocity of around 140 (or 5 m/s), we canexpect the model to underestimate the vibration amplitude sinceat lower flow velocities the near-field noise becomes smaller andis of the same order of magnitude as far-field noise and/or struc-tural borne vibration. This suggests that near-field turbulences areindeed the primary excitation mechanism for the fuel pins.

Based on the explanation in Section 3.2, we can give a morecomplete description of the vibration using the distribution of theamplitude. The histograms of the displacement are shown in Fig. 5for the 3 different flow speeds and one measurement point alongthe fuel rod near the center (at 77 cm). At this mid-point the dis-placements are maximal since the first mode is dominant. If wecompare different histograms, it is clear that the standard devi-ation (Fig. 5) agrees well with the time-averaged calculation ofthe vibration amplitude (Fig. 4). Differences can be explained byhow averaging happens in both methods: time-averaging is lessrobust than fitting a normal distribution. The increase in standarddeviation is proportional to the velocity to the power 1.5 as before.

These vibration amplitude values should be checked with thedesign constraints of the nuclear reactor fuel assembly. If these val-ues do not meet the requirements, efforts should be made to alterthe vibration amplitudes.

Another remark here is that for positions near the extremities

of the fuel pin these distributions will not adhere to a Gaussianmodel. This results from the fact that, due to the fixation technique,

tions. Right: Comparison of the reduced amplitude from the time-averaged analysis

B. De Pauw et al. / Nuclear Engineering and Design 284 (2015) 19–26 23

Table 1Eigenfrequencies and damping ratios for the first 6 modes and for 6 different flow conditions.

Flow (m/s) Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

3 ω [Hz] 5.53 ± 0.04 21.82 ± 0.04 45.91 ± 0.03 81.1 ± 0.1 125.6 ± 0.3 172.2 ± 0.1� [%] 11.1 ± 1.9 3.0 ± 0.2 1.4 ± 0.1 1.7 ± 0.2 2.0 ± 0.2 1.4 ± 0.1

3.75 ω [Hz] 5.61 ± 0.04 21.80 ± 0.03 45.82 ± 0.04 80.67 ± 0.05 125.5 ± 0.1 172.2 ± 0.2� [%] 12.6 ± 2.3 3.3 ± 0.2 1.6 ± 0.1 1.8 ± 0.1 2.1 ± 0.30 1.7 ± 0.3

4.5 ω [Hz] 5.68 ± 0.05 21.85 ± 0.05 45.74 ± 0.05 80.70 ± 0.05 124.8 ± 0.1 170.9 ± 0.2� [%] 12.7 ± 2.3 3.8 ± 0.2 1.8 ± 0.2 1.8 ± 0.1 2.3 ± 0.1 2.3 ± 0.20

5.25 ω [Hz] 5.68 ± 0.07 21.87 ± 0.06 45.72 ± 0.02 80.6 ± 0.1 124.7 ± 0.1 171.1 ± 0.2� [%] 12.9 ± 2.0 4.2 ± 0.2 2.0 ± 0.1 1.8 ± 0.1 2.3 ± 0.2 2.7 ± 0.1

6 ω [Hz] 5.78 ± 0.06 22.12 ± 0.04 45.73 ± 0.02 80.48 ± 0.03 123.9 ± 0.1 170.5 ± 0.2� [%] 12.7 ± 1.8 3.2 ± 0.8 2.2 ± 0.1 2.5 ± 0.2 2.6 ± 0.1 2.2 ± 0.2

6.75 ω [Hz] 6.00 ± 0.01 21.90 ± 0.09 45.62 ± 0.02 80.27 ± 0.02 123.6 ± 0.1 170.2 ± 0.1� [%] 12.2 ± 0.9 4.9 ± 0.2 2.2 ± 0.2 2.3 ± 0.1 2.2 ± 0.2 2.5 ± 0.1

Fta

ta

4

emflflmtt(sat(

ig. 6. Excerpt of time-domain data near center position. The random nature of theurbulence excitation can cause large fluctuations in signal level. These temporarynd localized fluctuations deform the mode shape.

he fuel pins are allowed limited movement at the extremities (seelso Fig. 2).

.2. Modal analysis results

We measured the transverse velocities in 12 points over thentire length of the fuel rod for 6 different flow speeds. With ourethod we can identify the first 6 modes of the system in each

ow condition (see Figs. 7 and 8). This identification took place atow speeds close to the Burgreen ratio of 1 without any additionaleans of excitation. The fuel pin vibration was solely the result of

he fluid flow. For each mode 3 modal parameters are estimated:he eigen-frequency (ω), the damping ratio (�) and the mode shape ). The eigenfrequency and damping ratio for the first 6 modes areummarized in Table 1. The mode shapes resulting from the modal

nalysis algorithm are shown in Fig. 7. For the lowest order modehe shape seems to differ somewhat from what one would expecti.e. classical beam modes). This feature results from the unsteady

Fig. 7. Modes of vibration of the analyzed fuel pin.

Fig. 8. A stabilization diagram resulting from the modal analysis algorithm. The first6 modes are identified as stable poles.

manner with which the fuel pins are excited by turbulent pressurefluctuations and from the internal structure of the fuel pins. Therandom nature of the turbulence and hence of the load on the fuelpin are illustrated with an excerpt of the time-domain data in Fig. 6.

During pre-processing the corrolelogram approach reduces thenoise when obtaining the power spectrum of the fuel pin responses.The exponential window used in this approach has a decay factorof about 0.23 in order to retain 1% of the amplitude at the end ofthe window. After pre-processing the modal parameters are esti-mated using a least-squares curve fitting technique based on themodal model. This fit is performed for every model order. The pre-set number of maximum evaluated model order should be chosenin accordance with the number of modes one expects. The presenceof noise can lead to modeling errors and hence the maximum modelorder should be chosen sufficiently high to allow for a general betterfit. However, an excessive model order introduces computationalpoles. In our setup we evaluated the bandwidth from 0.1 Hz up to200 Hz with a maximum model order of 32. From this estimationthe mode shapes (i.e. operational deflection shapes) can be deter-mined based on a singular value decomposition of the residues.Finally we evaluate the clusters and eliminate all unstable poles.The remaining set of solutions together with the raw data and finalcurve fit is called a stabilization diagram. Such a diagram is shownin Fig. 8. The averaged values of the remaining solutions yield thefinal values of the modal parameter estimation. These values arelisted for the first 6 modes and for 6 different flow conditions inTable 1.

We can compare the results of the (operational) modal analy-sis technique to traditional techniques and evaluate its accuracyand precision. To do so we compare the modal analysis techniquewith a traditional spectrum based local maxima technique to esti-

mate the eigenfrequencies of the system and with a bandwidthmethod to estimate the damping ratio. In order to evaluate theestimation of the modal parameters only we compare both tech-niques after exactly the same pre-processing and data segmenting.

24 B. De Pauw et al. / Nuclear Engineering and Design 284 (2015) 19–26

F maxima and with bandwidth method (red) for one flow condition. The modal analysist s to color in this figure legend, the reader is referred to the web version of this article.)

MwtwoudenHttubtddbtatt

tl

tTtpmftmwctd

aa

ig. 9. Comparison of our modal analysis technique (blue) with a traditional localechniques are more accurate and more precise. (For interpretation of the reference

oreover, the results for the local maxima and bandwidth methodere further improved by averaging spectral lines since these

echniques are very sensitive to noise. In Fig. 9 we show the resultsith the expectation value and uncertainty of the estimated res-

nance frequencies and damping ratios for both techniques. Thencertainty on the estimated values is calculated from the standardeviation of the estimation results for every data segment. Thexpectation values of the resonance frequencies for both tech-iques are statistically equivalent (i.e. within the uncertainty).owever, when comparing the uncertainty in Fig. 9, the error of

he local maxima technique is an order of magnitude larger thanhe error of our method. For the damping ratio estimation the val-es are not statistically equivalent. For the higher order modes, theandwidth method damping ratio estimate tends to be lower thanhat obtained with our technique. This difference stems from theependence of the resonance peak on the amplitude (needed toetermine the −3 dB points). The lower signal-to-noise ratio com-ined with the averaging of spectral lines results in overestimatinghe amplitude of the resonance peak. This feature contributes to theccuracy of the bandwidth based damping estimates. The uncer-ainty on the damping ratios is about 3.5 times smaller for ourechnique.

We therefore conclude that our modal analysis technique fea-ures an improved accuracy and precision compared to traditionalocal maxima and bandwidth methods.

The modal analysis method can be used to study the effect ofhe increased flow speed on the modal parameters of the system.able 1 and Sections 4.3 and 4.4 show the effect of flow speed onhe damped frequency of the fundamental mode. Following theroposed model in Chen (1985) and Paidoussis (2004) the funda-ental frequency should decrease with increasing velocity for a

uel pin clamped at both ends. This is due to the centrifugal forcehat acts as a compressive axial force on the cylinder. The experi-

ental data however shows an increase in frequency with velocityhile for the higher order modes the frequency decreases. In this

ase the limited mobility of the extremities of the fuel pin makeshe centrifugal force acts more as a follower force on the normal

rag force.

The modal parameters can be compared with those obtained inir and in stagnant water. This comparison is shown in Fig. 10 as

normalized overlay of the resulting power spectra under these

Fig. 10. Comparison of the normalized power spectra for air, stagnant water andflow conditions. Filling the test section with water and/or initiating flow increasesdamping while decreasing the eigenfrequency of e.g. the third mode.

three conditions. It is clear from the figure that the introductionof the water yields a so called added mass effect as seen from thedecrease of the eigen-frequency. This effect is further influencedwith the introduction of the fluid flow. Also, viscous damping anddrag contribute to the total damping of the fuel pin when subjectedto water and/or fluid flow. This damping and added mass effect as afunction of mode number and mean flow velocity are the subjectsof the following two sections.

4.3. Damping analysis and critical points estimation

The (normal) modal damping ratio � can be written as � =�v + �d + �c + �s where �v is the damping from viscous effects, �dthe damping induced by normal drag, �c the damping associatedwith the Coriolis force and �s the structural damping. The dampingvalues resulting from the modal analysis should mainly be viscousdamping and damping by the normal drag force. Structural damp-ing is usually much smaller and can be neglected while the Coriolisforce should be gyroscopic for the fuel pins clamped at both endsand should therefore not dissipate energy. Fig. 11 shows the damp-ing values for the first mode for 6 different flow conditions and

stagnant water. In air, the structural damping for the first mode isclose to 0.1%. Adding water to the systems increases this value bya factor 3 because of viscous damping. The introduction of waterflow further increases the damping to 11.1% for the slowest flow

B. De Pauw et al. / Nuclear Engineerin

Fo

sted(Cdic>eCbw

FmtmDa

ig. 11. Damping ratio of the first vibration mode as function flow velocity. By meansf cubic extrapolation a possible critical velocity can be estimated at 9.5 m/s.

peed tested. The damping ratio increases with the flow speed upo 5.25 m/s, after which it decreases again. For pins clamped at bothnds, the dominant fluid force is the centrifugal force and hence theamping should increase with flow velocity because of increasednormal) drag forces. Given the little mobility of the pin ends, theoriolis force may not be gyroscopic and hence contribute to theamping. Above 5.25 m/s the fuel pin experiences an impending

nstability such as divergence or flutter. Whilst one could expect theritical flow velocities leading to instabilities to be much larger (e.g.30 m/s), the internals of the fuel pin can cause nonlinear damping

ffects that decrease the damping and the critical flow velocities.ubic extrapolation of the data indicates that a fluid-elastic insta-ility can take place at a critical velocity just below 9.6 m/s (butith an uncertainty of the order of 2 m/s).

ig. 12. Top: The added mass or mass of displaced fluid compared to the fuel pinass as a function of vibration. middle: The added mass as a function of vibra-

ion frequency for different flow conditions. Except for the first mode, the addedass increases with increasing flow speed i.e. more water is dragged along. Bottom:ynamic amplification factor Cm as a function of vibration frequency. The inversemplification for the first mode is clearly visible here.

g and Design 284 (2015) 19–26 25

4.4. Added mass evaluation

The added mass values are an indication for the amount of fluidthat the structure seemingly drags along while vibrating. The areaunder the mode shapes of the fuel rod allow estimating how theadded mass values change as a function of mode number. In thecase of a uniform and ideally clamped fuel pin with a length of1.4 m the trend equals 1.4/2n. One should nevertheless be carefulwhen considering the amplitude of each vibration mode. In thefirst two graphs of Fig. 12 we show the mass of the displaced fluid(calculated from air versus quiescent fluid vibration modes) andthe added mass (calculated from quiescent fluid verses fluid flowvibration modes) for the first 4 modes.

We expect that mf and Cmmf should decrease with increasingfrequency (or mode number). Consequently the dynamic amplifica-tion factor Cm will increase with increasing frequency for relativelylow displaced fluid masses and small frequency shifts. This increaseis shown in the bottom graph of Fig. 12. The trend should continueup to a critical frequency related to the Mach number, which inthis case will be far beyond the analyzed frequency range. Sincethe length of the fuel pin is much larger than its diameter, a 2-Dapproximation is sufficiently accurate. The added mass values ofthe first mode decrease with velocity, while they increase for thehigher order modes. This illustrates the effect that the small allowedmobility of the fuel pin extremities makes the centrifugal force actas a follower force of the normal drag forces, effectively decreasingthe drag for the first mode.

This concept of added mass is crucial when adding fuel pins tothe setup since small pitch lengths lead to high added mass effects.The study of multiple fuel pins is part of future research.

5. Conclusions

We have described a new method to analyze the flow-inducedvibration of a nuclear fuel pin in conditions that mimic thoseencountered in the fuel assembly of the future MYRRHA reactor.The modal analysis technique allowed investigating the flow-induced vibration in operational conditions without any othermeans of excitation with higher accuracy and precision comparedto traditional techniques. The reduced amplitude of the vibrationswas measured to lie between 2% - 10%, and the trend as a functionof flow velocity agrees well with the theoretical model.

We have shown that the random nature of the turbulence exci-tation and the internal structure of the fuel pins yield mode shapesthat differ from the classical ones. Using the estimated modalparameters we studied the damping and added mass as a functionof mean flow velocity. This indicated that due to the support of thefuel pin ends and the resulting displacement, the fuel pin exhibitsa particular behavior with increasing velocity and a fluid-elasticinstability could possibly arise. This effect is also confirmed by theadded mass calculations (based on the frequency shifts). These con-clusions may affect the design of the fuel pins and fuel assembly.

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