43.6 Practical Example

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43.6.1 Expansion-Type Muffler for Reciprocating Compressor

Consider a double-acting (i.e., fluid on both sides of the piston in the cylinder) reciprocating compressor

for supplying high-pressure air to a machine shop of a factory, for example.

The specifications of the reciprocating compressor follow:

* Delivery pressure: 6.9 £ 105 Pa

* Rotational speed of driving shaft: 600 rpm

* Power of driving shaft: 450 kW

* Diameter of inlet pipe: 380 mm

Pressure pulsations of 10 and 20 Hz are produced by the compressor due to the rotational speed, as seen

in Figure 43.5. The pressure wave from the inlet propagates the free space and can the damage nearby

private houses. Wooden doors with glass paneling, wooden sliding-doors, and leaves of plants and foliage,

and have been found to vibrate due to low-frequency audible sound. An attenuation of 15 to 20 dB was

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Design of Reactive Mufflers 43-13

© 2005 by Taylor & Francis Group, LLC

necessary at the frequencies 10 and 20 Hz. A muffler using an expansion and tail pipe type was suggested

to handle the problem. The reflection coefficient of the tail pipe is approximately R ¼ 21 and ka ¼ 2

pfa=c ¼ 2p £ 20 £ 0:19=345 ¼ 0:0692:

IL ¼ 20 log10lcos kl2 cos kl3 2 m sin kl2 sin kl3l

where

k ¼ 2pf =c; wave number (1/m)

l2 ¼ the length of the chamber (m)

l3 ¼ the length of the tail pipe (m)

m ¼ the expansion ratio of the cross section

between the chamber and inlet

With kl2 ¼ kl3 ¼ p=2; we have

IL ¼ 20 log10lml

We need m . 10 in order to satisfy the desired value

of IL. For 20 Hz, we use kl2 ¼ kl3 ¼ p: When kl2 ¼

kl3 ¼ p=2 at f ¼ 10 Hz, we have IL ¼ 0. Then,

using kl2 ¼ kl3 ¼ p=2 at frequency 15 Hz, we can

satisfy the IL condition of 20 dB at both frequencies.

Hence, l2 ¼ l3 ¼ 345=ð4 £ 15Þ ¼ 5:75 (m) is

chosen at a speed of sound c ¼ 345 m/s.

The noise spectrum at the inlet of the reciprocating

compressor under study and insertion loss

of the muffler design in this example are shown

in Figure 43.5.

The diameters or the lengths of the chamber and

the tail pipe are properly selected, as shown and in

Figure 43.6. At 10 Hz, IL is determined as

FIGURE 43.5 Noise spectrum at inlet of reciprocating compressor and insertion loss of designed muffler.

Sound wave

To compressor

Expansion chamber

φ1600 × 4500

φ380 × 6000

Filter

Tail pipe Air flow

FIGURE 43.6 The muffler designed for the noise

control of a reciprocating compressor.

43-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

indicated below:

IL ¼ 20 log10lcosð2p £ 10 £ 4:5=345Þcosð2p £ 10 £ 6=345Þ 2 ð1:6=0:38Þ2sinð2p £ 10 £ 4:5=345Þ

£ sinð2p £ 10 £ 6=345Þl

¼ 20 log10l0:6825 £ 0:4600 2 17:728 £ 0:7308 £ 0:8879l ¼ 21:0 ðdBÞ

Similarly, at 20 Hz, we have

IL ¼ 20 log10lcosð2p £ 20 £ 4:5=345Þcosð2p £ 20 £ 6=345Þ 2 ð1:6=0:38Þ2sinð2p £ 20 £ 4:5=345Þ

£ sinð2p £ 20 £ 6=345Þl

¼ 20 log10lð20:0682Þ £ ð20:5767Þ 2 17:728 £ 0:9977 £ 0:8170l ¼ 23:2 ðdBÞ

Clearly, the attenuation at both frequencies satisfies the desired lower limit of 20 dB. Calculated values of

IL at low frequencies are shown by a curved continuous line in Figure 43.5.

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