43.5 Application Range of Model

Back

43.5.1 Condition for Approximation of Plane Wave

The frequency range where the approximation of a plane wave is valid is given by

fc , 1:22

c

D ð43:26Þ

where

fc ¼ critical frequency of plane wave (Hz)

c ¼ speed of sound (m/sec)

D ¼ diameter of muffler (m)

It is seen that the expansion ratio of an open area of a pipe increases with IL or TL. However, the

application range of the analytical model decreases with increasing diameter of chamber.

43.5.2 Effect of Temperature

Under conditions of high-temperature and high-speed gas flow, as in an engine exhaust system, the

primary effect of a change in pipe temperature is the corresponding change in the speed of sound, which

is proportional to the square root of the absolute temperature. In the design of a reactive muffler, it is

necessary to use the actual speed of sound in the gas inside the pipe. The most accurate values available

for density ðr0Þ and the speed of sound ðc0Þ at each element should be used in calculating the impedance

43-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

TABLE 43.1 Transmission Loss of Reactive Mufflers and Insertion Loss of Reactive Mufflers

Muffler

(see Figure 43.4)

TL (dB) Application Limits and Comments

Transmission loss of reactive mufflers

(a)

TL ¼ 10 log10 1 þ

kS

4a

􀀏 􀀐2

( )

a: radius of orifice

a=l , 0:1; R ¼ 0

(b)

TL ¼ 20 log10

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

ð1 þ R3 e22jkl3Þðcos kl2 þ jm21 sin kl2 Þ

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

þ m32ð1 2 R3 e22jkl3 Þðj sin kl2 þ m21 cos kl2 Þ

m10 ¼ S1 =S0 ; m21 ¼ S2 =S1 ; m32 ¼ S3 =S2 ; R0 ; R3:

the reflection coefficient of the open end,

l3 ¼ length of the tail pipe

f , 1:22c=D

(1) when R0 ¼ R3 ¼ 0; S0 ¼ S1 ¼ S3

TL ¼ 10 log10 1 þ

1

4

m21 2

1

m21

􀀏 􀀐2

sin kl2

􀀘 􀀙

(2) when R0 ¼ R3 ¼ 21; S0 ¼ S1 ¼ S3

TL ¼ 10 log10 1 þ

m2

21 2 1

m2

21

􀁻 !

{ðm2

21 þ 1Þsin2 kl3 2 1}sin2 kl2 þ

1

2

1

m21

2 m21

􀀏 􀀐

sin 2kl2 sin 2kl3

" #

(c)

TL ¼10 log10 2 cos kðl1 2 l11 2 l22 Þ 2

m 2 1

m

sin kðl1 2 l11 2 l22 Þðtan kl11 þ tan kl22 Þ

􀀘 􀀙2 􀀒

þ m þ

1

m

􀀏 􀀐

sinkðl1 2 l11 2 l22Þ þ ðm 2 1Þcosðl1 2 l11 2 l22 Þðtan kl11 þ tan kl22Þ

􀀘

2 ðm 2 1Þ2

m

tan kl11 tan kl22 sin kðl1 2 l11 2 l22 Þ

􀀙2 􀀓

M ¼ S1=S0

R < 0

(d) TL ¼ 10 log10 1 þ

1

4

m

kS2

C0

2 cot kl

0

BBB@

1

CCCA

2 8>><

>>:

9>>=

>>;

C0 ¼ NCi ; N: number of holes, Ci ¼ 2pa2i

=ðlb þ paiÞ; lb ; lb: thickness of the pipe, ai: radius of a hole,

m ¼ S12 =S

R < 0

(continued on next page)

Design of Reactive Mufflers 43-7

© 2005 by Taylor & Francis Group, LLC

TABLE 43.1 (continued)

Muffler

(see Figure 43.4)

TL (dB) Application Limits and Comments

(e) TL ¼ 10 log10 1 þ

1

4

ffiffiffiffiffiffi

pC0V

S

f

fr

2

fr

f

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

2 2

6664

3

7775

fr ¼

c

2p

ffiffiffiffiffi

C0

V

r

;

C0 ¼

2pa2

2lb þ pa

lb: length of the neck or thickness of the pipe, a: radius of the neck or hole

R < 0; lb ,, l Resonator size ,, l

(f) TL ¼ 10 log10 1 þ

m2

4

tan klb 2

Sb

kV

Sb

kV

tan klb þ 1

0

BB@

1

CCA

8>><

>>:

9>>=

>>;

m ¼ Sb =S

R < 0; lb ,, l Resonator size ,, l

(g) TL ¼ 20 log10

1

16m2 ½4mðm þ 1Þ2 cos 2kðl þ lc Þ 2 4mðm 2 1Þ2 cos 2kðl 2 lc Þ􀀉

􀀈 􀀈 􀀈

þ j 2ðm2 þ 1Þðm þ 1Þ2sin 2kðl þ lc Þ 2 2ðm2 þ 1Þðm 2 1Þ2 sin2kðl 2 lc Þ

n

24ðm2 2 1Þ2 sin 2klc

o􀀈 􀀈 􀀈

m ¼ S2 =S1

R < 0

(h) TL ¼ 10 log10 {cos 2kl 2 ðm 2 1Þsin 2klc tan klc }2 þ

j

2

m þ

1

m

􀀏 􀀐

sin 2kl

􀀘 􀀈 􀀈 􀀈 􀀈

þðm 2 1Þtan klc m þ

1

m

􀀏 􀀐

cos 2kl 2 m 2

1

m

􀀏 􀀏 􀀐􀀐􀀙2

􀀈 􀀈 􀀈 􀀈 􀀈

m ¼ S2=S1

R < 0

43-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

(i)

TL ¼ 10 log10

1

4

A1 þ jB1

A2 þ jB2

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

2 ( )

A1 ¼ Y3 X2

1 þ Z0 Y 2

3 þ Z0 ðX1 þ X3 Þ2

B1 ¼ X1 Y 2

3 þ X1 X3 ðX1 þ X3Þ

A2 ¼ Y3 X2

1 cos kl þ Z0 X1 Y3 sin kl

B2 ¼ X1 Y 2

3 þ X1 X3 ðX1 þ X3 Þcos kl 2 Z0 X1 ðX1 þ X3 Þsin kl

X1 ¼

vr

C0

2

rc 2

vV1

;

X2 ¼

vr

C0

2

rc2

vV2

X3 ¼

Z2

0 ðX2 cos 2kl þ

1

2

Z0 sin2 kl Þ

ðX2 sin kl 2 Z0 cos kl Þ2 þ X2

2 cos2 kl

Y3 ¼

Z0 X2

2

ðX2 sin kl 2 Z0 cos kl Þ2 þ X2

2 cos2 kl

Z0 ¼

rc

S0

;

C0 ¼

2pa2

2lb þ pa

lb: thickness of the pipe, a: radius of hole

R < 0 Resonator size ,, l

(j)

TL ¼ 10 log10 cos kl þ

rc

4S0 X

m þ

1

m

􀀏 􀀐

sin kl 2

rc

4S0 X

m 2

1

m

􀀏 􀀐

cos 2klb sin kl

􀀏 􀀐2 􀀘

þ

1

2

m þ

1

m

􀀏 􀀐

sin kl þ

rc

4S0 X

m 2

1

m

􀀏 􀀐

sin 2klb sin kl 2

rc

2S0 X

cos kl

􀀏 􀀐2 􀀙

X ¼

vr

C0

2

rc2

vV

;

m ¼

S

S0

R < 0

(continued on next page)

Design of Reactive Mufflers 43-9

© 2005 by Taylor & Francis Group, LLC

TABLE 43.1 (continued)

Muffler

(see Figure 43.4)

IL (dB) Application Limits and Comments

Insertion loss of reactive mufflers

(b) IL ¼ 10 log10

1

m30

£

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

m10

1 2 R0 e22jkl0

{jð1 þ R3 e22jkl3 ðsin kl1 cos kl2 þ m21 cos kl1 sin kl2Þ:

þ m32 ð1 2 R3 e22jkl3 Þðm21 cos kl1 cos kl2 2 sinkl1 sinkl2Þ}

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

2

f , 1:22c=D; R is plotted in Fig.43.1

ð1Þ IL ¼ 10 log10 {1 þ ðm2

21 2 1Þ} 1 2

m2

21 þ 1

m2

21

sin2 kl1

􀁻 !

sin2 kl2

􀀈 􀀈 􀀈 􀀈 􀀈

þ

1

2

m21 2

1

m21

􀀏 􀀐

sin 2kl1 sin 2kl2

􀀈 􀀈 􀀈 􀀈

R0 ¼ R3 ¼ 0; S0 ¼ S1 ¼ S3

ð2Þ IL ¼ 10 log10

cos kl1

cos kl0

{cos kl2 cos kl3 2 m21 sin kl2 sinkl3 2 tan kl1ðcos kl2 sin kl3

􀀏􀀈 􀀈 􀀈 􀀈

þ

1

m21

sin kl2 cos kl3 Þ}

􀀐􀀈 􀀈 􀀈 􀀈

2

R0 ¼ R3 ¼ 21; S0 ¼ S1 ¼ S3

(c) IL ¼ 20 log10

cos kl1 cos kl2 2 m sin kl1 sin kl2

cos kl11 cos kl22

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 m

¼

S1=S0

R ¼ 21

(d) IL ¼ 20 log10 cos2 kl2 þ

m

kS2

C0

2 cot kl

sin2kl2 þ

m

kS2

C0

2 cot kl

0

BBB@ 1 CCCA

2

sin2 kl2

2

6664

3

7775

C0 ¼ NCi ; Ci ¼ 2pa2i

=ðlb þ paiÞ, N: number of holes, lb: thickness of the

pipe, ai: radius of a hole, m ¼ S12 =S

R ¼ 21; kl0 ,, 1

43-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

(e)

IL ¼ 10 log10

ffiffiffiffiffiffi

pC0V

S

f

fr

2

fr

f

sin 2kl2 þ

C0 V

S2

f

fr

2

fr

f

􀀏 􀀐2 sin2 kl2 þ cos2 kl2

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

fr ¼

c

2p

ffiffiffiffiffi

C0

V

r

;

C0 ¼

2pa2

2lb þ pa

lb: length of the neck or thickness of pipe, a: radius of the neck or hole

R ¼ 21; kl0 ,, 1

(f)

IL ¼ 10 log10 cos2 kl2 þ m sin 2kl2

Sb

kV

tan klb þ 1

tan klb 2

Sb

kV

2

664

þ m

Sb

kV

tan klb þ 1

tan klb 2

Sb

kV

0

BB@

1

CCA

2

sin2 kl2

3

775

fr ¼

c

2p

ffiffiffiffiffi

C0

V

r

;

C0 ¼

2pa2

2lb þ pa

lb: length of the neck or thickness of the pipe, a: radius of the neck or hole

R ¼ 21; kl0 ,, 1

(k) IL ¼ 20 log10

􀀈 􀀈

ðcos kl1 cos kl11 2 m1 sin kl1 sin kl11Þ þ ðcos kl2 cos kl22

2 m2 sin kl2 sin kl22Þ þ · · · þ ðcos kli cos klii 2 mi sin kli sin klii Þ

􀀈 􀀈

R ¼ 21; kl0 ,, 1

(l) IL ¼ 20 log10

cos kL1 cos kl1 2 m sin kL1 sin kl1

cos kl11 þ

cos kL2 cos kl2 2 m sin kL2 sin kl2

cos kl12 cos kl21

􀀘􀀈 􀀈 􀀈 􀀈

þ · · · þ

cos kLn cos kln 2 m sin kLn sin kln

cos klðn21Þ2

􀀙􀀈 􀀈 􀀈 􀀈

R ¼ 21; kl0 ,, 1

A is the radius of tube in orifice hole or diameter of side branch, c is the sound speed, C0 is the conductivity, D is the diameter of chamber, f is the frequency, fr is the resonant frequency of

the resonator, k ¼ 2pf =c is the wave number, L is the length, m ¼ Si =Siþ1 is the ratio of the cross section, IL is the insertion loss, N is the number of holes, R is the reflection coefficient, S is the

cross section, TL is the transmission loss, V is the volume of chamber, Z is the acoustic impedance, r is the density, l is the wavelength, v ¼ 2pf ; angular frequency.

Design of Reactive Mufflers 43-11

© 2005 by Taylor & Francis Group, LLC

of the elements. The impedance, z; is given by

z ¼ 2j

r0c0

S

1

kl ð43:27Þ

where

S ¼ the cross-sectional open area of pipe

k ¼ 2pf =c0; wave number

l ¼ the length of the pipe element

Note that the impedance of the resonator chamber is proportional to r0c2

0 : However, c2

0 is proportional

to the absolute temperature of gas ðTÞ and r0 is proportional to 1=T: Hence, the chamber impedance is

independent of temperature. The connector impedance is a function of T; but in most cases the

connector will be at the pipe temperature. For a resonator-type muffler, a temperature difference between

the pipe and chamber is expected to have little effect on the performance of the muffler.

43.5.3 Effect of Gas Flow in Pipe

Under conditions of high-temperature and high-speed gas flow in a pipe, the pressure amplitude in the

pipe is large, and is larger than what is predicted by theory. Analysis by the characteristic curve method is

desirable under such conditions.

In a reactive muffler where the pipe flow passes through a sudden pipe expansion or an orifice, the

computed transmission loss or insertion loss tends to be an overestimate because of new noise that is

generated due to the resulting irregular air-flow within the muffler.

(a) Orifice (b) Single-expansion-chamber (c) Single-expansion-chamber

with insertion pipes

(d) Single-expansion-chamber

with perforated pipe

2af

dφ

S

S

S

S S

S

S0

l0

l11 l22

l1

l2

S1

S0

l0

l1

l2

l3

S0

S1 S3

S2

S

S0

S2

l2

l0

l

(i) Double-resonators (j) Combined resonator and

expansion chamber

(k) Multistage expansion-chambers (l) Multistage expansion-chambers

with insertion pipes

V1

V1

lb

l

S

S0 S0

l0

S0

S1

l1

S2 Si

l2 l11 li

lii

S0

S0

l0

S0

S0

S1

l11

l1

l21

l12

l22

lii

L1 L2

V2

l

S0

S

S

S

S

S

S S

S S

S Sb lb

S1 S1 S1 S1 S1

S2 S2 S2 S2

l 2lc l l l

lc lc

l2

l2

l2

lb

l0

l0

l0

lb

λ

s

s

V

(e) Helmholtz resonator (f) Helmholtz resonator

with the long neck

(g) Double-expansion-chambers (h) Double-expansion-chambers

with an insertion pipe

V

V

a

4

> 

FIGURE 43.4 Sketches of the 12 principal structures of reactive mufflers.

43-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

43.5.4 Effect of Friction Loss in Pipe

When an acoustic wave propagates in a pipe, it will attenuate due to viscous friction. The effect is large for

long pipes of small diameter. Friction damping in a pipe may be incorporated into the propagation

constant, g; such that

g ¼ d þ jk ð43:28Þ

where d is the attenuation constant per unit length of pipe. By substituting Equation 43.28 into Equation

43.3 and Equation 43.4, we obtain

j ¼ ðA e2gl 2 B eglÞe jvt ð43:29Þ

p ¼ 2rc2kðA e2gl þ B eglÞe jvt ð43:30Þ

Empirical formulas are given below for two cases of the attenuation coefficient d [3].

(1) The formula for seamless steel or chloride-ethylene pipes (regression formula when the inside

roughness is 4 to 8 mm and length under 3 m) is

d ¼ 26; 100l20:5 m

rcd ð43:31Þ

where

l ¼ wavelength of sound (m)

m ¼ viscosity of gas in the pipe (Pa sec)

r ¼ density of gas (kg/m3)

d ¼ diameter of the pipe (m)

(2) The equations for lining with glass wool are

d2 ¼ 2491l20:476 rcd

m

􀀏 􀀐21:068

d6 ¼ 5175l20:476 rcd

m

􀀏 􀀐21:303

d6 ¼ 11596l20:476 rcd

m

􀀏 􀀐21:270

ð43:32Þ

The suffix of d gives the thickness of absorbing material in mm.

Навигация