8.6 DISSIPATIVE MUFFLERS

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Dissipative mufflers or silencers differ from reactive mufflers in that noise

reduction in a dissipative muffler is achieved primarily by attenuation of the

acoustic energy within the lining or other elements within the muffler. The

dissipative muffler may also reflect some of the acoustic energy back to the

source, similar to the action of the reactive muffler. Some configurations of

dissipative mufflers are shown in Fig. 8-17.

Dissipative mufflers usually have wide-band noise reduction characteristics.

The sharp peaks and valleys in the transmission loss curves that are

found for reactive mufflers are usually not present for dissipative mufflers.

As a result of this characteristic, dissipative mufflers are useful for solving

noise control problems involving continuous noise spectra, such as fan

noise, intake and exhaust noise from gas turbines, and noise through access

openings in acoustic enclosures.

For the dissipative muffler shown in Fig. 8-18, the instantaneous

acoustic pressure and instantaneous volume flow, including the effect of

acoustic energy attenuation, may be written as follows:

pрx; tЮ ј Ae_x e jр!t_kxЮ ю Bex e jр!tюkxЮ (8-148)

Uрx; tЮ ј рSA=_ocЮ e_xe jр!t_kxЮ _ рSB=_ocЮ exe jр!tюkxЮ (8-149)

The quantity  is the attenuation coefficient for the muffler lining, and the

quantities A and B are constants to be determined from the boundary conditions.

The first term on the right side of Eqs (8-148) and (8-149) represents

the sound wave traveling in the юx direction, and the second term represents

the sound wave traveling in the _x direction.

Suppose we take the origin рx ј 0Ю at the interface of the inlet tube and

the muffler. At this point, the acoustic pressure on the inlet tube side p1р0; tЮ and on the muffler side p2р0; tЮ must be equal. Also, the volumetric flow

quantities U1р0; tЮ and U2р0; tЮ must be the same at the interface. These two

conditions yield the following relationships between the constants:

A1 ю B1 ј A2 ю B2 (8-150)

A1 _ B1 ј mрA2 ю B2Ю (8-151)

The quantity m is the area ratio, m ј S2=S1. It is assumed that the inlet and

outlet tubes have the same dimensions, S1 ј S3.

At the other end of the muffler рx ј LЮ, we may equate the instantaneous

acoustic pressures p2рL; tЮ and p3рL; tЮ. The volume flow quantities

U2рL; tЮ and U3рL; tЮ are also equal to the exit interface of the muffler. In

calculating the transmission loss for the muffler, it is assumed that there are

Silencer Design 377

Copyright © 2003 Marcel Dekker, Inc.

378 Chapter 8

FIGURE 8-17 Configurations for dissipative mufflers: (a) circular lined chamber,

(b) rectangular lined chamber, and (c) baffle-type muffler.

FIGURE 8-18 Nomenclature for dissipative muffler.

Copyright © 2003 Marcel Dekker, Inc.

no reflected sound waves in the exit tube, B3 ј 0. The application of these

conditions yields two additional relationships between the constants:

A2 e_L e_jkL юB2 eL ejkL ј A3 (8-152)

mрA2 e_L e_jkL _B2 eL ejkLЮ ј A3 (8-153)

The sound power transmission coefficient is the ratio of the power

transmitted to the power incident on the muffler:

at ј

Wtr

Win ј

S3Itr

S1Iin ј

p2

tr

p2

in ј jA3j2

jA1j2 (8-154)

If we solve Eqs (8-150) through (8-153) simultaneously, we obtain the

following expression for the reciprocal of the sound power transmission

coefficient for the dissipative muffler:

1=at ј C2

1 cos2рkLЮюC2

2 sin2рkLЮ (8-155)

The constants are given by the following expressions:

C1 ј coshрLЮ ю 1

2 рmю1=mЮ sinhрLЮ (8-156)

C2 ј sinhрLЮ ю 1

2 рmю1=mЮ coshрLЮ (8-157)

k ј

2_fc

c ј wave number (8-158)

The effect of the absorptive lining may be investigated by considering

two limiting cases for Eq. (8-155). First, for small attenuation рL_1Ю, or

for L _ 0:20 for practical purposes, the hyperbolic functions approach the

following limiting values within 2%:

coshрLЮ!1 and sinhрLЮ!L (for L < 0:2Ю

(8-159)

If we make the substitutions fromEq. (8-159) into Eq. (8-155), we obtain the

following expression for the small-attenuation limit:

1=at ј Ѕ1 ю 1

2 рmю1=mЮL_2 ю 1

4 рm_1=mЮ2 sin2рkLЮ (8-160)

Representative values for the transmission loss for small attenuation are

listed in Table 8-2.

At the other limit of very large attenuation рL _ 1Ю, or for L    5 for

practical purposes, the hyperbolic functions approach the following limiting

values within 1%:

sinhрLЮ _ coshрLЮ ! 1

2 eL (for L        5Ю (8-161)

Silencer Design 379

Copyright © 2003 Marcel Dekker, Inc.

If we make the substitutions from Eq. (8-161) into Eq. (8-155), we obtain the

following expression for the large attenuation limit:

1=at ј 1

4 e2LЅ1 ю 1

2 рm ю 1=mЮ_2 (8-162)

Taking log10 of both sides of Eq. (8-162) and multiplying by 10, we obtain

the transmission loss expression for the large-attenuation limit:

TL ј 8:6859L ю 20 log10Ѕ1

2 ю 1

4 рm ю 1=mЮ_ (8-163)

The first term in Eq. (8-163) represents the attenuation provided by the

lining, and the second term represents the effect of reflection of acoustic

energy back to the source as a result of the change in cross-sectional area of

the flow passage.

Example 8-10. A dissipative muffler has a length of 825mm (32.48 in) and

a diameter of 446mm (17.55 in). The diameter of the inlet and outlet tubes is

152mm (6.00 in). The attenuation coefficient is 1.25 Np/m or

(8.6859Ю ј 10:86 dB/m. The gas flowing through the muffler is air at

600K (6208F) and 110 kPa (15.96 psia), for which the density is 0.639 kg/

m3 (0.0399 lbm=ft3Ю and the sonic velocity is 491.0 m/s (1611 ft/sec). The

frequency of the sound being transmitted is 250 Hz. Determine the transmission

loss for the muffler.

Let us first calculate the pertinent dimensionless parameters:

L ј р1:25Юр0:825Ю ј 1:0313

kL ј

2_fL

c ј р2_Юр250Юр0:825Ю

р491:0Ю ј 2:6393 rad

m ј S2=S1 ј рD2=D1Ю2 ј р446=152Ю2 ј 8:6096

380 Chapter 8

TABLE 8-2 Transmission Loss for a Dissipative Silencer

with Small Values of Attenuation рL_0.2)a

kL ј 1

2 _ kL ј _

Attenuation _L ј 0 _L ј 0:20 _L ј 0 _L ј 0:20

1=at 4.516 5.546 1.000 2.031

TL, dB 6.5 7.4 0.0 3.1

aThe area ratio for data in this table is m ј 4. The quantity

k ј 2_ f =c ј wave number.

Copyright © 2003 Marcel Dekker, Inc.

The coefficients defined by Eqs (8-156) and (8-157) may be determined:

C1 ј coshр1:0313Ю ю 1

2 р8:6096 ю 1=8:6096Ю sinhр1:0313Ю

C1 ј 1:5805 ю 5:3402 ј 6:9207

C2 ј sinhр1:0313Ю ю 1

2 р8:6096 ю 1=8:6096Ю coshр1:0313Ю

C2 ј 1:2240 ю 6:8958 ј 8:1198

The reciprocal of the sound power transmission coefficient may be

found from Eq. (8-155):

1=at ј р6:9207Ю2 cos2р2:6393Ю ю р8:1198Ю2 sin2р2:6393Ю

1=at ј 36:796 ю 15:280 ј 52:076

The transmission loss is as follows:

TL ј 10 log10р52:076Ю ј 17:2dB

The reciprocal of the sound power transmission coefficient for an

unlined expansion chamber muffler having the same area ratio as the muffler

in this example may be found from Eq. (8-129):

1=at ј 1 ю 1

4 р8:6096 _ 1=8:6096Ю2 sin2р2:6393Ю ј 5:180

The transmission loss for the unlined expansion chamber muffler is as follows:

TL ј 10 log10р5:180Ю ј 7:1dB

The attenuation by the liner increases the transmission loss for the muffler

by 10.1 dB in this case.

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