MASTER - Digital Library/67531/metadc173319/m2/1/high... · DYNAMIC CHARACTERISTICS OF HEAT EXCHANGER TUBES ... The tubes comprising the tube bundle of a shell and tube heat exchanger

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AIJL--04-39

DE84 014844

ARGONNE NATIONAL LABORATORY9700 South Cass Avenue

Argonne, Illinois 60439

DYNAMIC CHARACTERISTICS OF HEAT EXCHANGER TUBESVIBRATING IN A TUBE SUPPORT PLATE INACTIVE MODE

by

J. A. Jendrzejczyk

Components Technology Division

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States

Government. Neither the United States Government nor any agency thereof, nor any of their

employees, makes any warranty, express or implied, or assumes any legal liability or responsi-

bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or

process disclosed, or represents that its use would not infringe privately owned rights. Refer-

ence herein to any specific commercial product, process, or service by trade name, trademark,

manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-

mendation, or favoring by the United States Government or any agency thereof. The views

and opinions of authors expressed herein do not necessarily state or reflect those of the

United States Government or any agency thereof.

June 1984

MASTER

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3

TABLE OF CONTENTS

Page

ABSTRACT ............................................................. 9

I. INTRODUCTION ................................................... 9

II. TEST DESCRIPTION ............................................... 11

A. Test Hardware.............................................. 11

B. Test Parameters and Conditions...........,................ 17

C. Test Procedures................. "..... ......... . ...... . 18

D. Data Acquisition and Analysis............................. 21

III. RESULTS FOR A TUBE WITHOUT A CENTER TSP........................ 24

IV. TSP ACTIVE M ODE................................................ 27

V. TSP INACTIVE MODE - A SERIES ................................... 29

A) Representative Results: Test A4 in Water.................. 31

B) Calculation of Damping and Frequency....................... 34

C) Comparison of Measured and Calculated Damping andFrequency." " " " " " " " "........""...... . .. . . .. . .. . . .. ... ... ., 38

VI. TSP INACTIVE MODE - B SERIES................................... 50

A. Theoretical Calculations................................... 50

B. Measured Quantities........................... ............ .61

VII. CONCLUDING REMARKS............................................. 64

ACKNOWLEDGMENTS ...................................................... 73

REFERENCES ........................................................... 74

APPENDIX: ANALYSES .... ...................... ...... .. ............. 75

4

FIGURES

Page

1 Tube Vibration in TSP Inactive and Active Modes.............. 12

2 Test Fixture................................................. 13

3 Photograph of Test Fixture.................................. 14

4 Closeup Photograph of Center TSP and Base.................... 15

5 Photograph of TSPs and MainBlock............................ 16

6 Tube Displacement in Air for a 0.061 cm (0.024 in.)Diametral Clearance and a 2.54 cm (1.0 in.) Wide TSP......... 19

7 Tube Displacement Response in Water for a 0.061 cm(0.024 in.) Diametral Clearance and a 2.54 cm (1.0 in.)Wide TSP ..................................................... 20

8 Kinematic Viscosity Plotted as a Function of Temperature

for #150 Oil1................................................. 22

9 Data Acquisition and Analysis................................ 23

10 Damping of a Tube in Air and Liquid for a TSP Active Mode.... 30

11 RMS Tube Displacements and Pressure for Test A4 [0.061 cm(0.024 in.) Diametral Clearance] with a 1.27 cm(0.5 in.) TSP ................................................ 32

12 Force-Displacement Response, Displacement and Pressure TimeHistories for Test A4 with a 1.27 cm (0.5 in.) TSP........... 33

13 Tube Displacement-Force Response for Test A4 in Water........ 35

14 Damping Ratio as a Function of Force for Test A4............. 36

15 Frequency as a Function of Force for Test A4............'.... 37

16 Calculated Damping Ratio as a Function of Baffle Width....... '9

17 Calculated Frequency as a Function of Baffle Width........... 40

18 Calculated Damping Ratio as a Function of Tube-BaffleDiametral Clearance... ....................................... 41

19 Calculated Frequency as a Function of Tibe-Baffle DiametralClearance0.................................................... 42

20 Measured Damping Ratio vs. Baffle Width in Water with VariousDiametral Clearances for a Force of 0.498 N (0.112 lb)....... 43

5

21 Theoretical and Measured Damping Ratios with a 0.0203 cm

(0.008 in.) Diametral Clearance.............................. 44





24 Theoretical and Measured Damping Ratios with a 0.061 cm(0.024 in.) Diametral Clearance.............................. 47



26 Sensitivity of Damping Ratio to Variations in DiametralClearance at Small Nominal Clearances........................ 49

27 Sensitivity of Damping Ratio to Variations in DiametralClearance at Large Nominal Clearances........................ 51

28 Measured Frequency vs. Baffle Width in Water for VariousDiametral Clearances and a Force of 0.0498 N (0.112 lb)...... 52

29 Theoretical and Measured Frequencies with a 0.0203 cm


30 Theoretical and Measured Frequencies with a 0.033 cm(0.013 in.) Diametral Clearance.............................. 54




34 Displacement vs. Frequency Response for Test A3 with a1.27 cm (0.5 in.) TSP in 48*C (118'F) Oi .................... 58

35 Theoretical Damping Ratios as a Function of Baffle Widthwith a 0.0397 cm (0.0156 in.) Diametral Clearance in(1) Water at 21.1 C (70*F), (2) Oil at 65.5*C (150*F),(3) Oil at 48*C (120 F), (4) Oil at 26.7*C (80*F), and(5) Oil at 21 C (70*F) ....................................... 59

36 Theoretical Damping Ratios as a Function of Baffl Iidthwith a 0.0794 cm (0.0313 in.) Diametral Clearance if(1) Water at 21.1 C (7 0 *F), (2) Oil at 65.5*C (15 0 F),(3) Oil at 48*C (120*F), (4) Oil at 26.7 C (80 F), and(5) Oil at 210C (70 *F)....................................... 60

6/

37 Frequency vs. Baffle Width for Test B2 in 65.5*C (150*F)Oil................... ........................... , ........ .. 62

38 Tube Displacement as a Function of Excitati _ Force forTest B2 with a 0.65 cm (0.25 in.) TSP........................ 63

39 Damping Ratio as a Fugct on of Baffle Width or Test Bi inWater, v = 1.01 x 10 m /s (1.57 x 10 in. /sec)........... 65

40 Damping Ratio as a Function of Baffle Width for Test Bi in65.5*C (150*F) Oil, v = 4.46 x 10- m2/s (0.0691 in. 2 /sec)... 66

41 Damping Ratio as a Function of Bgffle Width for Teit Bl in48*C (1200F) Oil, v = 9.85 x 10 m /s (0.1526 in. /sec)..... 67

42 Displacement vs. Frequency for Test Bl with a 1.27 cm(0.5 in.) in 26.7 C (80 F) Oil............................... 68

43 Damping Ratios as a Fgnc ion of Baffle idth2 for Test B2 inWater, v = 1.01 x 10~ m /s (1.57 x 10 in.2/sec)........... 69

44 Damping Ratios as a Function of Ba fl Width for Test B2 in65.50C (150*F) Oil, v = 4.46 x 10~ m /s (0.0691 in. /sec)... 70

45 Damping Ratio as a Function of B ff1e Width for Te t B2 in48C (120*F) Oil, V = 9.85 x 10~ m Is (0.1526 in. /sec)..... 71

46 Damping Ratio as a Function of Ba fl Width for Tes B2 in26.7*C (80*F) Oil, v = 2.48 x 10~ m /s (0.3845 in. /sec).... 72

A.1 A Fixed-Fixed Tube with a Baffle Plate Support............... 76

A.2 Real and Imaginary Values of H for a Cylinder Vibrating inan Infinite Fluid... ........................................ 80

A.3 Added Mass and Damping Parameters, Re[H'] and Im[H']

(Ref. A.2)........o........................................... 81

9

DYNAMIC CHARACTERISTICS OF HEAT EXCHANGER TUBESVIBRATING IN A TUBE SUPPORT PLATE INACTIVE MODE

by

J. A. Jendrzejczyk

ABSTRACT

Tubes in shell-and-tube heat exchangers, including nuclear

plant steam generators, derive their support from longitudinally

positioned tube support plates (TSPs). Typically there is a

clearance between the tube and TSP hole. Depending on design

and fabrication tolerances, the tube may or may not contact all

of the TSPs. Noncontact results in an inactive TSP which can

lead to detrimental flow induced tube vibrations under certain

conditions dependent on the resulting tube-TSP interaction

dynamics and the fluid excitation forces. The purpose of this

study is to investigate the tube-TSP interaction dynamics.

Results of an experimental study of damping and natural

frequency as functions of tube-TSP diametral clearance and TSP

thickness are reported. Calculated values of damping ratio and

frequency of a tube vibrating within an inactive TSP are also

presented together with a comparison of calculated and

experimental quantities.

I. INTRODUCT ION

The tubes comprising the tube bundle of a shell and tube heat exchanger

derive their support from tube support plates (also called baffle plates;

the designations 'tube support plate' and 'baffle' are used interchangeably

in this report) positioned along their '.ength. The tube support plates

(TSPs) can also serve the function of inducing crossflow, thereby,

increasing turbulence and heat transfer, as they direct the shellside flow

through the exchanger. To allow for ease of fabrication and, in many cases,

to allow differential thermal expensior between the tube bundle and shell,

there is a clearance between the tube and TSP hole. In particular, the TEMA

(Tubular Exchanger Manufacturers Acoociation) Standards states the following

with regard to the drilled holes in baffles and tube support plates [1]:

"Where the maximum unsupported tube length is (36) inches or

less, or for tubes larger in diameter than 1-1/4 inches O.D.,

standard tube holes are drilled 1/32" over the O.D. of the

tubes. Where the unsupported tube length exceeds 36" for tubes

10

1-1/4 inches diameter and smaller, standard tube holes are

drilled 1/64" over the O.D. of the tubes."

TEMA recommendations with regard to baffle or support plate thickness are

dependent on shell diameter and span lengths as indicated in Table 1.

Table 1. TEMA recommended TSP thickness [1]

Tube Support Plate Thickness (in.)

NominalShell

I.D.(in.)

Distance between adjacent full diameter baffles, supports or theunsupported tube length between other type baffles (in.)

Over 24to 36 Inc.

Over 36to 48 Inc.

Over 48to 60 Inc.

8-14 1/8 3/16 1/4 3/8 3/8

15-28 3/16 1/4 3/8 3/8 1/2

29-38 1/4 5/16 3/8 1/2 5/8

39-60 1/4 3/8 1/2 5/8 5/8

Recommended Recommended clearance4-- clearance 1/4ic

1/32 inch 1/64 inch

The TEMA Standards have been prepared for general process plant heat

exchangers employing segmental or multi-segmental TSPs. Typically, nuclear

plant steam generators must be designed for special, usually more severe,

service. In such cases, while rEMA Standards might be consulted for general

guidance, the TSPs are generally of special design and often thicker than

TEMA Standards specify. As examples, Lhe TSP of a PWR (pressurized water

reactor) steam generator is 1.91 cm (0.75 in.) thick while the TSP of a

recently designed LMFBR (liquid metal fast breeder reactor) plant steam

generator is 3.81 cm (1.5 in.) thick.

In the design of heat exchangers to avoid detrimental tube vibrations,

knowledge of the tube's vibrational characteristics ik crucial. The

vibrational characteristics include natural frequencies, modes, and

damping. Traditionally natural frequencies a1ld modes have been computed

based on the assumption that the TSPs are effective in providing support for

24 andUnder

Over 60

11

the tubes. In particular, it is typically assumed that the TSPs act as

knife edge supports forcing a vibration node at the TSP location. This is a

reasonable assumption in many cases, particularly, those designs involving

relatively long spans and small clearances [0.0397 cm (0.0156 in.)].

However, for the case of relatively short spans and large clearances

[0.0794 cm (0.03125 in.)], the probability of a tube "floating" in a TSP

hole is increased. In such a case, the TSP is effectively inactive. Since

a tube will generally respond in its lowest mode, it will tend to vibrate

within the TSP hole as illustrated schematically in Fig. la. A recent tube

failure in a pressurized water reactor plant steam generator has been

attributed to just such a vibration mode.

Obviously, the particular mode of motion under consideration will be

amplitude limited by the size of the radial clearance. However, particu-

larly for the case of dense, viscous fluids, the motion will also be

influenced by the damping associated with the fluid in the finite length

annulus defined by the TSP hole and the tube O.D. This paper reports the

results of an experimental study of the effects of TSP thickness, tube-to-

TSP hole clearance, and excitation force level, on the effective modal

damping factor and natural frequency. The experimental results are also

compared with theoretical predictions.

II. TEST DESCRIPTION

A. Test Hardware

The test fixture (Fig. 2) consists of a steel tube clamped at each end

in what can be considered active tube support plates (TSP). A TSP with

clearance is centered between the two end support plates. All three TSPs,

and the excitation coil and displacement transducers shown schematically in

Fig. 2, are mounte4 to a 3.81 cm (1.5 in.) thick, 10.2 cm (4 in.) wide, by

132 cm (52 in.) long steel base which provides rigidity. Figure 3 is a

photograph of the complete test fixture.

Pairs of displacement transducers (D1 ,D2) and pressure transducers

(P1 ,P2) are orthogonally mounted in the center section of the center TSP to

measure tube to tube support plate displacement and fluid pressure within

the gap. The center tube support plate (Figs. 4 and 5) width can be varied

from 0.318 cm (0.125 in.) to 5.08 cm (2 in.) with sections which are

designed to be installed or removed without disrupting the tube-TSP radial

alignment. The base of the center TSP is designed with adjustments to

accurately locate the plate in the desired position.

Excitation coils provide a sinusoidal force to the 1.6 cm (0.628 in.)

O.D., 0.277 cm (0.109 in.) wall, by 122 cm (48 in.) long steel tube.

Depending on the phase between each set of coils, the first or second tube

bending mode can be excited. First mode would be representative of an

12

INACTIVE T.SSP.

ACTIVE T.S.P.

ACTIVE T.S.P.

(a) TSP Inactive Mode

ACTIVE T.S.P.

ACTIVE T.S.P.

ACTIVE T.S.P.

(b) TSP Active Mode

Fig. 1. Tube Vibration in TSP Inactive and Active Modes.

ACTIVE TUBESUPPORT PLATE

TUBE

PRESSURETRANSDUCER # 2

PRESSURETRANSDUCER #1

DISPLACEMENTTRANSDUCER #2 DISPLACEMENT

TRANSDUCER #4

ACTIVE TUBE INACTIVE TUBESUPPORT PLATE SUPPORT PLATE

DISPLACEMENTTRANSDUCER #1

DISPLACEMENTTRANSDUCER #3

EXCITATION COILS

STEEL BASE

Fig. 2. Test Fixture.

t :. ,

.m4 '9 VM I

22 A 'Y

Fig. 3. Photograph of Test Fixture (ANL Neg. No. 113-84-73).

15

4

4*

4s

r

A

f# r 1

'~5Z

4'

Fig. 4. Closeup Photograph of Center TSP and Base (ANL Neg. No. 113-84-71).

Iy

a

,j a

\ r t, et . r ,

4, , .pe

"U

4i

N'C $SI

r =. *

itlp 27

Fig. 5. Photograph of TSPs and Main Block (ANL Neg. No. 113-84-70).

r

2004i

17

inactive TSP and second mode of an active TSP; see Fig. 1. The excitation

coils provide tube motion in the horizontal (D2,P2) direction. Displacement

transducers 3 and 4 are used to determine tube response when the tube is

excited in the second bending (active TSP) mode.

B. Test Parameters and Conditions

Two series of tests were conducted in air, water, and oil (Table 2).

The dimensions in the A series are typical of those used in steam generators

of light water and breeder reactors. The B series test dimensions were

Table 2. Test matrix

Tube-TSPTest Diametral Clearance TSP Width Fluid

Al 0.0203 cm (0.008 in.) 1.27 cm (1/2 in.) Water, Air1.91 cm (3/4 in.)2.54 cm (1 in.)

3.18 cm (1-1/4 in.)3.81 cm (1-1/2 in.)4.45 cm (1-3/4 in.)5.08 cm (2 in.)

A2 0.033 cm (0.013 in.) Same as Al Water, Air

A3 0.0457 cm (0.018 in.) Same as Al Water, Air, Oil

A4 0.061 cm (0.024 in.) Same as Al Water, Air

AS 0.0813 cm (0.032 in.) Same as Al Water, Air

B1 0.0397 cm (0.0156 in.) 0.0635 cm (1/4 in.) Water, Air, Oil0.953 cm (3/8 in.)1.27 cm (1/2 in.)1.59 cm (5/8 in.)

B2 0.0794 cm (0.0313 in.) 0.318 cm (1/8 in.) Water, Air, Oil0.476 cm (3/16 in.)0.635 cm (1/4 in.)0.794 cm (5/16 in.),0.953 cm (3/8 in.)1.27 cm (1/2 in.)

18

obtained from the Standards of Tubular Exchanger Manufacturers Association,

Mechanical Standards, TEMA Class R Heat Exchangers, Section 5 of Ref. 1, and

are representative of heat exchangers used in the process industries

(Table 1).

C. Test Procedures

Testing at a given diametral clearance commences wit's the installation

of a combination of TSP segments to form a TSP of the desired width.

Alignment of the tube-TSP is accomplished with careful adjustment of the

jackscrews at the support plate base, using displacement transducers D1 and

D2 to measure the position of the tube within the TSP.

After alignment, a small force is applied to the tube. The forcing

frequency is varied to allow measurement of the displacement-frequency

response of the tube in air. The force is then increased and another

displacement-frequency response is measured. Force is increased in small

increments until the tube impacts the TSP. A typical response plot in air

with a 0.061 cm (0.024 in.) diametral clearance is shown in Fig. 6. The

excitation force for each coil pair is listed in Table 3.

Table 3. Excitation force levels for in-air tests (Fig. 6)

Coil Current Forcefor One Coil Pair for One Coil Pair

F1 2.0 Amps 0.0356 N (0.008 lb)

F2 3.0 Amps 0.133 N (0.030 lb)

F3 4.0 Amps 0.285 N (0.640 lb)

F4 5.0 Amps 0.498 N (0.112 lb)

After obtaining a satisfactory response in air, the vessel is filled

with room temperature water [26.6*C (80*F)] and a similar set of

displacement-frequency response curves are plotted (Fig. 7) for the force

levels listed in Table 4. The displacement limiting effect of the TSP is

readily seen in the higher force level responses of Figs. 6 and 7.

The response curves are used to verify the experimental setup

(diametral clearance, angular alignment) and to calculate the modal damping

ratio in air. In water, swept sine response was measured - force levels

similar to those shown in Table 4. At each force level, the frequency was

adjusted to the peak on the displacement-frequency response curve and the

19

F4

0.00254 cr' F3(0.001 in)

F2

FI

40 50 60 70

Hz

Fig. 6. Tube Displacement in Air for a0.061 cm (0.024 in.) DiametralClearance and a 2.54 cm (1.0 in.)Wide TSP.

20

0.00254 cm(0.001 in j

FIO

F9F5 F6

20 30 40 50 60

Hz

Fig. 7. Tube Displacement Response in Water for a0.061 cm (0.024 in.) Diametral Clearanceand a 2.54 cm (1.0 in.) Wide TSP.

21

Table 4. Excitation force levels for in-water tests (Fig. 7)

Coil Current/Coil Pair Force/Coil Pair

F1 3 amps 0.133 N (0.030 ib)

F2 4 amps 0.285 N (0.064 lb)

F3 5 amps 0.49N N (0.112 lb)

F4 6 amps 0.796 N (0.179 lb)

F5 7 amps 1.09 N (0.245 lb)

F6 8 amps 1.48 N (0.332 lb)

F7 9 amps 1.92 N (0.432 lb)

F8 10 amps 2.40 N (0.540 lb)

F9 11 amps 3.02 N (0.680 lb)

F1 0 12 amps 3.52 N (0.792 li)

displacements (D1 ,D2) and pressures (P1 ,P2) recorded and analyzed.

initial series of tests was performed with a 5.08 cm (2 in.) wide TSP.

subsequent tests, TSP segments are removed, and measurements are taken

each of the TSP widths so obtained. The TSP widths which were tested

shown in Table 2.

The

For

for

are

A series of tests was conducted in #150 oil to study the effect of

viscosity on tube-TSP response. Oil temperature thus viscosity, was varied

by either cooling the oil with cold water, or heating it with steam,

circulated through several coils of copper tubing placed in the vessel. A

small propeller-type mixer was used to circulate the oil and assure a

homogeneous temperature. The effect of temperature on the kinematic

viscosity of the oil is shown in Fig. 8.

D. Data Acquisition and Analysis

The data process scheme and swept sine servo force control system is

shown in block diagram form in Fig. 9. Displacements and pre sures are

analyzed with a H.P. 5451C FFT (Fast Fourier Transform) analyzer end are

also recorded on magnetic tape. The majority of the rms (root-mean--square)

displacements and pressures were acquired from on-line data reduction.

22

040 60 80 100 120 140 160

5

2

N

~ -0

1

180

OIL TEMPERATURE,*F

Fig. 8. Kinematic Viscosity Plotted as a Function of Temperaturefor #150 oil.

I I I I I I6

5

4

3

2

0)

I--

U,.

m& m -

DISPLACEMEN

2

3

4

PRESSURE

2

T

IowF.F.T.

ANALYZER

TAPERECORDER

-~ RMS DISPLACEMENTS

-RMS PRESSURE

REAL TIME PLOTS

POWEREXCITATION

COILSAMP

SWEEP SERVO TRACKING

OSCILLATOR SYSTEM

TRACKING X - Y TUBE DISPLACEMENTDISPLACEMENT - FILTERPRESPONSE - FREQUENCY

PLPLOTS2Fi4 2 PLOTS

Fig. 9. Data Acquisition and Analysis.

JI L

m

R

1

24

Modal damping ratios were calculated using the bandwidth method [2] and

averaging the values measured at 0.707 and 0.5 of the maximum displacement

except at large values where only the 0.707 value was used. Force level is

servo controlled by a swept sine data analysis system which maintains a

constant current to the excitation coils as frequency is varied. A feedback

voltage proportional to current is derived by resistor R (see Fig. 9) which

is in series with the excitation coils. This voltage is filtered by a

tracking filter (#1 in Fig. 9) to eliminate any extraneous noise and is used

to control the output of the servo.

III. RESULTS FOR A TUBE WITHOUT A CENTER TSP

Tests of the tube without a center TSP were conducted in air and water

to determine tube natural frequency and damping. The results, summarized in

Table 5, are from measurements taken with displacement transducer 3.

Table 5. Measured values of first mode frequency and dampingin air and water without a center baffle

Force Level Air Water(For Coil Pair) f(f

1i (Hz) 1if (Hz) Cl

0.0356 N 55.7 0.964% 49.3 1.14%(0.008 lb)

0.142 N 55.7 0.941% 49.0 1.10%(0.032 lb)

0.285 N 55.0 0.927% 49.1 1.10%(0.064 lb)

0.498 N 55.3 0.922% 49.1 1.10%(0.112 lb)

0.796 lb 55.7 0.915% 48.9 1.10%(0.179 lb)

1.09 N - - 48.6 1.11%(0.245 lb)

Average 55.5 0.934% 49.0 1.11%

The first mode natural frequency of the tubeby using Eq. (A.4) of the Appendix. With n = 1,

in air may be calculated

25

(A )2 E 0.5

= - 2 [m](1)

The equivalent viscous damping coefficient for the first mode is calculatedwith Eq. (A.5),

(Ca)l = 2i1 mw1 . (2)

In Eqs. (1) and (2), w1 is the first mode circular natural frequency, (A1&)is the first modal constant, R is the tube length between the end TSPs, E isthe bending modulus, I is the bending moment of inertia, m is the mass perunit length, l is the first mode damping ratio, and (Ca)l is the first modeviscous damping coefficient in air. The following values apply to the testset-up:

( X1 )2 = 22.37

X = 121.9 cm (48 in.)

E = 1.857 x 107 N/cm2 (26.83 x 106 lb/in. 2)

I = 2.597 x 10-1 cm4 (6.246 x 10-3 in.4)

m = 8.979 x 10-5 N sect(1.302 x 10-4 lb sec2 ,

cm in.

= 0.00934, average measured value in air (see Table 5).

Using the above values in Eqs. (1) and (2), the calculated value of

wl is 348.72 rad/sec or 55.5 Hz and (Ca)l is 5.848 x 10-4 N sec2

cm(8.481 x 104 lb sec

in.

The first mode natural frequency and damping in water may be calculated

with Eqs. (A.7) and (A.8), respectively, of the Appendix,

(01R)2 El 0,5

= 2 CMMD + m (3)

and

(Ca)1 _1 (MD (4)1 2w(CMMD + m) 2 MMD + m)[

where CM is the added mass coefficient (CM = Re[H], see Fig. A.2 of the

26

Appendix), MD is the mass per unit length of the displaced fluid. Im[H] and

Re[H] are functions of wl, tube radius r, and fluid kinematic viscosity, V

(see Fig. A.2 of the Appendix).

The average first mode frequency in water is 49 Hz, or 308 rad/sec, as

given in Table 5. Using this value with r = 0.798 cm (0.314 in.) and

v = 1.01 x 10-5 m2/s (1.57 x 10-2 in. 2/sec), the nondimensional number

S(= wr2/v) from Fig. A.2 is calculated to be 1.934 x 104. From Fig. A.2 we

obtain

Re[H] = CM = 1

and

-Im[H] = 0.027

Using MD = 1.99 10-5 N-sec 2 /cm 2 (2.89 x 10-5 lb-sec2 /in. 2 ), a value of g =

315.45 rad/sec or f1 = 50.21 Hz and .l = 1.09% is calculated. These values

compare favorably with the measured values given in Table 5.

Equations (3) and (4) were also used to calculate the first mode

natural frequency and damping for the tube vibrating in oil at various

temperatures. The viscosity-temperature relationship of the oil is derived

from Fig. 8. Table 6 is a summary of the values of Re[H] and -Im[H],

determined from Fig. A.2 and used in the calculations.

Table 6. Values of Re[H] and -Im[H] for a tube vibrating in oilas determined from Fig. A.2

Oil Temperature V Re[H] -Im[H]

26.70C (80*F) 2.48 x 10~4 m2/s (0.3845 in.2/sec) 1.06 0.220

480C (118*F) 9.85 x 10-5 m2/s (0.1526 in.2/sec) 1.04 0.108

65.5*C (150*I' 4.45 x 10-5 m2/s (0.069. in.2/sec) 1.01 0.080

The calculated vales of frequency are 48.73 Hz, 48.83 Hz, and

48.99 Hz, and the calculated values of damping ratio are 0.0268, 0.0177, and

0.0154, at oil temperatures of 26.70C (80*F), 48C (118*F) and 65.6*C

(150*F), respectively. It should be noted that the Re[H] values are not

very sensitive to fluid viscosity (Appendix, Fig. A.2) for the range of "S"

tested.

27

Measured first mode natural frequencies and damping at different force

levels are given in Table 7. The measured values compare favorably with the

calculated values, considering the sensitivity of the damping ratio to oil

temperature.

Table 7. Measured values of first mode frequency andwithout a center baffle

damping in oil

26.7 C (80*F) Oil 48 C (118 F) Oil 65.5*C (150*F) Oil

Fre- Fre- Fre-quency, Damping quency, Damping quency, Damping

Force Hz Ratio Hz Ratio Hz Ratio

0.0356 N 49.8 2.70% 50.1 2.05% 49.7 1.94%(0.008 lb)

0.0133 N 49.7 2.79% 50.3 2.18% 49.8 1.66%(0.030 lb)

0.285 N 49.5 2.74% 50.0 2.10% 49.5 1.83%(0.064 lb)

0.498 N 49.6 2.74% 49.9 2.43% 49.5 1.88%(0.112 lb)

0.796 N 49.3 2.29% 49.8 2.31% 49.5 2.16%(0.179 lb)

1.09 N 49.4 2.50% 49.9 2.34% 49.5 2.04(0.245 lb)

1.48 N 49.3 2.61% 50 2.41% 49.5 2.00%(0.332 lb)

IV. TSP ACTIVE MODE

A TSP active mode (Fig. lb) is simulated by reversing the phase of one

set of driving coils and exciting the tube at its second natural fre-

quency. The experimental air and water data presented are for a 3.81 cm

(1-1/4 in.) wide TSP with a 0.061 cm (0.024 in.) diametral clearance. The

oil data were acquired using a 1.27 cm (0.5 in.) wide TSP with a 0.0457 cm

(0.018 in.) diametral clearance. Since the nodal point is centered within

the TSP, tube displacement within the TSP is very small, therefore, the air,

water, and oil data can be compared

28

The measured frequency and damping in air and water at force levels

from 0.0365 N (0.008 lb to 0.498 N (0.112 lb) per coil pair are presented

in Table 8. A calculated value of 152.9 Hz compares well with an average

measured value of 154.2 Hz for the second mode natural frequency in air.

The tube frequency and damping in water can be calculated using Eqs. (3) and

(4) along with the measured in-air damping. For water, calculations yield a

frequency of 138.2 Hz and a damping ratio of 0.667%.

Table 8. Measured TSP activein air and water

(second mode) frequencies and damping

Force Level Air Water(For Coil Pair) f2 (Hz) f2 (Hz)

0.0356 N 154.4 0.512% 139.2 0.792%(0.008 lb)

0.133 N 154.1 0.554% 139.0 0.723%(0.030 lb)

0.285 N 154.2 0.598% 139.2 0.782%(0.064 lb)

0.498 N 153.9 0.679% 138.8 0.955%(0.112 lb)

Average 154.2 0.586% 139.0 0.813%

Table 9 summarizes the measured

tube submerged in oil at three temperate

and damping are, respectively, 139.1

139.5 Hz and 0.0235 for 26*C (78.8*F

frequency and damping values of the

tures. Calculated values of frequency

Hz and 0.030 for 21*C (69.8*F) oil,

') oil, and 139.6 Hz and 0.0144 for

47.7 C (118 F) oil. Considering experimental error, the measured and

calculated frequencies are in good agreement. There is a larger difference

between the calculated and measured damping. This difference is due

primarily to the temperature distribution of the oil in the test vessel.

The quantity -Im[H] (Appendix Fig. A.2) is very sensitive to the kinematic

viscosity of the fluid, which is a function of temperature. Re[H] is much

less sensitive to viscosity, therefore the difference between the calculated

and measured frequencies is smaller. Oil temperature is only monitored

close to the center TSP, as the temperature of the oil in the TSP-tube

annulus is most critical.

29

Table 9. Measured values of TSP active (second mode) frequencies

and damping in oil

21 C (69.8 F) 26.6*C (78.8*F) 47.7*C (118*F)Force Level 01 Oil Oil

(For Coil Pair)f2 C2 f2 C2 f22 2

0.285 N 136.3 Hz 3.93% 137.0 Hz 2.74% 136.8 Hz 1.56%(0.064 lb)

0.49 N 136.3 Hz 4.26% 137.3 Hz 2.8% 136.8 Hz 1.66%(0.112 lb)

0.798 N 136.4 Hz 3.99% 137.3 Hz 2.69% 136.9 Hz 1.72%(0.179 lb)

1.09 N 136.4 Hz 3.28% 137.2 Hz 2.67% 136.9 Hz 1.73%(0.245 lb)

1.48 N 136.6 Hz 3.26% 137.3 Hz 2.64% 137.1 Hz 1.69%(0.332 lb)

1.92 N 136.6 Hz 3.29% 137.3 Hz 2.65% 137.0 Hz 1.75%(0.432 1)

2.40 N 136.3 Hz 3.59% 137.3 Hz 2.50% 137.2 Hz 1.93%(0.54 lb)

3.02 N 136.3 Hz 3.43% 137.3 Hz 2.49% 137.6 Hz 2.05%(0.68 lb)

*Oil viscosity from Fig. 8:

T, C ()F) v, m2/s (in.2/sec)

21.0 (69.8) 3.62 x 104 (0.5616)26.0 (78.8) 2.48 x 10~ (0.3845)47.7 (118.0) 9.85 x 10-5 (0.1526)

Damping as a function of excitation force is shown in Fig. 10 for the

various tests in air, water and oil. Damping values are seen to vary only

slightly with force amplitude.

V. TSP INACTIVE MODE - A SERIES

When the tube vibrates without contacting the center TSP, the condition

is defined as an inactive TSP. A series of tests was conducted (Tests Al to

3

30

0.25POUNDS

0.50 0.75

H-

21 C OIL

26 C OIL

48 C OIL

H 20.-- AIR

2 3FORCE /COIL PAIR, N

Fig. 10. Damping of a Tube in Air and Liquid for a TSP Active Mode.

0.07

0.06 -

0.05

0.04

a 0.03

p 0.02

0.01

0.000 4

1 1

I I I

I I

I

31

A5 in Table 2) to study the tube response with TSP widths typical of PWR and

breeder reactor steam generators. Measured quantities are compared to

calculated data to develop confidence in the accuracy of the governing

equations.

A. Representative Results: Test A4 in Water

Space limitations precluded presenting detailed data from all the test

cases. Therefore, test A4, with a tube-TSP diametral clearance of 0.061 cm

(0.024 in.), was selected as a representative case for presentation in

detail. RMS tube displacements and pressures for Test A4 with a 1.27 cm

(0.5 in.) wide TSP are shown in Fig. 11. Tube-TSP contact occurs above a

force level of 0.796 N (0.179 lb). Below that value the force-displacement

nonlinearity arises from the increase of damping with an increase of excita-

tion force. Above that value the tube contacts with the TSP, limiting the

displacement to a value approaching the diametral clearance. As the force

is increased, tube motion gets more orbital, as can be seen in the rise of

Dl for a force level greater than 1.48 N (0.332 lb). This change in motion

acts to lower the pressure P2 as tube motion is no longer along the P2-D2

central axis. If the tube were to be perfectly aligned within the TSP, and

limited in direction to only the P2-D2 axis, no orbiting would be expected.

The nonlinear tube-TSP system can be studied further with the aid of

Fig. 12. Figure 12 shows the force-displacement response, and displacement

and pressure time histories for force levels ranging from 0.285 N

(0.064 lb), which is below the level required for tube-TSP contact, to a

force level of 3.02 N (0.68 lb), which is considerably higher than that

required for contact. Displacement and pressure time histories are recorded

at frequencies corresponding to the peak, or what would be the peak, of the

respective displacement-frequency response curves.

At an input force level of 0.285 N (0.064 lb) per coil pair, the tube

response can be considered linear. The displacement time history is

sinusoidal and there is only a very small fluid pressure developed in the

tube-TSP gap.

When the force level is increased to 0.796 N (0.179 lb), the resulting

tube motion brings it very close to the TSP, but without contact. A peak-

to-peak pressure of 1.816 x 104 Pa (2.6 psi) is developed in the tube-TSP

gap. The positive pressure is developed as the tube is approaching the

pressure transducer (P2), which is mounted flush with the wall of the TSP.

As the tube comes to a stop and changes direction, the pressure decreases

and becomes negative.

Increasing the force to 1.48 N (0.332 lb) per coil pair results in

tube-TSP contact. The slight "flattening" of the displacement is a result

of the motion limiting effect of the TSP. The slight rise in the flat top

POUNDS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.714- 1- --

D2

12

0 10 /- .

X/ N\

v//

WA

C.et)4[.01c (.2 i. Daera l/rne

.2./ /W6 C/

CV) 4

2A

1.0 2.0 3.0

FORCE/COIL PAIR, N

Fig. 11. RMS Tube Displacements and Pressure~ for Test A4 [0.061 cm (0.024 in.) Diametral Clearance]

with a 1.27 cm (0.5 in.) TSP.

Wt-j

33

FORCE LEVEL

-- DIS PLACEMENT, D 2

-PRESSURE, P2

I I I I I

/

I I 1

I1

10.254 cm(0.01 In)

30 40 50 60Hz

I I I I

1 0.254 cm

(0.10 in)

6.984 X 103 Pa( 1.0 psi)

PHASING TSP

TUBE

P2 .- D2

ARROW INDICATESTUBE MOTION

IL L~~

0.796N(0.179*)

1.48N(0.3320)

2.4N(0.54 ")

z

3.02N W(0.68#)<

c

Fig. 12. Force-Displacement-Response, Displacement and Pressure TimeHistories for Test A4 with a 1.27 cm (0.5 in.) TSP.

0.285N(0.064 " )

0 0.01 0.02 0.03 0.04 0.05

TIME ,SECONDS

\\

F / I I

34

part of the frequency-displacement response curve can be attributed to a

slight misalignment of the tube within the TSP. Peak-to-peak pressure

amplitude is now 4.539 x 104 Pa (6.5 psi) with the same basic shape as

occurred with a force level of 0.796 N (0.179 lb), but with higher frequency

noise between pulses. This high frequency noise can be attributed to the

excitation of vibration modes in the TSP base mounting when impacting takes

place.

A force input of 2.4 N (0.54 lb) results in a wider top of the

frequency-displacement plot. A dwell time, at which the tube remains in

contact with the TSP, can clearly be seen on the displacement time

history. This dwell time results in a change in the gap pressure in that

there are now two pressure pulses per cycle. The positive pressure pulse is

the result of the tube approaching the TSP. The pressure then decreases and

remains low until the tube travels away from the TSP; this produces a

suction effect which results in a negative pressure pulse.

The frequency-displacement response at 3.02 N (0.68 lb) force per coil

pair yields a still wider top. ine solid line represents the response when

the frequency is swept from a low to a high value while the dashed response

is for a high to low frequency sweep. From this, the system nonlinearity

can clearly be seen. The pressure pulse amplitude is slightly higher than

that at 2.4 N (0.54 lb) but its basic shape remains the same. The

displacement-time history remains unchanged except for a slight addition of

tube higher mode response.

Measured tube displacement-force response (Test A4) in water at TSP

widths ranging from 1.27 cm (0.5 in.) to 5.08 cm (2 in.) with a constant

diametral tube-TSP clearance of 0.061 cm (0.024 in.) is shown in Fig. 13.

TSP width effect on tube displacement can be readily seen. This is

primarily the result of fluid damping in the tube-TSP clearance. Fluid

damping increases with both TSP width and excitation force level as shown in

Fig. 14. For the same series of tests, the measured frequencies are shown

in Fig. 15. Frequency decreases with increasing TSP width but is nearly

independent of excitation force level.

Tube dynamics, such as resonant frequency and damping, play an

important role in the stability of heat exchanger and steam generator tube

arrays [2]. Figures 13, 14, and 15, from Test A4, clearly show the

influence of TSP width on tube dynamics. Test A4 is for a constant

diametral clearance of 0.061 cm (0.024 in.). Similar results are obtained

from tests with other values of clearance.

B. Calculation of Damping and Frequency

Damping .. 3 frequency of a tube with an inactive TSP may be calculated

with Eqs. (A.ll) and (A.12), respectively, of the Appendix,

35

POUNDS0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.0 2.0FORCE/COIL PAIR,N

3.0 4.0

Fig. 13. Tube Displacement-Force Response for Test A4 in Water.

-J

C,)

CI-

C,)

LU

w

I -

9

8

7

6

5

4

3

2

0

I.7c (.in.

X1.27 cm (0.75in.)- .......----- 2.54 cm ( in.

- ' 3 I ~ m I . 5 i)do , 3 .8 1 c m 1 1 .5 in .) -

. 4.45 m (I.75 in.)

5.08 cm 2 in.)

s -

0

36

POUNDS

0.3 0.4 0.5

5.08 cm

0.6

(2 in.)

4.45 cm (1.75 in.)

3.81 cm (1.5 in.)

2.54 cm(1I.00 in. )

1.91 cm (0.75 in.)

1.27 cm (0.5 in.)

VA /-5.08 cm (2 in.)

!.0

AIR

2.0 3.0

FORCE /COIL PAIR,N

Fig. 14. Damping Ratio as a Function of Force for Test A4.

0.I 0.20.08

0.07

0.06

0.7

0;

z-

0.05

0.04

0.03

0.02

0.01

0.000

1 l

'/

.

I II

37

0 0.152

50

48,

46

44

pounds

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2FORCE /COIL PAIR, N

Fig. 15. Frequency as a Function of Force for TesL A4.

N

U

z

0W

.27 cm (0.5 in.)1.91 cm (0.75 in.)

2.54 cm (1,0in.)

3.18 cm (1.25 in.)

? - / 1 C

3.E1 cm (.5 in.)

S4.45 cm (1.75 in.)

5.08 cm (2.0 in.)

42

400 3 4

1

l I I I 1 I I

I I

i

38

0.5

n R2E I

wn R2 M' 2((5

m + CMMD+M(

and

1 an D mV 2

nM'4 2 ) 2w - Im[H] + 2 R ;(6)

m+CMMD+ R2

see Appendix for nomenclature. Values of Re[H'] and -Im[H'] are determined

from Fig. A.3 of the Appendix.

Equations (5) and (6) were evaluated for a series of tube-TSP diametral

clearances and TSP widths. The effect of TSP width on damping and frequency

for tube-TSP diametral clearances of 0.0152 cm (0.006 in.) to 0.0813 cm

(0.032 in.) is shown in Figs. 16 and 17. From these figures, it can be

seen, as one would intuitively expect, that the TSP width has a greater

influence on damping and frequency at .iall diametral clearances. For

convenience, the data plotted in Figs. 16 and 17 are replotted in Figs. 18

and 19 using the diametral clearance as the abscissa.

C. Comparison of Measured and Calculated Damping and Frequency

The series of tests (Al-A5) described above provide data to test the

validity of Eqs. (5) and (6). The tests were conducted in air and water; in

addition, Test A3 was also conducted in oil at various temperatures to vary

the fluid viscosity.

Measured in-water damping ratio is plotted as a function of TSP width

for various diametral clearances and an input force level of 0.498 N

(0.112 lb) per coil pair in Fig. 20. The plotted values are from one set of

data (viz., one force level) and are not averaged. This data can be

compared to the calculated damping ratios of Fig. 16. With fluid in the

tube-TSP clearance, damping is a function of excitation force (Fig. 14). To

obtain a more meaningful comparison between measured and calculated quanti-

ties, the ranges of measured damping for excitation forces are compared to

the calculated values. These comparisons are made in Figs. 21 to 25. It

will be observed that there is a larger difference between the calculated

and measured damping at the small diametral clearances. This difference is

due to the sensitivity of the measurements to small errors in the tube-TSP

dimensions and alignment. In Fig. 26, a nominal tube-TSP clearance of

0.0203 cm (0.008 in.) is assumed, the damping ratio is then calculated for a

+0.0051 cm (0.002 in.) increase in diametral clearance and also a +0.0127 cm

(0.005 in.) increase in diametral clearance. It can be seen that the

39

inches

2

3 4 5BAFFLE WIDTH, cm

Fig. 16. Calculated Damping Ratio as a Function cf Baffle Width.

0

1.0

3

Lnf

wi

Oz-Q:

Z 0.I

DIAMETRAL CLEARANCE

0.0152 cm (0.006 in.)0.0203 cm (0.008 in.)

0.0254 cm (0.01 in.)

0.033 cm (0.013 in.)

0 0457 cm (0 018 in)l0.061 cm (0.4 in.

0.061 cm (0.024 in.)

0.0813 cm (0.032 ;n.)

0.010 2 6 7 8 9

-

' 1

I

I I I

1 II II I

I

350

48

46

44

l I

-

_

-

-

-

-

-

-

0.0152

6

(0.008 in.)

cm (0.006 in.)

7 8BAFFLE WIDTH, cm

Fig. 17. Calculated Frequency as a Function of Baffle Width.

0

40

inches

2

DIAMETRALCLEARANCE

0.0813 cm (0.032 in.)

0.061 cm (0.024 in.)

0.0457 cm (0.018 in.)

0.033 cm (0.013 in.)

0.0254 cm (0.010 in.)

N

V

c.)

Li..

42

40

38

36

34

32

300 2 3 4 5 9

II I l

I

0.0203 cm

I I I I

I

41

0.01 L0.00254

inches

0.01

I I I I I I I I I I

0.1

I

BAFFLE WIDTH

i

2.54 cm (1.0 in.)1.91 cm (0.75 in.)

1.27 cm (0.50 in.)

0.0254

cm (2.0 in.)cm (1.75 in.)

cm (1.5 in.)cm (1.25 in.)

0.254DIAMETRAL CLEARANCE, cm

Fig. 18. Calculated Damping Ratio as a Function of Tube-BaffleDiametral Clearance.

0.001l.Vr

z

0

0.I

_ 1 1 1 l w

5.084.453.813.18

42

inches0.001

I I I I I I I

BAFFLE1.27

0.01

.E WIDTHcm (0.5 in.)

I1 II1-1,-I50

48

46

44

42N

40

:38

-r

I I I I 1 I 15.08 cm (2.0 in.)

I I I I 11111

0.00254

I I i i

0.0254DIAMETRAL CLEARANCE, cm

Fig. 19. Calculated Frequency as a Function ofClearance.

Tube--Baffle Diametral

0.I

I I I 1- --1

1.91 cm (0.75 in.)

2.54 cm (!.0 in.)

3.18 cm (1.25 in.)

3.81 cm (1.5 in.)

4.45 cm (1.75 in.)

36

34

32

30PIP.

oo

0.254

11F

l l l l

III

w

43

inches2

0 I

3

2 3 4 5 6 7 8 9BAFFLE WIDTH, cm

Fig. 20. Measured Damping Ratio vs. Baffle Width in Water with VariousDiametral Clearances for a Force of 0.498 N (0.112 lb).

01.0

0.

6-J

O

Z

Q

DIAMETRAL CLEARANCE

0.0203 cm (0.008 in.)

0.033 cm (0.013 in.)

0.0457 cm (0.018 in.)

. A 0.0 6 1 cm (0.024 in.)

0.0813 cm (0.032 in.)

I I0.01 - -

i

T

I T I

I I I I I I

44

INCHES

1.0

0.I

0.01

0 2

0 2 3 4 5 6

BAFFLE WIDTH, cm

Fig. 21. Theoretical and Measured Damping Ratios with a 0.0203 cm(0.008 in.) D!.ametral Clearance.

0I-

a-

_II

-

-

ICALCULATED -

I MEASURED -

- DIAMETRAL CLEARANCE -

0.0203 cm (0.008 in)

7

I

I

45

INCHES

2

2 4 5 6 7

cm

3

BAFFLE WIDTH,

Fig. 22. Theoretical and Measured Damping Ratios with a 0.0320 cm(0.013 in.) Diametral Clearance.

01.0

-

za-

CD

0.1

-- CALCULATED

I MEASURED

DIAMETRAL CLEARANCE

II

III0.01

0

1 l

I

0.03

I I I

I

46

INCHES

2

2 3 4

BAFFLE WIDTH, cm

5 6 7

Fig. 23. Theoretical and Measured Damping Ratios with a 0.0457 cm

(0.018 in.) Diametral Clearance.

01.0

0.:a-

0

-- CALCULATED

- 1I ME ASUR ED-- DIAMETRAL CLEARANCE-

ML_ 0 .0 4 5 7 c m ( 0 .0 1 8 i n )

I-____________________L-t I

0.010

I

r

1

47

INCHES

0 I 21.0

--- CALCULATED

I MEASURED

DIAMETRAL CLEARANCE

- 0.061 cm (0.024 in)

0.1

0.01 I0 I 2 3 4 5 6 7

BAFFLE WIDTH, cm

Fig. 24. Theoretical and Measured Damping Ratios with a 0.061 cm(0.024 in.) Diametral Clearance.

48

INCHES

1

3 4

BAFFLE WIDTH, cm

Fig. 25. Theoretical and Measured Damping(0.032 in.) Diametral Clearance.

Ratios with a 0.0813 cm

01.0

2

DIAMETRAL CLEARANCE

- 0.0813 cm (0.032 in)- CALCULATED -IMEASURED

01-

a-0

0.1

0.010 1 C. 5 C 7

SI I

I MEASIAT 0.C

3

0.0203 cm (0.008 in.)

0.0254 cm (0.010 in.)

0.033 cm (0.013 in.)

DIAMETRALCLEARANCE

JRED VALUE)203 cm (0.008 in)

0 2 3 4 5 6 7 8 9BAFFLE WIDTH, cm

Fig. 26. Sensitivity of Damping Ratio to Variations t.n DiametralClearance at Small Nominal Clearances.

49

inches0

1.02

z

0

0.I

0.01 -- _ _ _ 1 __ _ 1 _ _ __ - 1 _ _ 1 1 1 1 _ _ _

i

f

1

50

measured data fits in well between the 0.0254 cm (0.010 in.) diametral

clearance and the 0.033 cm (0.013 in.) diametral clearance. Likewise, the

results of a 0.0102 cm (0.004 in.) dimensional change on a nominal

0.0813 cm (0.032 in.) diametral clearance is shown in Fig. 27. These

results serve to demonstrate that a small dimensional or alignment error has

considerably more effect on damping when the tube-TSP clearances are small.

Measured tube first mode fending frequency is plotted as a function of

TSP width for an input force of 0.498 N (0.112 lb) at various diametral

clearances in Fig. 28. This frequency data can be compared to the calcu-

lated frequencies of Fig. 17. Since the data used to plot Fig. 28 was

obtained from only one series of tests (viz., one force level) and not

averaged, the comparison will not be as meaningful as that when a series of

averaged data is used. Averaged measured frequencies are used for the

comparisons in Figs. 29 to 33.

Allowing for slight variation in tube mounting (fixed-fixed end

conditions) due to having to remove the tube to change diametral clearance,

the data compares well with the exception of the 0.0457 cm (0.018 in.)

diametral clearance where a slight misalignment problem occurred.

Several tests (A3) were conducted in oil. With the smallest TSP width

(1.27 cm, 0.5 in.), the measured and calculated damping ratios were very

large. Figure 34 is a typical displacement-frequency response for Test A3

with a 1.91 cm (0.75 in.) TSP. No quantitative data could be generated so

testing in oil was discontinued for the A series of tests.

VI. TSP INACTIVE MODE - B SERIES

A. Theoretical Calculations

In this series of tests, the dimensions of TSP width and tube-TSP

diametral clearances were chosen to be similar to those of typical process

industry heat exchangers as recommended by TEMA [Ref. 1 and Table 1]. Tests

were conducted in water and in oil at four different temperatures

(viscosities).

Theoretical values of damping are calculated by use of Eq. (6) for room

temperature water and oil at 210C (70 F), 26.7*C (80*F), 480C (120*F), and

65.5*C (150*F). Kinematic viscosity, Re[H1, -Im[H], Re[H'], and -Im[H'],

are given, for the various temperatures, in Table 10. The calculated

damping ratios for a 0.0397 cm (0.0156 in.) diametral clearance in liquid at

various viscosities are shown in Fig. 35. Figure 36 is for a 0.0794 cm

(0.03125 in.) diametral clearance and the same liquid viscosities.

From Figs. 35 and 36 the significant effect of the viscosity of the

fluid in the tube-TSP can be seen. The damping ratio for a given TSP width

51

inches

2

2 3 4

3

5 6 7 8 9

BAFFLE WIDTH, cm

Fig. 27. Sensitivity of Damping Ratio to Variations in DiametralClearance at Large Nominal Clearances.

01.0

0.1

0

I MEASURED VALUE0.0813 cm (0.032 In)

DIAMETRALCLEARANCE

0.0711 cm (0.028 in.)

0.0813 cm (0.032 in.)0.0914 cm (0.036 in.

0.010 I

I-

--- -

i

i

I I I

)

52

INCHES

50

48

46

44

42

40

38

36

34

0 2

0 2 4 5 63

BAFFLE WIDTH, cm

Fig. 28. Measured Frequency vs. Baffle Width in Water for VariousDiametral Clearances and a Force of 0.0 #98 N (0.112 lb).

z

L&

0.0813 cm _

(0.032 in.)

0.061 cm(0.024 in.) ~~

\Q.Q457 cm- (0.018 in) -

0.033 c m- (0.013 in) -

0.0203 cm(0.008 in)

7

I

I I II I I

I

53

INCHES

250

48

46

44

- CALC

0 0 MEANS

DIAMETRAL

0.0203 cm0

0

0

0

I I

2 3 4 5 6

BAFFLE WIDTH, cm

Fig. 29. Theoretical and Measured Frequencies with

(0.008 in.) Diametral Clearance.a 0.0203 cm

0

0

ELATED

URED

CLEARANCE

(0.008 in)

N

zZU

W

LU

42

40

38

36

34

320 7

I I

l l 1 l l

I

I I

I I I

I

54

INCHES

2

0

CA0

0 ME

0 DIAMETRA

0.033 cm

0

0

2 3 4

LCULATED

ASSURED

L CLEARANCE

(0.013 in)

0

5 6

BAFFLE WIDTH, cm

Fig. 30. Theoretical and Measured irequcacies with a 0.033 cm(0.013 in.) Diarmetral Clearance.

50

48

46

N%

S.

LV

Z

Li

44

42

0

0

40

38

36

-

I

I J 1

1

55

0

DIAM

0.04

INCHES

2

0

0

0

- CALCULATED o

MEASURED

ETRAL CLEARANCE

57 cm (0.018 in)

2 3 4 5 6 7

BAFFLE WIDTH, cm

Fig. 31. Theoretical and Measured Frequencies with a 0.0457 cm(0.018 in.) Diametral Clearance.

050

48 L

46k -V-

LL441-

42

400

I I

I

I I

I

56

INCHES

2

--

0DIAMETRAL CI

0.061 cm ((

0

- CALCULATED

MEASURED 0

2 3 4 5

LEARANCE

0.024 in)

6

BAFFLE WIDTH, cm


050

48

46

44

Z

ciLA.

420 7

l 1

I

T T

I

57

INCHES

o

-- CALCULATED

0 MEASURED

DIAMETRAL CLEARANCE

0.083 cm

2

(0.032 in)

3 4 65

BAFFLE WIDTH, cm


050

48

2

46VZ

U-W

O: 44 -

42

400 7

1 l

I

I I I I I I

I

I

1.01

I

I-

0.5

I

z

-J0

C!)

3,02 N (0.68#)!.92N (0.4321)

1 .09 N (O.245#)

0.498 N (0.112#)

-

10 20 30 40 50

FREQUENCY, Hz

Fig. 34. Displacement vs. Frequency Response for Test A3 with a 1.27 cm (0.5 in.)

TSP in 48*C (118 F) Oil.

0 60

59

INCHES

1.00.1.0w

1 __

0.01 1'2.54

I a I I i

I.0

BAFFLE WIDTH, cm

Fig. 35. Theoretical Damping Ratios as a Function of Baffle Width with a0.0397 cm (0.0156 in.) Diametral Clearance in (1) Water at 21.1*C(70 F), (2) Oil at 65.5*C (150*F), (3) Oil at 48 C (120*F),(4) Oil at 26.7 C (80*F), and (5) Oil at 21*C (70*F).

.I

17

0

0

0.I

534 2

2

v w N AN I v mrI I H

I

60

INCHES

I.0I.' I IIF

1.0

BAFFLE WIDTH, cm

Fig. 36. Theoretical Damping Ratios as a Function of Baffle Width with a0.0794 cm (0.0313 in.) Diametral Clearance in (1) Water at 21.1 C(70*F), (2) Oil at 65.5*C (150 F), (3) Oil at 48*C (120 F),(4) Oil at 26.7 C (80 F), and (5) Oil at 21 C (70*F).

0.Iin -

- 1

_ 0O.'0

0.012.54

- m m m maI

61

Table 10. Re[H], -Im[H], Re[H'], and -Im[H'] values in viscous fluids

Fluid V Re[H] -Im[H] Re[H'] -Im[H']

Water 1.0 x 10-6 m2/s 1.0 0.027 1.2 1.4(0.00157 in.2/sec)

21 C (70*F) 3.62 x 10~4 m2/s 1.07 0.30 1.2 230Oil (0.5616 in. 2/sec)

26.7*C (80*F) 2.48 x 10~4 m2/s 1.06 0.22 1.2 160Oil (0.3845 in. 2/sec)

48 C (120*F) 9.85 x 10-5 m2/s 1.04 0.108 1.2 72Oil (0.1526 in.2/sec)

65.5*C (150 F) 4.46 x 10-5 m2/s 1.01 0.08 1.2 43Oil (0.0691 in.2/sec)

is considerably larger with the 0.0397 cm (0.0156 in.) diametral clearance

than with the 0.0794 cm (0.03125 in.) diametral clearance.

First mode tube resonant frequency can also be calculated with

Eq. (5). However, it is insensitive to the viscosity within the tube-TSP

gap. This is due to the fact that Re[H'] remains essentially constant (see

Table 10) for the liquid viscosities tested. The only effect of liquid

viscosity is that from Re[H] which is associated with the tube vibrating in

an "infinite" fluid.

B. Measured Quantities

Figure 37 is a typical plot of tube resonant frequency vs. TSP width

for a tube-TSP diametral clearance of 0.0794 cm (0.03125 in.) in 65.5*C

(150*F) oil [4.46 x 10-5 m2/s (0.0691 in.2/sec), viscosity] at an input

force level per coil pair of 0.796 N (0.179 lb). Figure 37 allows for

comparison of measured and calculated values. Considering experimental

tolerances, the calculated and measured frequencies are in reasonable

agreement. The decrease in frequency due to increasing TSP width is

considerably less in high viscosity fluids such as oils.

Tube displacement for a given input force is limited by increased tube-

TSP gap damping at the higher fluid viscosities (Fig. 38). Overall, the

displacements for the B series of tests were kept to lower values to avoid

damage to the tube or TSP resulting from impact of the tube with a narrow

TSP. Also, the maximum excitation force was restricted to avoid overheating

of the excitation coils.

INCHES

0.2 0.3 0.4

1

0

0 0

-- CALCULATED

o MEASURED

I.00.5

BAFFLE WIDTH, cm

Fig. 37. Frequency vs. Baffle Width for Test B2 in 65.5 C (150*F) Oil.

051

0.I

50k-

0.5 0.6 0.7

49 1-

N

Z-W

LL.

48 1-

47 1-

46

O'

2.0L.5

T

i

I I 1 I I I

II I

FORCE,0 0.2 1.0 2.0 3.0

26.7*C WATER12(800 F)

10

WLJ

W 8

48*C OIL

m 6 (118* F)

65.6*C OIL

S 4_ (150* F) 26.7*C OI L -

(80*F )

0

5 10 15

FORCE, N

Fig. 38. Tube Displacement as a Function of Excitation Force for Test B2 with a 0.65 cm (0.25 in.) TSP.

G'W&

64

Measured and calculated damping ratios for a diametral clearance of

0.0397 cm (0.0156 in.) are compared in Figs. 39, 40 and 41. For water in

the gap, Fig. 39, the mea:;ured damping ratios are somewhat lower than the

calculated. This can be attributed to: 1) The excitation force levels are

limited, therefore, the maximum displacements at which damping is measured

are lower and the tube does not impact or traverse the total gap; 2) Any end

effect of the TSP becomes more pronounced at small TSP widths.

In oil (Figs. 40 mnd 41), there is a considerable difference between

the measured and calculated damping ratios. The difference is due to the

sensitivity of dampirg at small tube-TSP clearance to system parameters.

For these tests, the actual diametral clearance was slightly larger, which

resulted in a smaller measured value of damping. The damping ratio for

26.7*C (80*F) oil could not be measured because of the large values

encountered. Typical displacement-frequency response curves for 26.7*C

(80*F) oil with a 1.27 cm (0.5 in.) TSP are given in Fig. 42. Due to the

very large damping, the tube responds as a rigid body.

Measured and calculated comparisons for a diametral clearance of

0.0794 cm (0.03125 in.) are made in Figs. 43 to 46. The discrepancy between

the measured and calculated values for the test in water are the same as

previously mentioned. For the oil tests, the calculated and measured

damping ratios are in good agreement, with the exception of a few points.

in the later case, the variations are attributed a nonuniform oil

temperature profile within the vessel. Also, because of the large diametral

clearance, tube response is less sensitive to system parameters.

VII. CONCLUDING REMARKS

Theoretical pr-dictions, as outlined in the Appendix, yield damping

ratios and frequencies which are, within experimental accuracy, in good

agreement with measured quantities. In cases where there is a possibility

of designing a steam generator or heat exchanger with TSP inactive supports,

prediction of tube response characteristics (damping ratio and frequency)

can be useful in determining if a fluidelastic instability could occur and

in predicting response to turbulent buffeting. A method of theoretical

prediction would also be useful in the following:

1) To determine the dynamic response modification of a tube or tube

array due to an increase of tube-TSP clearances when chemical cleaning is

performed to remove corrosion products in steam generators. Conceivably, in

a steam generator, if some initial tube-TSP wear occurred before significant

corrosion product buildup, removal of the corrosion product and/or the

effects of the cleaning process itself could result in a larger diametral

clearance than when the steam generator was first put into service. This

65

INCHES0.I

1.0 r-

-I---

I.O

CALCULATED

RANGE OF MEASUREDVALUES

14

- -o- AVERAGE MEASUREDVALUES

--00/

I I I I I I I

.254 1.0

BAFFLE WIDTH, cm

Fig. 39. Damping Ratio as a Fugction of Baffle Width or Test B1 inWater, v = 1.01 x 10~ m /s (1.57 x 10- 3 in. /sec).

0

c.

z

C-

0

0.I

0.01 - i

(6

INCHES0.1 1.0

1.0

0

0.1

0

CALCULATEDRANGE OF MEASUREDVALUES

-o - AVERAGE MEASUREDVALUES

0.01.254 1.0

BAFFLE WIDTH, cm

Fig. 40. Damping Ratio as a Function of Baffle Width for Test B1 in

65.5 C (150*F) Oil, v = 4.46 x 10~ m2/s (0.0691 in.2/sec).

67

INCH ES0.

I.0

.254

I I 1_

-- CALCULATEDRANGE OF MEASUREVA LUE S

-o- AVERAGE MEASUREDVALUES

1.0

BAFFLE WIDTH, cm

Fig. 41. Damping Ratio as a Function of Bgffle Width for Test B1 in480C (120*F) Oil, v = 9.85 x 10~ m2/s (0.1526 in.2/sec).

'I

D

0

Os'Z

&..A

0.01

-

i.0

i

-1

4 -- 3.02 N (0.68# )

m 1 .92 N (0.432 )

cc 3 -- 1.09 N (0.245# )~~

W 0.49 8 N (0. 112 *

W co-

m

0

S0 10 20 30 40 50 60

FREQUENCY, Hz

Fig. 42. Displacement vs. Frequency for Test Bi with a 1.27 cm (0.5 in.) in 26.7C (80*F) Oil.

69

INCHES

--- -n- T TT

1.0

- - CALCULATED

RANGE OF MEASUREDVA L U ES

-0- AVERAGE MEASUREDVALUES

00e/ 00

-won-

I I I YI I 1

1.0

BAFFLE WIDTH, cm

Fig. 43. Damping Ratios as a Function of Baffle Width for Test B2 inWater, v = 1.01 x 10-6 m/s (1.57 x 10-3 in.2/sec).

0.i.0 r

0

a-

zc

0

0.I

0.01.254

II I

1

70

INCHES

0.1 1.0

1.0 I iFFF1F

CALCULATED

- RANGE OF MEASUREDVALUES


0

a-

0.01.254 1.0

BAFFLE WIDTH, cm

Fig. 44. Damping Ratios as a Function of Baffle Width for Test B2 in65.50C (150*F) Oil, v = 4.46 x 10-5 m2/s (0.0691 in.2/sec).

71

INCH ES0.1 1.0

-- CALCULATED



0

0.1

.254 1.0

BAFFLE WIDTH, cm

Fig. 45. Damping Ratio as a Function of Baffle Width for Te t B2 in

480C (120 F) Oil, v = 9.85 x 10- 5 m2/s (0.1526 in. /sec).

72

INCHES0.1 1.0

1.0

? 0.1

-- CALCULATED



0.01.254 1.0

BAFFLE WIDTH, cm

Fig. 46. Damping Ratio as a Function of Ba fle Width for Tes B2 in26.7 C (80 F) Oil, v = 2.48 x 10 m2/s (0.3845 in. /sec).

73

larger clearance could increase the possibility of detrimental tube

vibrations, for example, by permitting vibration in a TSP inactive mode.

2) In some cases, when a tube vibration problem exists iin a heat

exchanger or steam generator, individual tubes are expanded to reduce tube-

TSP clearances. A method to predict resulting tube dynamics would aid in

determining the amount of expansion required.

Although the majority of this report relates to system parameters such

as tube natural frequencies and damping, it is expected that the pressure

measurements may prove useful in correlating tube motion (when impacting) to

tube-TSP wear. A large fluctuation of pressure within the tube-TSP annulus

would be helpful in "washing" wear particles from within the annulus.

ACKNOWLEDGMENTS

This work was performed under the sponsorship of the Office of Power

Conversion Technology, U. S. Department of Energy.

The author wishes to express his gratitude to Dr. S. S. Chen, for

providing the Appendix of this report; to Dr. M. W. Wambsganss, for his

guidance, review, and helpful comments; and to Mr. W. A. Ellis, who provided

help with the test hardware and the performance of the tests.

74

REFERENCES

1. Standards of Tubular Exchanger Manufacturers Associates, Sixth Edition,1978.

2. Structural Damping, ed. J. E. Ruzicka, ASME Publication (1959),

3. Chen, S. S., and Jendrzejczyk, J. A., "Stability of Tube Arrays inCrossflow," Nucl. Eng. Des., 75(3), 351-374 (June 1983).

75

APPENDIX: ANALYSIS

Consider a clamped tube with a baffle plate at midspan (Fig. A.1). We

wish to calculate the natural frequencies and damping values under different

conditions.

The equation of motion for the tube is

4

EI 4 + [Ca + Cv + C 6()] aax

2+ [m + CMMd + M'6(2)] 2 = 0 , (A.1)

at

where El = flexural rigidity, u = tube displacement, x = axial coordinate,

t = time, Ca = viscous damping coefficient in air, Cv = viscous damping of

infinite fluid, Cv= fluid gap damping, CM = added mass coefficient in

infinite fluid, m = tube mass per unit length, Md = displaced mass of fluid

per unit length, M' = added mass of fluid gap. Equation (A.1) can be used

for different conditions as follow:

(a) Vibration in Air

For a tube vibrating in air, C Cv M zM' 0, and thev v d

equation of motion can be written as

4 2EI 4 + Caat + m 2 = 0 . (A.2)

ax at

For a fixed-fixed tube, the orthonormal mode shape is

n(x) = coshanx - cosanx - a (sinhanx - sina x)

xiI = 4.73004 , a = 0.9825C2 , (A.3)

X2L = 7.85320 , a2 = 1.007773

The natural frequencies and in-air viscous damping coefficient are

(an ))2 0.5= n 2 (E) (A.4)w m

Ca = 2mwn n. A5

(A.5)

76

2L

A AK B

1/2 1/2-

h

BAFFLE PLATE

Fig. A.l. A Fixed-Fixed Tube with a Baffle Plate Support.

77

(b) Oscillations in Water with No Tube Support

The equation of motion is

4

EI + (Ca + Cv) a + (m + CMMd)ax

2

ate

(A.6)

The modal damping and natural frequency can be analyzed in a straightforward

manner [Ref. A.1] to obtain

(X )2 0.5n - 2 C Ed + m

_______ 1 MdS= Ca -L j+Im(H)

= 2(m + CMMd n 2 m + CMMd

(A.7)

(A.8)

where

CM = Re(H)

41Z1(a)M = 1 + a a

M=1+aK (a)0

2 20wr

a = (1 + i) /r = tube radius

v = fluid kinetic viscosity

Note that there are two terms in the damping:

structural damping and the other is associated with

(c) Oscillations in Fluid with Tube Support

The full equation given in Eq. (A.1)

case. Let

u = q(t) 4n(x)

Using Eqs. (A.1) and (A.9) yields

d2d 2+2 4o +w q=0

dt2 n ndt n

one is associated

fluid damping.

with

has to be used in this

(A.9)

(A.10)

78

n (X 2 E0.5

Wn t2 M,2 R

m + CMMd +

___ ___ __MdCv?

n 2X 2 2- Im(h) +2Xw .M' ( ) n 2+C d

mMMd+

The added

given by

mass associated with the gap M' and damping coefficient Cv are

LM' = f (C - 1)Mddz

-L d(A.13)

(A.14)L

C f C'dzv -L v

where 2L

A.2]

is the tube support plate width and C and C' are given by [Ref.

C = R rr)Re(H')[1 - ch(z/R) (A.15)

C' = Md(R rn[-Im(H')J[1 - cosz ] (A.16)v d R r n coshLR

Substituting Eq. (A.15) into (A.13) and (A.16) into (A.14) and performing

the integrations yields

M' = 21d[K(R r )Re(H') - 1] (A.17)

Cv = 2UMdK(R r )w[-Im(H')] (A.18)

where,

K =1 - tanh (1)

where,

(A.1l)

(A.12)

79

H' a sinh a(a + a sinh a - 2 cosh a)

w (R -r)

a=(1+i) n 22v

Using Eqs. (A.ll) and (A.12), together with related equations, the natural

frequency and modal damping ratio can be calculated.

References

A.l Chen, S. S., Wambsganss, M. W., and Jendrzejczyk, J. A., "Added Massand Damping of a Vibrating Rod in Confined Viscous Fluids," J. Appl.Mech. 98(2), 325-329 (1976).

A.2 Mulcahy, T. M., "Fluid Forces on Rods Vibrating in Finite LengthAnnular Regions," J. Apple. Mech. 47, 234-240 (1980).

100

I0

I

10 100

S

1000

Real and Imaginary Values of H for a Cylinder Vibrating in an Infinite Fluid.

Re (H)

-Im (H)

SS =wr 2

- w= CIRCULARFREQUENCY

vE

"MON0.1

0.01

0.0010.I

0

10000 100000I

Fig. A.2.

81

- 1100.0

80.060.0

40.0

20.0

10.08.0

6.0

4.0

2.0

1.0

0.80.6

0.4

0.2

0.1 I I I I I I 10.4 0.6 i.0

I I I I

2.0 4.0 6.0

+1 V

Added Mass and Damping Parameters, Re[H'] and Im[H']

(Ref. A.2).

-Im(H)

1111-

Re(H')

0.I 0.2

I I I

Fig. A.3.

10.0

F

i i i i i . iI

-'"

I

82

Distribution for ANL-84-39

Internal:

E. S. BeckjordC. E. TillR. S, ZenoP. R. HuebotterG. S. RosenbergM. W. WambsganssA. R. BrunsvoldB. L. BoersR. D. CarlsonS. S. Chen

H. H. ChungH. Halle

J. A. Jendrzejczyk (72)T. M. MulcahyP. TurulaM. P. AgrestaS. K. ZussmanR. A. ValentinS. H. FistedisANL Patent Dept.ANL Contract FileANL Libraries (2)TIS Files (6)

External:

DOE-TIC, for distribution per UC-79k (126)Manager, Chicago Operations Office, DOEDirector, Technology Management Div., DOE-CHW. Frost, DOE-CHComponents Technology Division Review Committee:

D. J. Anthony, General Electric Co., Schenectady, NYA. A. Bishop, U. PittsburghB. A. Boley, Northwestern U.F. W. Buckman, Wood-Leaver & Assoc., Monroeville, Penn.R. Cohen, Purdue U.J. Weisman, U. Cincinnati

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