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Reactive Silencer Modeling by Transfer Matrix Method and

Experimental Study

OVIDIU VASILE*, KOLUMBAN VLADIMIR***Department of Mechanics

University POLITEHNICA of Bucharest

Splaiul Independentei 313, post code 060042, Bucharest**Acoustics and Vibration Laboratory

Research Institute for Construction Equipment and Technology ICECON S.A.

Pantelimon, 266, sector 2, post code 021652,CP 3-33, Bucharest

ROMANIA

[email protected], [email protected]

Abstract: - Thispaper investigates the acoustic performance of a reactive silencer for two special cases using

numerical and experimental techniques.

The plane wave based models such as the transfer matrix method (TMM) can offer fast initialprototype solutions for silencer designers. In this paper, the principles of TMM for predicting the transmissionloss (TL) of a silencer are briefly presented. The method is applied for each silencer configuration and thenumerical predictions are compared with the results obtained by means of an experimental set up. Onlystationary, non-dissipative silencers are considered. The acoustic performances of the experimental set up are

analyzed as compared with the theoretical results of the plane wave method approach. Comparison takes intoaccount the low frequency range where the firing engine frequency and its first order harmonics are the mainsources of noise.

Key-Words: -silencer, transmission loss, transfer matrix method

1 IntroductionA silencer is an important noise control elementfor reduction of machinery exhaust noise, fan noise,and other noise sources involving flow of a gas. Ingeneral, a silencer may be defined as an element inthe flow duct that acts to reduce the soundtransmitted along the duct while allowing free flowof the gas through the flow passage. A silencer maybe passive, where the sound is attenuated byreflection and absorption of the acoustic energywithin the element. An active silencer is one wherethe noise is canceled by electronic feed forward and

feedback techniques. In this paper, we will examinetwo cases of reactive (passive) silencers, also called

mufflers, using numerical and experimentaltechniques. The detailed design procedures formufflers are available in the literature (Munjal,1987). [1]

The expansion chamber muffler is a reactive-type

muffler, because the reduction of noise transmissionthrough the muffler is achieved by reflecting back to

the source a part of the energy entering the muffler.There is only a negligible amount of energydissipation within the muffler.

The expansion chamber muffler consists of oneor more chambers or expansion volumes which act

as resonators to provide an acoustic mismatch for

the acoustic energy being transmitted along the maintube.

Reactive silencers consist typically of severalpipe segments that interconnect a number of largerdiameter chambers. These silencers reduce the

radiated acoustical power primarily counting on tothe impedance mismatch, that is, by allowing the

acoustical impedance discontinuities to reflect soundback toward the source. Essentially, the mostrelevant discontinuities are commonly achieved by:(a) sudden cross-sectional change (expansions or

contractions), also by (b) wall property changes(transition from a rigid wall pipe to an equal-diameter absorbing wall pipe), or by (c) anycombination thereof.

The use of a silencer is prompted by the need toreduce the noise radiated from a source but in mostapplications the final selection is based on sometrade-offs among the predicted acousticalperformance, the mechanical performance, thevolume/weight ratio, and the cost of the resultingsystem.

The impact of the silencer upon the mechanical

performance of the source is determined consideringthe change in the silencer back pressure. For a

9th WSEAS Int. Conf. on ACOUSTICS & MUSIC: THEORY & APPLICATIONS (AMTA '08), Bucharest, Romania, June 24-26, 2008

ISBN: 978-960-6766-74-9 94 ISSN 1790-5095

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continuous-flow source, such as a fan or a gasturbine, the impact is determined from the increasein the average back pressure; by contrast, for anintermittent-flow source, such as a reciprocatingengine, the impact is a function of the increase in theexhaust manifold pressure when an exhaust valve isopen.

Most silencers are subject to volume/weightconstraints, which also influence the silencer designprocess. In addition, the initial purchase/installationcost and the periodic maintenance cost are otherimportant factors that influence the silencer

selection process.

2 Silencer Representation by Basic

Silencer ElementsEvery silencer can be divided into a number of

segments or elements each represented by a transfermatrix. The transfer matrices can then be combinedto obtain the system matrix in order to predict the

corresponding acoustical performance for thesilencer system.

The procedure is illustrated by considering thesilencer in Fig. 1a and 1b, which is divided into thebasic elements, labeled 1-13, indicated by thedashed lines.

Elements 1, 3, 5, 7, 9, 11 and 13 are simple pipes

of constant cross section.Elements 2, 6 and 10 are a simple area

expansion, elements 4, 8 and 12 are an areacontraction with an extended outlet pipe.

The thirteen elements are characterized by thetransfer matrices T

(1) through T

(13); therefore, the

system matrix T(S) for both particular cases of thewhole silencer is obtained by matrix multiplication.

- case a:( ) ( ) ( ) ( )521 TTTT S = (1)

- case b:

( ) ( ) ( ) ( )1321 TTTT S = (2)

The matrices for each of the above elements may

be derived from the formulas presented later in thissection.

Two cases of the most common reactive mufflerelements are discussed in this section and are used inillustrative design examples.

Transfer matrices for each of these elements havebeen derived over the last three decades by different

researchers, and explicit expressions for their four-pole parameters are given in detail by Munjal. [3, 5]

The variety of silencer elements and themultitude of elements per silencer result in

numerous silencer configurations.

a.

b.Fig. 1. Decomposition of silencers into basic

elements

3 The Transfer Matrix MethodThe transfer matrix method (TMM) use the transfermatrix of a silencer element as a function of theelement geometry, state variables of the medium,mean flow velocity, and properties of duct liners, if

any. The results presented below correspond to thelinear sound propagation of a plane wave in the

presence of a superimposed flow. In certain cases,the matrix may also be influenced by nonlineareffects, higher order modes, and temperature

gradients; these latter effects, which can be includedin special cases, are discussed qualitatively later in

this section, but they are excluded from theanalytical procedure described below. Thefollowing is a list of variables and parameters that

appear in most transfer matrix relations of reactiveelements: [3]

ip = acoustical pressure at ith location of elementiu = particle velocity at ith location of element

0 = mean density of gas, kg/m3

c = sound speed, m/s, = 273331

= absolute temperature, 273+= CK

iS = cross section of element at ith location, m2

il = length of element i, m

ii ScY =

A, B= amplitudes of right- and left- bound fields

( )

2

01 Mkk

c = assuming negligible frictional

energy loss along straight pipe segments

31

24

5

6

7

8

9

10

11

12

13

31

2 4

5


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cfck 20 ==

f 2=

f = frequency, Hz;

cVM =

V= mean flow velocity through S

ijT = (ij)th element of transmission matrix of

Symbols without subscripts, such as V, c, andM,describe quantities associated with the referenceduct.

For pipe with Uniform Cross Section theacoustical pressure and mass velocity fields pi,

iiuS0 in a pipe with uniform cross section iS

corresponding fieldspi+1, 10 +iiuS we have [3,4,5]

=

+

+

ii

i

ii

i

uS

p

TT

TT

uS

p

02221

1211

10

1

(3)

where the transmission matrix pipeT is given by

=

=

=

icic

i

iciicljMk

pipe

pipe

lklkY

j

lkjYlk

e

TT

TTT

ic

cossin

sincos

2221

1211

(4)

For Cross Sectional Discontinuities case, thetransition elements used in modeling the cross-sectional discontinuities are shown in the firstcolumn of Table 1. Using decreasing element-subscript values with distance from the noisesource, the cross-sectional areas upstream, at, anddownstream of the transition (S3, S2, and S1) arerelated through [5]

032211 =++ SSCSC (5)

Element Type C1 C2 K

-1 -1

2

13

1

S

S

-1 1

2

3

1 1

S

S

Table 1. Parameter values of transition elements

where the constants 1C and 2C are selected to

satisfy the compatibility of the cross-sectional areas

across the transition.Transfer matrices for cross-sectional

discontinuities (csd) in the presence of mean flow

that include terms proportional up to the fourthpower (M4) of the Mach number are presented in

reference [5].Components of the matrix Tcsdis presented below

[3]

=

2221

1211

csdcsd

csdcsd

csdTT

TTT (6)

where

( ) i

csd

csd

csd

csd

lkcotScjZ

YMSZSC

)KSSC(YMZSCT

YMSZSC

SCT

YkMT

T

022

333222

3111122222

332221

22

21

1112

111

=

+

+=

+

=

=

=

and 2l = length of the extended inlet/outlet

pipe, m

Letting length 2l tend to zero yields the transfer

matrix

10

1 11YKM (7)

The transfer matrix for a stationary medium isgiven by [3,4]

= 1

101

2Z

Tst (8)

where

( ) 2022 lkcotScjZ =

2l = length of the extended inlet/outlet pipe, m

For 02 =l the approximate end correction inlet

and outlet give by [4]

( )

( )

13

2

11111313

1

2

3311

3

8

3

8

R

kR,RRHS

R

kR,RRHS

outlet

inlet

=

=

(9)

l2

S2S1S3

l2

S2S3 S1


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are used for inlet and outlet ducts, respectively, Hbeing the Karal correction factor. A lengthcorrection is applied to both sides of the baffle hole.These corrections take into account the excitation ofhigher modes at the area discontinuities.

The transmission loss (TL) is given by [3,4,5]

220 22211311211

TTYY/TTlogTL

+++= (10)

The TL for a simple expansion chamber mufflerconfiguration may be obtained by substituting intoequation (10) the transmission matrices for theseelements obtained from equations (4) and (6).

In a simplified case Fig. 1 a, where the flow is oris assumed to be insignificant, the product of thesematrices is reduced to the expression [3]

+= kLsin

NN,logTL 2

21

250110 (11)

where k is the wave number, L is the chamber

length, andNis the area ratio given by 12 SSN = ,

where 2S is the area of cross section of the chamber

and 1S that of the tail pipe or exhaust pipe (in this

case is equal).

4 Case study

The tests configurations for the expansion chambermuffler are shown in Fig. 2, a, b. (case a and b)

a.

b.

Fig.2. Studied cases: a- single expansion chambermuffler; b- three expansion chambers muffler.

Dimensions:

0950131 ,ll == m, 030113 ,ll == m,

29507 ,l = m,

01210986542 ======== llllllll

01021 ,dddd ei ==== m; 040,D = mFor a simple expansion chamber Fig. 2 a,

1173 lllL ++= .

a.

b.Fig.3. Test stand for mufflers

PULSETM

3560B sound and vibration analysissystem from Brel&Kjr together with FFT andCPB Analysis Type 7700 software were used duringthe experimental tests for noise signal analysis.

A pink noise generator and a controlled level

sound source provided the acoustic excitation for thetest configuration.

High performance condenser microphones were

placed in M1 and M2 measurement positions forsound pressure signals capture.

Test results are presented below as TL spectralrepresentation derived from the M1 and M2 signalsspectra considering a low frequency range up to

about 2 kHz.Transmission loss values are given in dB and

they are higher than the predicted ones in thefrequency interval 200 Hz 1 kHz in both

configuration cases presented.

l1 l3 l7 l11 l13

di de

DSoundsource

Anechoic

termination

M1 M2

l1 l3 l7 l11 l13

di de

DSoundsource

Anechoic

termination

M1 M2

d2d1


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-10

-5

0

5

10

15

20

25

30

35

50

200

350

500

650

800

950

1100

1250

1400

1550

1700

1850

2000

2150

2300

Frequency [Hz]

TL[

dB]

-20

-10

0

10

20

30

40

50

60

70

50

200

350

500

650

800

950

1100

1250

1400

1550

1700

1850

2000

2150

2300

Frequency [Hz]

TL

[dB]

Fig.4. FFT - Test results for the single expansionchamber muffler

Fig.5. FFT - Test results for the three expansionchambers muffler

Fig.6. Predicted transmission loss for singleexpansion chambers muffler (TMM)

The predicted TL is presented for area ratio N=16and displayed versus the dimensionless quantity

kLq = (Fig. 7). The troughs of the TL curve

occur at nkL = , while the peaks occur at

2)12( = nkL , where ck = is the wave

number and nis an integer.

Fig.7. Predicted transmission loss for singleexpansion chambers muffler for area ratioN=16

Fig.8. Predicted transmission loss for threeexpansion chambers muffler (TMM) for

002.012108642 ====== llllll m, 004.095 == ll m

Fig.9. Predicted transmission loss for threeexpansion chambers muffler (TMM) for

approximate end correction

bafflellllllll ======== 1210986542

where [4]

))R(k0.038375)R(k0.001504

Rk0.03158(0.338898R

31

21

11

++

++=baffle


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Fig.10. Predicted transmission loss for three

expansion chambers muffler (TMM) forapproximate end correction

bafflellllllll ======== 1210986542

where 005768.0=baffle

5 Conclusion

A major disadvantage of the simple expansionchamber Fig. 2 a, is that in certain applications time-varying tones and their harmonics may alignsimultaneously with the periodic through and causea severe deterioration in acoustical performance.The problem may be partly solved by usingextended-tube (inlet and/or outlet) elements.

The model for three expansion chambers Fig. 2 b,computed without end corrections fails atconsiderably lower frequency while exhibiting a TLclose to that of a simple expansion chamber without

baffle in its valid frequency range.Transmission loss values obtained for both single

chamber and three chambers mufflers configurationsunder test are higher than the predicted ones over thefrequency interval 200 Hz 1 kHz.

The three chambers muffler configurationprovides considerably better TL values than the

single chamber muffler configuration.

References:[1] Randall F.B., Industrial Noise Control and

Acoustics, Marcel Dekker, Inc., 2001.[2] Gerges, S.N.Y, Jordan, R., Thime, F.A., Bento

Coelho, J.L., Arenas, J.P., Muffler Modeling byTransfer Matrix Method and ExperimentalVerification, J. Braz. Soc. Mech. Sci.& Eng., vol.27, no.2, Rio de Janeiro, Apr./June 2005, pp.132-140.

[3] Ver, I.L., Beranek, L.L, Noise and vibrationcontrol engineering, Second Edition, JohnWiley&Sons, Inc., 2006.

[4] Selamet A., Denia F.D., Besa A.J., Acousticbehavior of circular dual-chamber mufflers,Journal of Sound and Vibration, Vol. 265, 2003,pp. 967-985.

[5] Munjal, M.L., Acoustics of Ducts and Muffler,Wiley, New York, 1987.


ISBN: 978-960-6766-74-9 99 ISSN 1790-5095

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