[asme asme 2005 pressure vessels and piping conference - denver, colorado, usa (july 17–21, 2005)]...

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Proceedings of PVP2005 2005 ASME Pressure Vessels and Piping Division Conference

July 17-21, 2005, Denver, Colorado USA

PVP2005-71380

ACOUSTICALLY INDUCED VIBRATION OF DRUMS EXCITED BY ROTATING MACHINERY

Itsuro HAYASHI

CHIYODA ADVANCED SOLUTIONS CORPORATION 1-25, Shin-Urashima-cho 1-chome, Kanagawa-ku, Yokohama 221-0031, Japan

Tel: (81-45) 441-1283 Fax: (81-45) 441-1286 E-mail: [email protected]

Shijie GUO

EBARA RESEARCH CO., LTD 2-1, Honfujisawa 4-chome, Fujisawa-shi 251-8502, Japan

Proceedings of PVP20052005 ASME Pressure Vessels and Piping Division Conference

July 17-21, 2005, Denver, Colorado USA

PVP2005-71380

ABSTRACT Rotating machinery generates pressure pulsations, and the pulsations may cause severe vibrations of drums in high frequency region, resulting in material fatigue failure under certain conditions. Experiments and numerical simulations were performed to investigate the mechanism of the high frequency vibrations of the drums downstream of compressors. The results show that fatigue failure occurs when acoustic diametral modes of a drum are excited by pressure loading. In order to establish practical countermeasures against the vibrations, three-dimensional sound-structural coupled analysis as well as one-dimensional pulsation analysis were conducted. As a result, practical measures such as changing diameter, or thickness of the drums, applying restriction orifice are confirmed effective by using the approach proposed in this study. The validity of the simulation methods incorporating the sensitivity to the fluid conditions is shown. NOMENCLATURE A : cross-sectional area of a pipe [m2] K : bulk modulus of elasticicty of gas [Pa] Rcn : equivalent concentrated resistance

[kg/sm2] Rｐ : equivalent resistance of pipe [kg/sm3] c : speed of sound [m/s] p : pressure fluctuation[Pa]

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t : time [s] u : velocity fluctuation [m/s] x : distance along pipe [m] xi : spatial coordinates [m] xn : position of a concentrated resistance

[m] δ : delta function ρ : density of gas [kg/m3] ω : angular frequency [rad/s] C : geometrical coupling matrix FF : acoustic load vector FS : structural load vector KF : fluid stiffness matrix KS : structural stiffness matrix MF : fluid mass matrix MS : structural mass matrix p : vector of nodal pressures w : vector of nodal displacements INTRODUCTION Severe vibrations in piping systems or drums in high frequency region are frequently observed in process plants when diameters of the systems are large enough to excite acoustic resonance in diametral direction. High frequency vibrations of piping systems due to random excitations are well known to occur downstream of high pressure ratio reducing devices such as safety valves, depressurizing valves and restriction orifices. In particular, fatigue

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failure is observed in the case of complete coincidence in which in addition to a match between wave numbers also resonance of acoustic and mechanical frequencies occurs [1-3]. Circumferential stiffener rings are effective in piping systems to reduce vibration level, and their reduction effect can be evaluated by using sound-structural coupled analysis [4-5]. On the other hand, significant vibrations are usually excited by tonal pulsations from compressors. Piping vibrations due to reciprocating compressors can be evaluated with API618 [6]. One-dimensional modal analysis or transfer matrix method is usually applied in engineering [7]. However, in the high frequency region as is observed in centrifugal compressors, since the diametral acoustic modes dominate the vibration characteristics, three-dimensional sound analysis is required as well. In this study, acoustically induced vibration of drums was investigated by means of experiments and analysis to find effective and practical ways for vibration cut-down in engineering. The analysis in this study was a combination of one-dimensional pulsation analysis and three-dimensional sound-structural coupled analysis. CONDITION OF EXPERIMENT

VIBRATION TEST The experimental setup for vibration test is shown in Fig.1. The test drums were installed downstream of a compressor. The compressor gave pressure pulsations with a dominant frequency of 297Hz, which was the blade passing frequency. The specifications of the compressor and the test drums are listed in Table 1 and Table 2. The test conditions are listed in Table 3. Each case was performed under several conditions of water levels in the drum. A single-hole orifice plate having orifice bore equal to 127mm (open area ratio:25%) was installed at the drum inlet nozzle to investigate its vibration-reduction effect. Seven accelerometers were set on the circumference of the drum. The signals were recorded by a multi-channel Digital Audio Tape (DAT) recorder simultaneously.

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Table 1. Specifications of the compressor

Type Water Sealed Fluid Air Discharge Pressure 0.35 MPa Discharge Temperature 319-323 K Rotating Speed 890 rpm Blade Passing Frequency 297 Hz

Table 2. Specifications of the test drums Material SUS316L Diameter 1.15 m

No. 1

Wall thickness 0.006 m Material SS400 Diameter 1.0 m

No. 2

Wall thickness 0.006 m Material SUS316L Diameter 1.0 m

No. 3

Wall thickness 0.012 m

Table 3. Test conditions Case No. Drum No. Orifice Plate

1 1 N/A 2 2 N/A 3 3 N/A 4 1 Applied

SPEAKER TEST Speaker test was performed for No.1 and No.2 drums (Table 2) in order to investigate the acoustic resonant frequencies and the acoustic modes in the drums. The experimental setup for the speaker test is shown in Fig.2. A loud speaker was set at the inlet nozzle. It gave acoustic excitations at a frequency range of 200-600Hz. Eight microphones were set inside the drum around the circumference. The signals were recorded by a DAT simultaneously so that the acoustic mode shapes could be obtained. The water level was set at certain conditions during the speaker test. CONDITION OF NUMEERICAL SIMULATION

PULSATION ANALYSIS The equation of motion and the continuity equation for fluctuations of fluid for one-dimensional flow in a pipe are expressed as follows [7]:

( ncnp xxuRuR )xp

tu

−−−∂∂

−=∂∂ δρ (1)

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( )xAu

tp

KA

∂∂

−=∂∂

(2)

Transfer matrix method was applied for one-dimensional pulsation analysis to evaluate the effect of the orifice plate as a countermeasure. The piping model for the one-dimensional transfer matrix method is shown in Fig.3.

SOUND-STRUCTURAL COUPLED ANALYSIS The wave equation for acoustic analysis and the coupled equations between structural displacement and acoustic pressure are as follows:

(3)

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−−

F

S

FF

tSS

FF

pw

MKCCMK

2

2

ωω

(4)

In this study, Finite Element Method (FEM) was used for both structural and acoustic analyses. The structural FEM model of the drum is shown in Fig.4. Clamped boundary conditions were applied to the legs at the ends fixed to the ground. The solid model for acoustic analysis is shown in Fig.5. The pressure pulsation was loaded at the inlet nozzle. In the sound-structural coupled analysis, the two models were linked on the inside surface of the drum. Property constants for numerical simulations are shown in Table 4. Table 4. Property constants for numerical simulations Modulus of elasticity 197 GPa Poisson’s ratio 0.3 Speed of sound (Fluid) 340 – 360 m/s RESULTS

VIBRATION TEST Figure 6 shows spectra of displacement amplitude of the drum wall under pressure loading from the compressor. Remarkable peaks were observed at 297Hz, which was the blade passing frequency of the compressor. The maximum displacement amplitudes of the drum wall at 297Hz for drum No.1 are shown in Fig.7. The amplitude

02

22

2

2

=∂∂

−∂∂ pp

ixc

t

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changed with water level. The maximum amplitude of drum No.1 was 140µmp-p which may result stresses large enough to cause fatigue failure. Actually we have succeeded in confirming the generation of cracks at the toe of the weld between the shell and the leg after 60-hours endurance test for drum No.1. It means that there is a possibility of fatigue failure in certain conditions, even though the compressor is well-designed. On the other hand, the maximum amplitude of drum No.2, whose wall thickness was the same with that of drum No.1 was 40µmp-p. These results show that the diameter of a drum is a dominant factor to the vibration in high frequency region. The effects of an orifice plate installed at the inlet nozzle of the drum are shown in Fig.6 and Fig.7. A decrease of 67% in the maximum vibration level was obtained by applying the orifice plate. Displacement amplitudes measured by each accelerometer are shown in Fig.8. SPEAKER TEST Table 5 shows the acoustic resonant frequencies of drum No.1 obtained in the speaker test with the results of the acoustic modal analysis. Good agreement was obtained with an accuracy of 98.5%. Table 5. Acoustic resonant frequencies (Hz) of drum No.1 Water Level = 900 mm Water Level = 1200 mm Experiment Calculation Experiment Calculation

226 243 271 280

- 295 318 324

227 242

- 278 286 294 316 325

219 253 266

- 294

- 340

- 251 261 286 295 329 345

Table 6. Calculated acoustic resonant frequencies (Hz) of drum No.2 and drum No.3 at water level = 1200mm

Drum No.2 Drum No.3 284 286 338 355

278 314 331

In Table 5, an acoustic mode was observed within the frequency range of 294-295Hz, which was close to the blade passing frequency of the compressor. Figure 9 shows the circumferencial phase of the acoustic mode of 295Hz. The second order diametral mode was excited in this plane. The


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three-dimensional mode shape obtained by FEM analysis shows good agreement with that obtained in the speaker test (Fig. 10). The calculated resonant frequencies of drum No.2 and drum No.3 are shown in Table 6. There are no frequency matching between the blade passing frequency and the acoustic resonant frequencies of drum No.2 and drum No.3. Therefore, it can be said that the high level vibration of drum No.1 at the blade passing frequency was caused by acoustic resonance inside the drum. DISCUSSION

EFFECT OF ORIFICE PLATE Orifice plates are usually used to mitigate pressure pulsations from compressors. In this study, the mechanism of the pulsation reduction by an orifice plate was investigated in the high frequency region under the acoustic resonant condition. To understand the effect of the orifice plate to the acoustic modes in the drum, acoustic resonant frequencies are shown in Table 7. Table 7. Acoustic resonant frequencies of drum No.1. obtained by speaker test.

Acoustic resonant frequency [Hz] Water Level[mm] Without orifice plate With orifice plate

900 295 294 1000 295 296 1200 294 295

Table 7 shows that the second order diametral mode in the drum was not affected by the orifice plate. This means that the vibration-reduction by the orifice plate is not from the change of acoustic modes in the drum. Furthermore, the displacement amplitude of the drum wall in Fig. 8 shows that the vibration mode shape was not affected by the orifice plate. From these results, it can be said that the orifice plate may not strongly influence the three-dimensional acoustic characteristics in the drum and the structural characteristics of the drum. Therefore, the vibration-reduction by the orifice plate was evaluated by the one-dimensional pulsation analysis. Table 8 shows the results of the pulsation analysis for drum No.1 with and without an orifice plate. A decrease of 70% in the pressure pulsation and in the velocity fluctuation was obtained by applying the orifice plate. This reduction rate is similar to that of the vibration level of the drum wall. In the calculation, the orifice plate was modeled as concentrated resistance in the pipe. Therefore, it can be said that the vibration of a drum with an orifice plate can be roughly evaluated by use of one-dimensional pulsation analysis

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considering a damping effect at the orifice plate in the nozzle without considering the three-dimensional resonant condition in the drum. Further discussion is required in the future work to clarify the detailed mechanism of the pulsation reduction by an orifice plate with the pulsation source at the compressor being considered. Table 8. Effect of the orifice plate to the vibrations and pulsations of drum No.1 (Case 1 and Case 4). Each value is shown as a relative level to the original case (without orifice plate). Without

orifice plate

With orifice plate

Pressure pulsation in discharge pipe of compressor (Calculation)

100 31.4

Velocity fluctuation at drum inlet nozzle (Calculation)

100 27.5

Vibration level of the drum wall (Experiment)

100 32.8

EFFECT OF STRUCTURAL CHARACTERISTICS OF

DRUM The results of the speaker test shows that the high level vibration of drum No.1 is caused by the acoustic resonance in the drum. On the other hand, structural characteristics influence the dynamic response of the drum as well. Therefore, the three-dimensional structural analysis together with the acoustic analysis is required to evaluate the vibration of the drum. In order to investigate the effect of wall thickness of a drum, structural modal analysis was performed for drum No.1. Table 9 shows the mechanical natural frequencies of drum No.1 with different wall thickness. Table 9. Calculated mechanical natural frequencies (Hz) of drum No.1. Three cases of wall thickness were investigated. Wall thickness 6 mm

Wall thickness 12 mm

Wall thickness 18 mm

288 292 295 303 310

274 290 305 310 357

270 293 297 306 377

In the frequency range higher than 250Hz, the mode density is so high that the avoidance of frequency matching between loading pressure and mechanical natural frequencies is difficult in design. In such a case, the dynamic response to the pressure loading


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has to be investigated. The maximum displacement and stress amplitude for these cases were analyzed by sound-structural coupled analysis and the results are shown in Fig.11 and Fig.12. When the wall thickness is 18mm, mechanical natural frequencies of 293Hz and 297Hz are close to the pressure loading frequency (Table 9). However, maximum stress amplitude for the wall thickness of 18mm is lower than that of 12mm. This means that the dynamic response of the drum is not simply dominated by the mechanical natural frequencies. The structural mode shapes have to be considered as well in the evaluation. So as to discuss the dynamic response of the drum to the pressure loading, the deformation of drum No.1 under the pressure loading at 297Hz are shown in Fig.13. Higher order structural mode was excited by the pressure loading from the compressor, while the second order diametral acoustic mode was excited in this frequency (Fig.10). The detailed form can not be confirmed because of lack of the number of the accelerometers used in this experiment. However, the deformed shape seems to be plausible because it shows similar shape to those of the structural modes near the pressure loading frequency. The detailed validity of the simulation is discussed by the displacement amplitude described below. Sound-structural coupled analysis was carried out for drums No.1, No.2, and No.3 under the same pressure loading condition to find the practical countermeasures in engineering. Speed of sound was changed as a parameter in order to incorporate the change in process conditions of the gas. Figure 14 shows the maximum displacement amplitude of the drums for three cases. In the range of 340-360 m/s of sound speed, the vibration was decreased by 68% by changing the drum diameter from 1.15 m to 1.0 m. Further reduction of 86% was obtained by increasing the drum wall thickness from 6mm to 12mm in Case 3. The effect of these countermeasures and the validity of the evaluation methods were confirmed by the experimental results shown in Fig.15. In Fig.15, 85% reduction in the vibration is shown in Case 3 in the whole range of operating water level. On the basis of the result, it can be said that the practical countermeasures can be established by using the three-dimensional sound-structural coupled analysis taking into consideration the sensitivity to the fluid. SUMMARY AND CONCLUSION The one-dimensional pulsation analysis and the three-dimensional analysis proposed in this study are confirmed effective to evaluate high frequency vibration of drums in engineering. The high frequency

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vibration of the drum downstream of the compressor was found to be caused by the acoustic resonance in the drum. Fatigue failure may occur when acoustic diametral mode of a drum is excited. Using an orifice plate is an effective way to reduce vibration in high frequency, and its effect can be evaluated by the one-dimensional pulsation analysis. The influence of the structural characteristics of the drums to the vibration level can be estimated by use of the three-dimensional sound-structural coupled analysis with the sensitivity to the process conditions being considered. REFERENCES [1] Carucci, V. A., and Mueller, R. T., 1982,

“Acoustically Induced Piping Vibration in High Capacity Pressure Reducing Systems,” ASME Paper 82-WA/PVP-8, pp5-18

[2] Eisinger, F. L., 1996, “Designing Piping Systems

against Acoustically-Induced Structural Fatigue,” ASME PVP 328, pp397-404

[3] Eisinger, F. L., and Francis, J. T., 1999,

“Acoustically-Induced Structural Fatigue of Piping Systems,” ASME PVP 389, pp393-399

[4] Hayashi, I., Hioki, T., and Isobe, H., 2002,

“Investigation of Design Method for Piping Systems to Prevent Acoustic Fatigue in Process Plants,” ASME PVP 440, pp137-143

[5] Hayashi, I., Hioki, T., and Isobe, H., 2003,

“Evaluation of Acoustically Induced Vibration and Fatigue Failures in Process Piping Systems,” Internoise, 926, pp4080-4087.

[6] API Standard 618, Forth Edition, June 1995 [7] Matsuda, H., and Hayama, S., 1989, “A Method for

Calculating Pressure Pulsations Taking Dynamic Compressor-Piping Interaction into Account,” ASME PVP 154, pp17-23


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Figure 1. Experimental setup for vibration test

Figure 3. Piping model for one-dimensional

pulsation analysis.

LG

LG : Level Gage

Water supply

Restriction Orifice

Test Drum

M : Microphone

M M M

Sound Levelmeter

DAT Recorder

Multi channel

Loud Speaker

Power Amplifier

GraficEqualizer

GeneratorFFT Analizer

Multi channel

LG

LG : Level Gage

Water supply

Restriction Orifice

Test Drum

M : Microphone

M M M

Sound Levelmeter

DAT Recorder

Multi channel

Loud Speaker

Power Amplifier

GraficEqualizer


Multi channel

LG

LG : Level Gage

Water supply

Restriction Orifice

Test Drum

M : Microphone

M M M

Sound Levelmeter

DAT Recorder

Multi channel

Loud Speaker

Power Amplifier

GraficEqualizer


Multi channelP

P

Flow Orifice

FI

To Pit

Drain

P

T

LG

From Atmosphere

P

P : Pressure TransducerT : ThermometerLG : Level GageFI : Flowmeter

Pump

Water supply

Restriction Orifice

Compressor Test Drum

Discharge GasV : Accerelometer

V V V

Signal Conditioner

DAT Recorder

Multi channel

Multi channelPP

Flow Orifice

FI

To Pit

Drain

P

T

LG

From Atmosphere

P

P : Pressure TransducerT : ThermometerLG : Level GageFI : Flowmeter

Pump

Water supply

Restriction Orifice

Compressor Test Drum

Discharge GasV : Accerelometer

V V V

Signal Conditioner

DAT Recorder

Multi channel

Multi channel

Figure 2. Experimental setup for speaker test.

P001

P002

PS00

P003

RO00

P004

P005

PD00

P006

P007

PS01

BV00

Restriction Orifice

Test DrumCompressor

Pit

P001P001

P002P002

PS00PS00

P003P003

RO00RO00

P004P004

P005P005

PD00PD00

P006P006

P007P007

PS01PS01

BV00BV00

Restriction Orifice

Test DrumCompressor

Pit


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Figure 4. Three-dimensional structural shell

model of drum No.1 for structural FEM.

Inlet nozzle

Outlet nozzle

Inlet nozzle

Outlet nozzle

Inlet nozzle

Outlet nozzle

Figure 5. Three-dimensional acoustic solid model

of drum No.1 for acoustic FEM.

0

20

40

60

80

100

120

140

160

700 800 900 1000 1100 1200 1300Water Level　( mm )

Dis

plac

emen

t am

plitu

de　

(µm

p-p

)

Without Orifice Plate (Case 1)

With Orifice Plate (Case 4)

0

50

100

150

200

0 200 400 600 800

Frequency(Hz)

Dis

plac

emen

t Am

plitu

de( µ

mp-

p)

Without OrificePlate(Case1)With OrificePlate(Case4)

Figure 6. Measured spectra of displacement

amplitude of the drum wall for Case 1 and Case 4. Water level was 1000mm.The restriction orifice with 25% open area ratio was applied.

Figure 7. Measured maximum displacement

amplitude of the drum wall for Case 1 and Case 4. The restriction orifice with 25% open area ratio was applied.


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0

40

80

120

1600

45

90

135

180

225

270

315

Without Orifice Plate (Case 1)With Orifice Plate (Case 4)

Pressure Loading

Figure 8. Measured displacement amplitude of the drum wall for Case1 and Case 4.Water level was 1000mm.

The restriction orifice with 25% open area ratio was applied.

Figure 9. Measured circumferencial phase for the

acoustic mode of 295Hz for drum No.1 with water level 1200mm.

Figure 10. Three-dimensional mode shape for the acoustic mode of 295Hz for drum No.1 with water level 1200mm. Acoustic modal analysis was performed.

-180

-135

-90

-45

0

45

90

135

180

Pressure Loading


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Figure 11. Calculated maximum displacement

amplitude of the drum wall for drum No.1. Relative value to the case of 6mm wall thickness are shown.

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Wall thickness (mm)

Dis

plac

emen

t am

plitu

dedm

ax /

dmax

@t6

(-)

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Wall thickness (mm)

Stre

ss a

mpl

itude

smax

/ sm

ax@

t6 (-

)Figure 12. Calculated maximum stress amplitude in

the drum wall for drum No.1. Relative value to the case of 6mm wall thickness are shown.

Figure 13. Deformations of the drum wall for Case1 with water level 1000mm. The results of sound-structural coupled analysis and the experimental results measured by accelerometers are shown.

-550

600

1750

08 16

2432

40

48

56

64

72

80

88

96

104

112

120

128

136

144152

160168176184192

200208

216

224

232

240

248

256

264

272

280

288

296

304

312

320

328336

344 352

CalculationExperimentUndeformed

Inlet Nozzle



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0

0.2

0.4

0.6

0.8

1

1.2

280 300 320 340 360 380Speed of sound (m/s)

Dis

plac

emen

t am

plitu

dedm

ax/d

max

@dr

umN

o.1(

-)

Drum No.1

Drum No.2

Drum No.3

Figure 14. Calculated maximum displacement amplitude of the drum wall for drums No.1, No.2 and No.3.

Relative value to the maximum displacement of drum No.1 is shown. Speed of sound was changed to investigate the sensitivity to the fluid conditions.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

800 1000 1200Water Level (mm)

Dis

plac

emen

t am

plitu

dedm

ax /

dmax

@dr

umN

o.1(

-)

Drum No.1Drum No.2Drum No.3

Figure 15. Measured maximum displacement amplitude of the drum wall for drums No.1, No.2 and No.3.

Relative value to the maximum displacement of drum No.1 is shown.



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