8.5 EXPANSION CHAMBER MUFFLERS

Back

The expansion chamber muffler is a reactive-type muffler, because the

reduction of noise transmission through the muffler is achieved by reflecting

back to the source a portion of the energy entering the muffler. There

is generally a negligible amount of energy dissipation within the muffler.

The expansion chamber muffler consists of one or more chambers or

expansion volumes which act as resonators to provide an acoustic mismatch

for the acoustic energy being transmitted along the main tube.

Some typical configurations for the expansion chamber muffler are

shown in Fig. 8-14.

8.5.1 Transmission Loss for an Expansion Chamber

Mu\er

At the junction of the inlet tube and the expansion chamber, the instantaneous

acoustic pressure in the inlet tube and in the expansion chamber are

equal; and, similarly, the instantaneous acoustic pressures are equal at the

junction of the expansion chamber and the outlet tube for the expansion

chamber muffler. The instantaneous volumetric flow rates, UрtЮ ј SuрtЮ, are

equal on each side of the inlet and outlet junction. These conditions are the

same as those used for analysis of transmission of sound from medium 1

through medium 2 into medium 3, as discussed in Sec. 4.7. Instead of the

characteristic impedance Zo ј _oc, the acoustic impedance ZA ј _oc=S ј Zo=S appears in the final expression for the sound power transmission

coefficient. The gas is the same in all sections of the muffler, so that the

impedance ratios become area ratios:

Z1=Z3 ј ZA1=ZA3 ј S3=S1 _ _ (8-122)

Z1=Z2 ј ZA1=ZA2 ј S2=S1 _ m (8-123)

Z2

Z3 ј

ZA2

ZA3 ј

S3

S2 ј

S3=S1

S2=S1 ј

_

m

(8-124)

368 Chapter 8

Copyright © 2003 Marcel Dekker, Inc.

If these substitutions are made into Eq. (4-123), the following expression

is obtained for the sound power transmission coefficient (or the

reciprocal) for an expansion chamber muffler:

1

at ј р1 ю _Ю2 cos2рkLЮ ю рm ю _=mЮ2 sin2рkLЮ

р4_Ю

(8-125)

The quantity k ј 2_fL=c is the wave number and L is the length of the

expansion chamber.

The expression for the sound power transmission coefficient, Eq.

(8-125), may be written in a different form by substituting the trigonometric

identity:

cos2рkLЮ ј 1 _ sin2рkLЮ (8-126)

1

at ј р1 ю _Ю2 ю Ѕрm ю _=mЮ2 _ р1 ю _Ю2_ sin2рkLЮ

4_

(8-127)

The second term in the numerator of Eq. (8-127) may be rearranged as

follows:

1

at ј р1 ю _Ю2 ю Ѕрm _ _=mЮ2 _ р1 _ _Ю2_ sin2рkLЮ

4_

(8-128)

Silencer Design 369

FIGURE 8-14 Configurations for expansion chamber mufflers: (a) single expansion

volume and (b) double expansion volume.

Copyright © 2003 Marcel Dekker, Inc.

For the special case in which the inlet tube and the outlet tube have the

same cross-sectional area, _ ј S3=S1 ј 1, Eq. (8-128) reduces to the following

expression:

1=at ј 1 ю 1

4 рm _ 1=mЮ2 sin2рkLЮ (8-129)

The transmission loss obtained from Eq. (8-129) is plotted in Fig. 8-15.

It is noted from Eq. (8-128) that the transmission loss is a maximum

for the frequencies corresponding to the following condition:

kL ј

2_foL

c ј n _ 1

2

             

_ рn ј 1; 2; 3; . . .Ю (8-130)

fo ј

n _ 1

2

             

c

2L рn ј 1; 2; 3; . . .Ю (8-131)

The maximum transmission loss (which occurs at the frequency fo) may be

found by combining Eqs (8-128) and (8-130):

1

at

_ _

maxј р1 ю _Ю2 ю рm _ _=mЮ2 _ р1 _ _Ю2

р4_Ю

(8-132)

370 Chapter 8

FIGURE 8-15 Plot of the transmission loss as a function of frequency for a singlechamber

expansion chamber muffler.

Copyright © 2003 Marcel Dekker, Inc.

For the special case of _ ј 1, the maximum transmission loss expression is

as follows:

р1=atЮmax ј 1 ю 1

4 рm _ 1=mЮ2 (8-133)

We also observe that the transmission loss is a minimum for the frequencies

corresponding to the following condition:

kL ј n_ рn ј 1; 2; 3; . . .Ю (8-134)

fp ј

nc

2L рn ј 1; 2; 3; . . .Ю (8-135)

The minimum transmission loss expression is as follows:

1

at

_ _

minј р1 ю _Ю2

4_

(8-136)

For the special case of _ ј 1, (1=atЮmin ј 1 and TLmin ј 0 dB.

8.5.2 Design Procedure for Single-Expansion

Chamber Mu\ers

If the same parameters are given or known as in the case of the side-branch

muffler design given in Sec. 8.4.2, we may develop a similar design procedure

for the expansion chamber muffler with a single expansion volume.

D1. The resonant frequency (or center frequency) for the optimum

expansion chamber muffler is the arithmetic average of the low and high

operational frequencies, f1 and f2, of the muffler:

fo ј 1

2 р f1 ю f2Ю (8-137)

D2. The optimum length of the expansion chamber corresponds to the

situation in which the transmission loss is maximized at the center or resonant

frequency, as given by Eq. (8-131):

L ј

n _ 1

2

             

c

2 fo рn ј 1; 2; 3; . . .Ю (8-138)

D3. The value of the integer n may be estimated by satisfying the

following condition:

k2L _ k1L _ 1

2 _ (8-139)

If we substitute for the expansion chamber length from Eq. (8-138), the

following expression is obtained for the approximate value of the integer n:

n _ 1

2 ю

fo

2р f2 _ f1Ю

(8-140)

Silencer Design 371

Copyright © 2003 Marcel Dekker, Inc.

The quantities f1 and f2 are the lower and upper frequencies for the main

operational range of the muffler. Note that n must be an integer рn ј 1 or 2

or 3, etc.), so that Eq. (8-140) may not be satisfied exactly. As illustrated in

the following example, the transmission loss should be checked at the end

frequencies.

D4. The expansion ratio required to achieve the minimum allowable

transmission loss (which occurs at the end frequencies) may be found from

Eq. (8-128), using k1L ј 2_f1L=c. If practical dimensions cannot be

achieved in this design step, the use of multiple expansion chambers may

be required.

Example 8-8. A cylindrical air conditioning duct has a diameter of 250mm

(9.84 in). The fluid flowing in the duct is air at 108C (508F) and 105 kPa

(15.23 psia), for which the sonic velocity c ј 337:3 m/s (1107 ft/sec) and the

density _o ј 1:292 kg=m3 (0.0807 lbm=ft3). It is desired to design a single

expansion chamber muffler that has a minimum transmission loss of 6 dB

for the frequency range between 177 Hz and 354 Hz.

The center frequency for the muffler may be determined from Eq.

(8-137):

fo ј 1

2 р177 ю 354Ю ј 265:5Hz

The estimated value of the integer n may be found from Eq. (8-140):

n _ 1

2 ю р265:5Ю

р2Юр354 _ 177Ю ј 1:25

We could possibly use either n ј 1 or n ј 2.

Let us check the results when using n ј 2 first. The length of the

muffler is found from Eq. (8-138):

L ј

2 _ 1

2

             

р337:3Ю

р2Юр265:5Ю ј 0:953m р37:5 inЮ

At the lower frequency (177 Hz) in the operational range for the muffler, we

find the following value for k1L ј 2_f1L=c:

k1L ј р2_Юр177Юр0:953Ю

р337:3Ю ј 3:1416 rad ј _ rad

As shown by Eq. (8-134), the transmission loss for _ ј 1 and kL ј _ is

TL ј 0, so the value of n ј 2 cannot be used.

Let us try the other possibility, n ј 1. The required muffler length is

found from Eq. (8-138):

372 Chapter 8

Copyright © 2003 Marcel Dekker, Inc.

L ј

1 _ 1

2

             

р337:3Ю

р2Юр265:5Ю ј 0:318m ј 318mm р12:50 inЮ

For the lower frequency, f1, we find the following value for the parameter

k1L:

k1L ј р2_Юр177Юр0:318Ю

р337:3Ю ј 1:0472 rad ј 1

3_rad ј 608

The sound power transmission coefficient (or the reciprocal) at the

lower frequency corresponds to the minimum design transmission loss:

TLmin ј 6 dB ј 10log10р1=atЮ

1=at ј 100:60 ј 3:981

The required size of the expansion chamber may be found from Eq.

(8-129):

1=at ј 3:981 ј 1 ю 1

4 рm_1=mЮ2 sin2р_=3Ю

m_1=m ј р2Юр

ffiffiffiffiffiffiffiffiffiffiffi

p2:981Ю

sinр608Ю ј 3:9874

m2 _3:9874m_1 ј 0

If we solve for the area ratio, we obtain the following value:

m ј 1:9937юЅр1:9937Ю2 ю1_1=2 ј 4:224 ј S2=S1 ј рD2=D1Ю2

The required expansion chamber diameter is as follows:

D2 ј р250Юр4:224Ю1=2 ј 514mm р20:2inЮ

The maximum transmission loss for the muffler occurs at the center

frequency, fo ј 265:5 Hz. The value is determined from Eq. (8-133):

р1=atЮmax ј 1 ю 1

4 Ѕ4:224_р1=4:224Ю2_ ј 4:975

TLmax ј 10 log10р4:975Ю ј 7:0dB

8.5.3 Double-Chamber Mu\ers

For mufflers with two or more expansion chambers, the analysis is most

conveniently carried out using the transfer matrix approach (Beranek and

Veґ r, 1992). The results for the transmission loss for the double-chamber

muffler with an external connecting tube and expansion chambers having

equal lengths, as shown in Fig. 8-16(a), is given by the following expression

(Davis et al., 1954):

Silencer Design 373

Copyright © 2003 Marcel Dekker, Inc.

1

at ј

F2

1 ю F2

2

16m2 (8-141)

The quantities F1 and F2 are defined by the following expressions:

F1 ј рm ю 1Ю2 cosЅ2kрL1 ю L2Ю_ _ рm _ 1Ю2 cosЅ2kрL2 _ L1Ю_ (8-142)

F2 ј 1

2 рm ю 1=mЮfрm ю 1Ю2 sinЅ2kрL1 ю L2Ю_

_ рm _ 1Ю2 sinЅ2kрL2 _ L1Ю_g _ рm _ 1=mЮрm2 _ 1Ю sinр2kL1Ю

(8-143)

The quantity L1 is the half-length of the connecting tube (total length, 2L1),

L2 is the length of one expansion chamber, m ј S2=S1 ј cross-sectional

area ratio for chamber and inlet tube (assumed to have the same diameter

as the connecting tube), and k ј 2_f =c ј wave number.

The transmission loss for the double-chamber muffler is generally

larger than that for a single-chamber muffler. There is a low-frequency

region present, however, in which the TL is relatively small. The lowfrequency

pass band appears in the double-chamber muffler as a result of

resonance between the connecting tube and the expansion chambers. When

374 Chapter 8

FIGURE 8-16 Nomenclature for double-chamber expansion chamber mufflers: (a)

external connecting tube and (b) internal connecting tube.

Copyright © 2003 Marcel Dekker, Inc.

the connecting tube is made longer, the low-frequency pass band frequency

width is made smaller. The upper frequency (called the cut-off frequency) in

this pass band may be estimated from the following relationship:

fc ј

c

2_ЅmL1L2 юL2рL2 _L1Ю=3_1=2 (8-144)

The connecting tube length should be selected such that the primary

operating frequency range of the muffler lies above the cut-off frequency fc.

The maximum transmission loss in the first band above the pass band is

increased as the length of the connecting tube is increased; however, at

higher frequencies, there are regions of low TL with frequency width on

the order of 50Hz or more for long connecting tube lengths. These pass

bands would be objectionable if there were a significant fraction of the

sound energy incident on the muffler in this frequency range.

For the double-chamber muffler with an internal connecting tube, as

shown in Fig. 8-16(b), the following expression has been obtained for the

sound power transmission coefficient (or the reciprocal):

1=at ј G21

ю G22

(8-145)

The quantities G1 and G2 are defined by the following expressions:

G1 ј cosр2kL2Ю _ рm _ 1Ю sinр2kL2Ю tanрkL1Ю (8-146)

G2 ј 1

2 рm _ 1Ю tanрkL1ЮЅрm ю 1=mЮ cosр2kL2Ю _ рm _ 1=mЮ_

ю 1

2 рm ю 1=mЮ sinр2kL2Ю

(8-147)

The quantity L1 is the half-length of the connecting tube (total length, 2L1),

L2 is the length of one expansion chamber, m ј S2=S1 ј cross-sectional

area ratio for chamber and inlet tube (assumed to have the same diameter

as the connecting tube), and k ј 2_f =c ј wave number.

A low-pass band region is also present at frequencies below fc, given

by Eq. (8-144), as in the case of the muffler with the external connecting

tube. There is a significant difference between the performance of the two

mufflers, however, when the connecting tube length 2L1 is equal to the

expansion chamber length L2. The frequency band over which the muffler

has a high TL for the internal tube muffler is about twice that of the external

tube muffler, for the same dimensions. The largest attenuation occurs when

kL1 ј 1

2 _, since the term tanрkL1Ю in Eqs (8-146) and (8-147) becomes

infinitely large under this condition, or the sound power transmission coefficient

at approaches zero when kL1 ј 1

2 _.

Example 8-9. An expansion chamber muffler has two expansion chambers,

each having a length of 300mm (11.81 in) and a diameter of 200mm

Silencer Design 375

Copyright © 2003 Marcel Dekker, Inc.

(7.874 in). The connecting tube between the two chambers is an internal tube

having a diameter of 100mm (3.937 in) and a length of 200mm (7.874 in).

The inlet and outlet tubes for the muffler also have diameters of 100mm

(3.937 in). The gas flowing through the muffler is air at 108C (508F) and

105 kPa (15.23 psia), for which the sonic velocity c ј 337:3 m/s (1107 ft/sec)

and the density _o ј 1:292 kg=m3 (0.0807 lbm=ft3). Determine the transmission

loss for the muffler at a frequency of 250 Hz.

The area ratio for the muffler is as follows:

m ј S2=S1 ј рD2=D1Ю2 ј р200=100Ю2 ј 4:00

Let us calculate the following dimensionless parameters:

2kL2 ј р2Юр2_Юр250Юр0:300Ю

р337:3Ю ј 2:794 rad ј 160:18

kL1 ј р2_Юр250Юр0:100Ю

р337:3Ю ј 0:4657 rad ј 26:78

The parameters from Eqs (8-146) and (8-147) may now be calculated:

G1 ј cosр2:794Ю _ р4 _ 1Ю sinр2:794Ю tanр0:4657Ю ј _0:9403 _ 0:5133

G1 ј _1:4536

G2 ј 1

2 р4 _ 1Ю tanр0:4657ЮЅр4 ю 1=4Ю cosр2:794Ю _ р4 _ 1=4Ю_

ю 1

2 р4 ю 1=4Ю sinр2:794Ю ј р0:7538Юр_7:7461Ю ю 0:7235

G2 ј _5:1159

The reciprocal of the sound power transmission coefficient for the doublechamber

muffler with an internal connecting tube may be found from Eq.

(8-145):

1=at ј G21

ю G22

ј р_1:4536Ю2 ю р_5:1159Ю2 ј 28:285

The transmission loss for the muffler at 250 Hz is as follows:

TL ј 10 log10р28:285Ю ј 14:5dB

Let us check the cut-off frequency given by Eq. (8-144):

fc ј р337:3Ю

р2_ЮЅр4:00Юр0:300Юр0:100Ю ю р0:300Юр0:300 _ 0:100Ю_1=2 ј 126:5Hz

This frequency is well below the frequency (250 Hz) in the previous calculation.

376 Chapter 8

Copyright © 2003 Marcel Dekker, Inc.

Навигация