8.7 EVALUATION OF THE ATTENUATION COEFFICIENT

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The best source of information for the attenuation coefficient is experimental

data on the lining material. In some cases, this information is not available

from manufacturer’s data. For muffler design, we may need to estimate

the attenuation coefficient before the muffler is built. After the unit has been

constructed, the prototype should be tested to verify the design calculations.

8.7.1 Estimation of the Attenuation Coe⁄cient

The attenuation coefficient (uncorrected for random incidence end effects)

may be estimated from the following expression (Beranek, 1960):

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 ј

_fPw

2S2

_eY

2_

_ _1=2

Ѕр1 ю 2Ю1=2 _ 1_1=2 (8-164)

The quantities appearing in Eq. (8-164), which will be discussed in more

detail in the following section, are as follows:

Pw ј perimeter of the flow passage in the muffler

S2 ј open cross-sectional area of the muffler

_e ј effective density for the gas within the lining material

Y ј porosity of the lining

_ ј effective elasticity coefficient for the gas within the lining

material

The quantity is a dimensionless parameter defined by:

ј

Re

2_f _e

(8-165)

The quantity Re is the effective flow resistance per unit thickness for the

lining material.

Two general classes of acoustic lining materials are considered: (a) a

semirigid material and (b) a soft blanket material. A semirigid material is

one in which the solid portion of the material is relatively rigid, e.g., an

acoustic ceiling tile. A soft blanket material is one in which the solid portion

of the material is relatively flexible, e.g., a panel of glass fiber acoustic

material.

An alternative method, based on an empirical curve fit of attenuation

data, has also been developed (Beranek and Veґ r, 1992, p. 214). The attenuation

coefficients for mineral wool and for glass fibrous material have been

correlated by an expression in the following form, valid in the range

0:3 < o < 120,

 ј

_fPw

2Sco

a b

o (8-166)

The quantity co is the speed of sound at ambient temperature рToЮ for the

gas in the lining, and the quantity o is defined by the following expression:

o ј

R1рT=ToЮ_1:65

2_f _o

(8-167)

The quantity R1 is the specific flow resistance per unit thickness for the

acoustic material, T is the absolute temperature of the gas within the

acoustic material, and To is ambient temperature (300K). The temperature

ratio term corrects for density and viscosity variation with

382 Chapter 8

Copyright © 2003 Marcel Dekker, Inc.

temperature. The values for the regression constants a and b are given in

Table 8-3.

8.7.2 E!ective Density

The inertia effect of the fibers in a semirigid material is negligible, because

the fibers do not move appreciably with the gas when a sound wave passes

through the material. For the semirigid acoustic material, the effective density

is given by the following relationship:

_e ј _o s (8-168)

The quantity     s is called the structure factor (Zwikker and Kosten, 1949),

which takes into account the effect of cavities and pores that are perpendicular

to the direction of sound propagation in the material. The structure

factor may be approximated by the following relationship:

            s ј 1 ю 4:583р1 _ YЮ (6-169)

The quantity Y is the porosity of the material, defined as the ratio of the void

volume or volume occupied by the gas within the material to the total

volume of the lining material.

The inertia effect of the fibers in a soft blanket material at low frequencies

is small, because the fibers move along with the motion of the gas

within the material. At the limit of low frequencies, the effective density of a

soft blanket material is equal to the bulk density of the blanket material plus

the weighted density of the gas within the material. For high frequencies, the

inertia effect of the fibers is large and the fibers are not able to follow the

motion of the gas within the material. For the limit of high frequencies, the

effective density of soft blanket materials becomes the same as that of rigid

materials. The effective density for the soft blanket materials may be calculated

from the following relationship:

Silencer Design 383

TABLE 8-3 Values for the Regression

Constants in Eq. (8-166)

Material Rangea a b

Mineral wool o < 6:4 0.605 0.663

o

_ 6:4 0.810 0.502

Glass fiber o < 6:4 0.618 0.674

o

_ 6:4 0.919 0.458

aThe quantity o is defined by Eq. (8-167).

Copyright © 2003 Marcel Dekker, Inc.

_e ј

_o        sf1 ю ЅY ю р_m=_o   sЮ_ 21

g

1 ю 21

(8-170)

The quantity _o is the density of the gas within the material, _m is the bulk

density of the lining material, Y is the porosity for the lining material, and          s

is the structure factor. The quantity 1 is defined by the following expression:

1 ј

R1

2_f_m

(8-171)

We note that for small frequencies р 1!1Ю, the effective density from Eq.

(8-170) approaches the following limiting value:

_e ! _o sY ю _m (small frequencies) (8-172)

For large frequencies р 1 ! 0Ю, the effective density from Eq. (8-170)

approaches the other limiting value:

_e ! _o s (large frequencies) (8-173)

8.7.3 E!ective Elasticity Coe⁄cient

The quantity _ in Eq. (8-164) is the effective elasticity coefficient for the gas

within the lining material. The term _=Y is actually the effective bulk modulus

or compressibility coefficient for the gas. When the frequency of the

sound is less than about 100 Hz, there is sufficient time for energy to be

exchanged between the gas and the fibers of the lining material, such that the

compression and expansion of the gas takes place almost at constant temperature

(isothermally). For an ideal gas рP ј _RTЮ, the isothermal bulk

modulus may be evaluated (Van Wylen et al., 1994):

_T ј _

@P

@_

_ _

Tј _oRTo ј Po (8-174)

The quantity Po is the absolute pressure of the gas within the lining material.

On the other hand, when the frequency of the sound wave is greater

than about 1000 Hz, there is not sufficient time between cycles for significant

energy transfer to take place between the fibers and the gas, so the compression

and expansion process is practically adiabatic. For an ideal gas, the

adiabatic bulk modulus is given by the following relationship:

_s ј _Po (8-175)

The quantity _ is the specific heat ratio for the gas within the liner. For air

and most diatomic gases, _ ј 1:40:

384 Chapter 8

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The effective elasticity coefficient for the gas within the liner material

may be determined from the following relationship:

_ ј

Po (for f _ 100 HzЮ

Ѕр3 _ 2_Ю ю р_ _ 1Ю log10р f Ю_Po (for 100 Hz < f < 1000 HzЮ _Po (for f       1000 HzЮ

8<

:

(8-176)

8.7.4 E!ective Speci￠c Flow Resistance

For a semirigid material, the fibers of the material do not move appreciably

when a gas moves through the material. The velocity of the fibers is practically

zero, so the effective specific acoustic resistance for the semirigid material

is equal to the actual specific acoustic resistance of the lining (rayl/m):

Re ј R1 (semirigid material) (8-177)

For the soft blanket materials at low frequencies, the fibers move

along with the air, such that the effective specific acoustic resistance

approaches zero at low frequencies. For high frequencies, the fibers cannot

follow the motion of the air and remain more nearly stationary. In this case,

the effective specific resistance approaches the actual specific acoustic resistance.

The general expression for the effective acoustic resistance for a soft

blanket material is the following:

Re ј

R1

1 ю 21

(soft blanket material) (8-178)

The quantity 1 is defined by Eq. (8-171).

The specific acoustic resistance per unit thickness R1 must be determined

experimentally, in general, by measuring the pressure drop _p

through a material sample of known thickness _x and surface area S,

with a measured volume flow rate U through the material. The specific

acoustic resistance may be calculated from the following expression:

R1 ј

S_p

U _x

(8-179)

Correlations have been developed for some commonly used acoustic

lining materials. The specific acoustic resistance of bulk glass-fiber materials

has been correlated with the bulk density _m of the material and the diameter

of the fibers _f (Nichols, 1947):

R1 ј 3180р_mЮ1:53р_f Ю_2 (8-180)

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Equation (8-180) is not a dimensionless correlation. The constant 3180

applies only when the bulk density is in kg/m3 and the fiber diameter is in

mm. The units for the specific acoustic resistance are rayl/m ј (Pa-s/m)/m ј Pa-s/m2.

Another empirical correlation has been developed for several commercial

lining materials:

R1 ј Cmр_m=_ref Юn (8-181)

The quantity _m is the bulk density of the material and the reference density

_ref ј 16:018 kg=m3 ј 1:00 lbm=ft3. Values for the regression constants Cm

and n for several materials are given in Table 8-4.

386 Chapter 8

TABLE 8-4 Values of the Regression Coefficients Cm

and n in Eq. (8-181) for Various Commercial Acoustic

Materialsa

Material Cm n

Thermoflex 300b 27,500 1.45

Thermoflex 400b 20,300 1.45

Thermoflex 600b 46,300 1.45

Thermoflex 800b 16,100 1.45

Spincousticb 1,850 1.82

Spintex 305-3.5b 1,874 1.70

Spintex 305-4.5b 1,500 1.50

Aluminum wool, 30–50 mm diameter 42.2 3.123

Basalt wool, 4–7 mm diameter 1,418 1.55

Kaoline wool, 1–3 mm diameter 15,910 1.36

Kaowool blanket 1,800 1.48

Glass fiber, 15–20 mm diameter 267 1.83

Glass fiber, 3–7 mm diameter 5,890 1.40

Glass fiber, 2–4 mm diameter 9,340 1.56

aR1 ј Cmр_m=_ref Юn where _m is the bulk density of the

material, and the reference density

_ref ј 16:018 kg=m3 ј 1:00 lbm=ft3. The units for R1 are

rayl/m.

bMaterial originally manufactured by Johns Manville

Corporation.

Copyright © 2003 Marcel Dekker, Inc.

8.7.5 Correction for Random Incidence End

E!ects

The attenuation coefficient, calculated according to the previous correlations,

must be corrected for random incidence end effects, because the

sound wave is not always a plane wave within the muffler. The corrected

value of the attenuation coefficient is found by adding the random incidence

correction _рLЮ:

рLЮcorr ј L ю _рLЮ (8-182)

The end correction may be evaluated from the following expressions

(Beranek, 1960, p. 449):

(a) For рS2Ю1=2=_ ј f рS2Ю1=2=c _ 0:09:

_рLЮ ј 0 (8-183a)

(b) For 0:09 < рS2Ю1=2=_ < 1:00:

_рLЮ ј 0:5756f1 ю Ѕ1 ю 1:912 log10р

ffiffiffiffiffi

S2

p

=_Ю_1=3g (8-183b)

(c) For рS2Ю1=2=_    1:00:

_рLЮ ј 1:1513 (9-183c)

The quantity S2 is the open cross-sectional area of the muffler chamber and

_ is the wavelength of the sound wave in the muffler.

Example 8-11. A dissipative muffler having a length of 4.500m (14.76 ft)

and a diameter of 1.500m (4.92 ft) is lined with a Kaowool blanket material,

which is a soft blanket-type material. The unit specific acoustic resistance for

the material is R1 ј 180,000 rayl/m, the mean density of the material is

100 kg/m3 (6.24 lbm=ft3), and the porosity of the material is 0.960. The

fluid flowing through the muffler is air at 450K (1778C or 3508F) and

140 kPa (20.3 psia), for which the density is 1.084 kg/m3 (0.0677 lbm=ft3),

the sonic velocity is 425 m/s (1394 ft/sec), and the specific heat ratio

_ ј 1:40. The frequency of the sound wave in the muffler is 2 kHz.

Determine the attenuation coefficient.

First, we may calculate the attenuation coefficient, uncorrected for end

effects. The structure factor for the lining material may be estimated from

Eq. (8-169):

            s ј 1 ю р4:583Юр1 _ 0:960Ю ј 1:1833

The dimensionless parameter involving the specific acoustic resistance is

found from Eq. (8-171):

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1 ј

R1

2_f _m ј р180,000Ю

р2_Юр2000Юр100Ю ј 0:1432

The effective density for the gas in the soft blanket material is found

from Eq. (8-170):

_e ј р1:084Юр1:1833Юf1 ю Ѕ0:96 ю р100=1:2827Ю_р0:1432Ю2g

1 ю р0:1432Ю2

_e ј р1:2827Юр2:5666Ю ј 3:292 kg=m3 р0:2055 lbm=ft3Ю

The frequency of 2 kHz is greater than 1000 Hz, so the effective elastic

constant for the gas in the lining material is found from Eq. (8-176) for the

adiabatic case:

_ ј _Po ј р1:40Юр140Ю ј 196 kPa

The effective specific acoustic resistance for the soft blanket material is

found from Eq. (8-178):

Re ј

R1

1 ю 21

ј р180,000Ю

1 ю р0:1432Ю2 ј 176,380 rayl=m

The dimensionless parameter defined by Eq. (8-165) may be calculated:

ј

Re

2_f _e ј р180,000Ю

р2_Юр2000Юр3:292Ю ј 4:351

The cross-sectional area and perimeter of the muffler chamber may be

determined:

S2

Pw ј

1

4 _D22

_D2 ј 1

4D2 ј р1

4Юр1:500Ю ј 0:375m

The attenuation coefficient, uncorrected for end effects, may now be

found from Eq. (8-164):

 ј р_Юр2000Ю

р2Юр0:375Ю

р3:292Юр0:960Ю

р2Юр196Юр103Ю

_ _1=2

Ѕр1 ю 4:3512Ю1=2 _ 1_1=2

 ј р8378Юр0:002839Юр1:8613Ю ј 44:275Np=m

The wavelength of the sound wave in the muffler is found as follows:

_ ј c=f ј р425Ю=р2000Ю ј 0:2125m р8:37 inЮ

Let us calculate the dimensionless parameter involved in the end correction:

рS2Ю1=2=_ ј Ѕр1

4 _Юр1:500Ю2_1=2=р0:2125Ю ј р1:329Ю=р0:2125Ю ј 6:26 > 1

388 Chapter 8

Copyright © 2003 Marcel Dekker, Inc.

The end correction may be evaluated from Eq. (8-183c):

_рLЮ ј 1:1513

The attenuation coefficient, corrected for end effects, may be evaluated

from Eq. (8-182):

 ј 44:275юр1:1513Ю=р4:50Ю ј 44:275ю0:256 ј 44:531Np=m

The attenuation coefficient may also be expressed in decibel units:

8:6859 ј 386:8 dB=m

Let us evaluate the attenuation coefficient from Eq. (8-166) for comparison.

The density and sonic velocity for air at 300K and 101.3 kPa are

_o ј 1:177 kg=m3 and co ј 347:2m/s. The dimensionless parameter defined

by Eq. (8-167) may be calculated:

o ј р180,000Юр425=300Ю_1:65

р2_Юр2000Юр1:177Ю ј 6:850

If we take the constants from Table 8-3 for mineral wool, we may evaluate

the following term, for o > 6:4:

a b

o ј р0:810Юр6:850Ю0:502 ј 2:128

The attenuation coefficient may now be calculated from Eq. (8-166):

 ј р_Юр2000Юр2:128Ю

р2Юр0:375Юр347:2Ю ј 53:29Np=m

This value is about 20%larger than the value calculated through the more

exact procedure (44.53 Np/m). This difference is probably reasonable in

view of the scatter of experimental data on the attenuation coefficient

parameters.

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