Checking Heat Transfer Equipment for Noise and Vibration Problems
Checking Heat Transfer Equipment for Noise and Vibration Problems
Whenever a fluid flows across a tube bundle such as boiler tubes in an economizer, evapo- rator, or air heater, vortices are formed and shed beyond the wake beyond the tubes. This shedding on alternate sides of the tubes causes a harmonically varying force on the tube perpendicular to the normal flow of the fluid and a self-excited vibration results. If the frequency of the von Karman vortices, as they are called, coincides with the natural fre- quency of vibration of the tubes, resonance occurs and tubes vibrate, leading to leakage and damage at tube supports. Vortex shedding is most prevalent in the Reynolds number
from 300 to 2 × 10 5 . Boiler components such as superheaters, evaporators, economizers, and air heaters operate very much within this range. There is some evidence from the litera- ture that vortex shedding does not occur in two-phase flow and that it is a concern only in single-phase flows.
Another mechanism associated with vortex shedding is acoustic vibration, which is normal to both fluid flow and tube length. This is observed only with gases and vapors. Standing waves are formed inside the duct at a particular frequency, and the noise can be
a nuisance. When the boiler load changes, the noise typically goes away. Hence, in order to analyze tube bundle vibration and noise, three frequencies must
be computed: natural frequency of vibration of tubes, vortex shedding frequency, and acoustic frequency. When these are apart by at least 20%, vibration and noise may be absent. There are instances when these frequencies matched but no vibration issues were noticed!
Determining Natural Frequency of Vibration of Tubes
The natural frequency of transverse vibrations of a uniform beam supported at each end is given by [4]
4 05 f .
n = ( C/ 2 π ) EI/m/L ) (6.15)
where
f n is the natural frequency of vibration, 1/s
C is a factor determined by end condition of tube support, Table 6.6
E is Young’s modulus of elasticity, N/m 2 (186 to 200 × 10 9 N/m 2 for carbon steel)
I = (π/64)(d 4 −d 4 i ), m 4 L is the length of tube between supports, m m is the tube weight, kg/m
328 Steam Generators and Waste Heat Boilers: For Process and Plant Engineers
TABLE 6.6
C Values for Various Modes of Vibration
End Support Type
Both ends clamped 22.37 61.67 120.9 One clamped, one hinged
15.42 49.97 104.2 Both hinged
Acoustic Frequency f a
fi a s given by V / s λ (6.16) where V s is the velocity of sound at the air temperature in the duct in m/s = (ϑRT/MW) 0.5 .
ϑ is the ratio of specific heats = 1.4, R = 8.314 J/mol K, MW = molecular weight, T is gas temperature in K. Simplifying and using an MW of 29, we have V s = 20√T
Wavelength λ = 2w/n where w is the width of duct, m, and n is the mode of vibration.
Vortex Shedding Frequency f e
Strouhal number S is used to determine the vortex shedding frequency f e .
(6.17) where
fd e
d is the tube OD, m
V is the gas velocity in the tube bundle, m/s S is available in the form of charts (Figure 6.7 a through d). It typically ranges from 0.2 to
0.3. Note that these calculations give some idea whether vibrations are likely. One should rely more on field experience with certain tube spacing and geometry of the heat transfer equipment.
As mentioned earlier, if f e and f n are close by less than 20%, then bundle vibration is
likely. If f e and f a are within 20% of each other, noise vibration is likely.
Example 6.12
A tubular air heater in a boiler plant is 3.57 m wide, 3.81 m deep, and 4.11 high. 50.8 × 46.7 mm carbon steel tubes are arranged in an inline fashion with a transverse pitch of 89 mm and longitudinal pitch of 76 mm. There are 40 tubes wide and 60 tubes deep. 136,000 kg/h of air at an average temperature of 104°C flows across the tubes. Tubes are fixed at both ends. Weight of tube = 2.485 kg/m. Check if tube bundle vibrations are likely.
First determine the natural frequency of vibration in the first and second modes. 0 . 5 5
f n = ( . 22 37 2 / π ) 200 10 × 9 ×× π ( . 0 0508 4 4 − 4 . 0 0467 ) / . 2 485 4 11 / . = . c/s. In mode 2, 18 3
it will be (61.67/22.37) × 18.3 = 50.3 c/s Let us compute the vortex shedding frequency f e . From Figure 6.7a, for S T /d = 89/50.8 = 1.75 and S L /d = 76/50.8 = 1.5, S is 0.33. Compute the actual air velocity. Density at 104°C and at atmospheric pressure = ρ g =
12.17 MW/(t g + 273) = 12.17 × 28.84/(104 + 273) = 0.932 kg/m 3 .
Miscellaneous Boiler Calculations 329
0.5 S L /d=1.25
er (s) (in-line)
Strouhal numb
Flow 0.1 S
0.5 Unfinned tube bank Finned tube bank
er (s) stagger
Strouhal numb
Strouhal number for (a, b) inline bank of tubes. (Continued)
330 Steam Generators and Waste Heat Boilers: For Process and Plant Engineers
3.0 1.0 0.40 0.35 0.30 0.25 0.23 0.20 /d S T
0.72 S L /d
S T 2.6 0.20 0.26 0.27 0.19 0.23 0.19
FIGURE 6.7 (Continued)
Strouhal number for (c) Staggered, and (d) inline tube banks. Gas mass velocity = 136,000/[40 × (0.089 − 0.0508) × 4.11]/3600 = 6.01 kg/m 2 s
Air velocity = 6.01/0.932 = 6.45 m/s
Hence, from S = f e d/V, f e = 0.33 × 6.45/0.0508 = 41.9 c/s
Compute f a , the acoustic frequency
Sonic velocity V s = 20 × (104 + 273) 0.5 = 388 m/s.
Width w = 3.56 m, wavelength = 2 × 3.56 = 7.13 m, and f a = 388/7.13 = 54.4 c/s The results are summarized in Table 6.7. It may be seen that without baffles, f a and f e are
close. Hence, noise problems may arise, particularly as the load increases slightly. Then,
Miscellaneous Boiler Calculations 331
TABLE 6.7
Summary of Results
Mode of Vibration
18.3 50.3 f e 41.9 41.9
f a (Without baffle)
f a (With 1 baffle)
the gas velocity and f e will increase. If a baffle plate is inserted parallel to the tube length (dividing the duct width by half), then f a will double thus keeping f e and f a far apart.
Vortex shedding is unlikely to cause damage. This is due to the large mass of the system compared to the low energy in the gas stream (the analogy is a tall pole that will flex in the wind, but if it is short, it may not). Thus, using tube supports at intervals of 2–2.5 m will help reduce the vibration.
The tube natural vibration frequencies will be small in the first mode of vibration. The amplitude of vibration will be smaller at higher modes, and hence, the first mode is generally considered in the analysis.