42.6 Attenuation of Dissipative Mufflers

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42.6.1 Transmission Loss of Lined Expansion Chamber

The geometry and nomenclature for a dissipative muffler are given in Figure 42.15. For f , 1:2c=D; the

assumption of plane wave is acceptable where D ¼ the diameter of the muffler.

The transmission loss for the light lining in the chamber may be obtained using [8,9]:

TL ¼10 log10 coshðdele=2Þþ

m þ1

2m

sinhðdele=2Þ

􀀘 􀀙2

cos2kle þ sinhðdele=2Þþ

m þ1

2m

coshðdele=2Þ

􀀘 􀀙2

sin2kle

􀀒 􀀓

ð42:26Þ

FIGURE 42.11 Damping function KSL2 as a function

of dimensionless frequency, vl=nc:

TABLE 42.3 Formulas for Attenuation of Several Lined Ducts

See Figure 42.7 Low-Frequencies

Range:

vl

nc

, 1

Middle-Frequencies Range

(KySyly; see Figure 42.7)

High-Frequencies

Range:

vl

nc

. 5

(A) b ¼

4:34

nly

b ¼

8:7c

l2y

v ðKy Sy l2y

Þ b ¼ 21:4

c2 n

v2 l3y

(B) b ¼ 4:34

􀀄

1

ny l y þ

1

nx l x

􀀅

b ¼

8:7c

v

􀀄

Ky Sy l2y

l2y

þ

Kx Sx l 2

x

l 2

x

��

b ¼ 21:4

c2

v2

􀀄

ny

l3y

þ

nx

l 3

x

􀀅

(C) b ¼

8:7

nly

b ¼

34:7c

l2y

v

􀀄

Ky Sy l2y

4

􀀅

b ¼ 171

c2

v2

ny

l3y

(D) b ¼ 4:34

􀀄

2

ny ly þ

1

nx lx

􀀅

b ¼

8:7c

v

􀀄

Ky Sy l2y

l2y

þ

Kx Sx l 2

x

l 2

x

􀀅

b ¼ 21:4

c2

v2

􀀄

8ny

l3y

þ

nx

l 3

x

􀀅

(E) b ¼ 8:7

􀀄

1

ny ly þ

1

nx lx

􀀅

b ¼

34:7c

v

􀀄

Ky Sy l2y

4l2y

þ

Kx Sx l 2

x

4l 2

x

􀀅

b ¼

171c2

v2

􀀄

ny

l3y

þ

nx

l 3

x

􀀅

(F) b ¼

17:4

nl

b ¼

69:5c

4l 2v

KSl 2 b ¼

341c2 n

v2 l3

b is attenuation (dB/m), n is absorbing factor plotted in Figure 42.7, KySyly is damping function, plotted in Figure 42.8,

c is sound speed, l is the width of the duct, v ¼ 2pf : angular frequency, x; y: coordinates, see Figure 42.7.

Source: Bru¨el, P.V. 1951. Sound Insulation and Room Acoustics, Chapman & Hall, London, p.159. With permission.

42-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

FIGURE 42.12 Sketch of a typical lined bend with plane wave incidence. (Source: Beranek, L.L. Noise Reduction,

McGraw-Hill, 1960. With permission.)

Plane axial wave input

Random input

20

10

Z / l

0

0.1 0.2 0.5 1 2 5 10

F db

FIGURE 42.13 Insertion loss for lined bend. (The lining must extend two to four duct widths beyond the bend for

this data to be valid.) (Source: Beranek, L.L. Noise Reduction, McGraw-Hill, 1960. With permission.)

100

50

20

10

5

2

1

0.1 0.2 0.5 1 2 5

a = 2b d = 75 mm

150

300

600

d / λ

ATT (dB/m)

200

100 a=b

50

20

ATT (dB/m)

10

5

2

1

0.05 0.1 0.2 0.5 1

d / λ

2 5

d=50mm

100

200

400

FIGURE 42.14 Sound attenuation for a splitter duct. Each baffle is constructed with two sheets of perforated metal

filled with mineral wool, with about 100 to 140 kg/m3 gross density; a ¼ the width of the open space, b ¼ the width

of the baffle, d ¼ the center-to-center distance of baffles, l ¼ the wavelength of the sound.

Design of Absorption 42-13

© 2005 by Taylor & Francis Group, LLC

in which de ¼ the attenuation per unit length for the lined duct, which is given by the following equation:

20 log10ðdeleÞ¼

KlPle

S ð42:27Þ

The Kl values are obtained from the absorption coefficient, as shown earlier (see Figure 42.8). In

particular, de is given by

de ¼

1

le

100:05Kl Ple =S ð42:28Þ

where m ¼ the ratio of the area of expanded or lined sections to the area of inlet or outlet sections of

muffler; k ¼2pf =c; and le ¼ the length of the muffler.

The transmission loss for the case of a thick lining of glass wool in the chamber is obtained using the

empirical formula [10]

TL ¼ 10 log10 1 þ

1

2

agmkle

􀀘 􀀙2 􀀒 􀀓

ð42:29Þ

where

ag ¼ the coefficient, which is obtained from Table 42.4, using the filling volume and the density of

glass wool

m ¼ the ratio of the area of expanded or lined sections to the area of inlet or outlet sections of muffler

k ¼ 2pf =c

le ¼ the length of muffler

42.6.2 Transmission Loss of a Plenum Chamber

The geometry and nomenclature for a plenum chamber are given in Figure 42.16. A plenum chamber is

similar in many ways to a lined expansion chamber. The main difference is that the inlet and outlet of a

plenum chamber are not located in line. Generally, there is an offset to direct transmission of

sound. Sound is reflected at the square-cornered bend as the cross section dimension of the duct is

TABLE 42.4 Filled up Factor of Glass Wool, ag

Vg/V 0.05 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

ag 0.106 0.124 0.288 0.365 0.529 0.677 0.794 0.885 0.935 0.960 0.987 1.0

Vg ¼ Filled up volume (factors of 100 kg/m3), V ¼ Volume of chamber

FIGURE 42.15 A dissipative muffler.

42-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

sufficiently large. Particularly at high frequencies,

almost all of the sound energy may reflect many

times off the lined sides when propagating from

the inlet to the outlet. The transmission loss of a

single plenum chamber can be obtained approximately

from [11]:

TL ¼ 10 log10 Sw

cos u

2pd2 þ

1

R

􀀘 􀀏 􀀐􀀙

ð42:30Þ

where

Sw ¼ lW ¼ area of the inlet and outlet

d ¼ {ðL 2 lÞ2 þ H2}1=2 ¼ the slant distance

from inlet to outlet

cos u ¼ H=d

R ¼ a=ð1 2 amÞ

a ¼ the total lined area in chamber times

absorption coefficient

am ¼ the statistical absorption coefficient of the

lining

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