Reactive Silencer Modeling by ransfer Matrix Method and Experimental Study OVIDIU VASILE *, KOLUMBAN VLADIMIR ** * Department of Mechanics University POLIEHNICA of Bucharest Splaiul Independentei, post code 64, Bucharest ** Acoustics and Vibration Laboratory Research Institute for Construction Equipment and echnology ICECON S.A. Pantelimon, 66, sector, post code 65,CP -, Bucharest ROMANIA vasile@cat.mec.pub.ro, vkolumban@icecon.ro Abstract: - his paper investigates the acoustic performance of a reactive silencer for two special cases using numerical and experimental techniques. he plane wave based models such as the transfer matrix method (MM) can offer fast initial prototype solutions for silencer designers. In this paper, the principles of MM for predicting the transmission loss (L) of a silencer are briefly presented. he method is applied for each silencer configuration and the numerical predictions are compared with the results obtained by means of an experimental set up. Only stationary, non-dissipative silencers are considered. he acoustic performances of the experimental set up are analyzed as compared with the theoretical results of the plane wave method approach. Comparison takes into account the low frequency range where the firing engine frequency and its first order harmonics are the main sources of noise. Key-Words: - silencer, transmission loss, transfer matrix method Introduction A silencer is an important noise control element for reduction of machinery exhaust noise, fan noise, and other noise sources involving flow of a gas. In general, a silencer may be defined as an element in the flow duct that acts to reduce the sound transmitted along the duct while allowing free flow of the gas through the flow passage. A silencer may be passive, where the sound is attenuated by reflection and absorption of the acoustic energy within the element. An active silencer is one where the noise is canceled by electronic feed forward and feedback techniques. In this paper, we will examine two cases of reactive (passive) silencers, also called mufflers, using numerical and experimental techniques. he detailed design procedures for mufflers are available in the literature (Munjal, 987). [] he 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 part of the energy entering the muffler. here is only a negligible amount of energy dissipation within the muffler. he 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. Reactive silencers consist typically of several pipe segments that interconnect a number of larger diameter chambers. hese silencers reduce the radiated acoustical power primarily counting on to the impedance mismatch, that is, by allowing the acoustical impedance discontinuities to reflect sound back toward the source. Essentially, the most relevant 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 equaldiameter absorbing wall pipe), or by (c) any combination thereof. he use of a silencer is prompted by the need to reduce the noise radiated from a source but in most applications the final selection is based on some trade-offs among the predicted acoustical performance, the mechanical performance, the volume/weight ratio, and the cost of the resulting system. he impact of the silencer upon the mechanical performance of the source is determined considering the change in the silencer back pressure. For a ISBN: 978-96-6766-74-9 94 ISSN 79-595
continuous-flow source, such as a fan or a gas turbine, the impact is determined from the increase in the average back pressure; by contrast, for an intermittent-flow source, such as a reciprocating engine, the impact is a function of the increase in the exhaust manifold pressure when an exhaust valve is open. Most silencers are subject to volume/weight constraints, which also influence the silencer design process. In addition, the initial purchase/installation cost and the periodic maintenance cost are other important factors that influence the silencer selection process. Silencer Representation by Basic Silencer Elements Every silencer can be divided into a number of segments or elements each represented by a transfer matrix. he transfer matrices can then be combined to obtain the system matrix in order to predict the corresponding acoustical performance for the silencer system. he procedure is illustrated by considering the silencer in Fig. a and b, which is divided into the basic elements, labeled -, indicated by the dashed lines. Elements,, 5, 7, 9, and are simple pipes of constant cross section. Elements, 6 and are a simple area expansion, elements 4, 8 and are an area contraction with an extended outlet pipe. he thirteen elements are characterized by the transfer matrices () through () ; therefore, the system matrix (S) for both particular cases of the whole silencer is obtained by matrix multiplication. - case a: ( ) ( ) ( ) ( 5) S () - case b: ( ) ( ) ( ) ( ) S () he matrices for each of the above elements may be derived from the formulas presented later in this section. wo cases of the most common reactive muffler elements are discussed in this section and are used in illustrative design examples. ransfer matrices for each of these elements have been derived over the last three decades by different researchers, and explicit expressions for their fourpole parameters are given in detail by Munjal. [, 5] he variety of silencer elements and the multitude of elements per silencer result in numerous silencer configurations. a. b. Fig.. Decomposition of silencers into basic elements he ransfer Matrix Method he transfer matrix method (MM) use the transfer matrix of a silencer element as a function of the element geometry, state variables of the medium, mean flow velocity, and properties of duct liners, if any. he results presented below correspond to the linear sound propagation of a plane wave in the presence of a superimposed flow. In certain cases, the matrix may also be influenced by nonlinear effects, higher order modes, and temperature gradients; these latter effects, which can be included in special cases, are discussed qualitatively later in this section, but they are excluded from the analytical procedure described below. he following is a list of variables and parameters that appear in most transfer matrix relations of reactive elements: [] p i acoustical pressure at i th location of element u i particle velocity at i th location of element ρ mean density of gas, kg/m c sound speed, m/s, θ 7 θ absolute temperature, K C+ 7 S i cross section of element at i th location, m l length of element i, m i 4 4 5 6 Y i c S i A, B amplitudes of right- and left- bound fields k c k ( M ) assuming negligible frictional energy loss along straight pipe segments 7 8 9 5 ISBN: 978-96-6766-74-9 95 ISSN 79-595
k ω c πf c ω πf f frequency, Hz; M V c V mean flow velocity through S (ij) th element of transmission matrix of ij Symbols without subscripts, such as V, c, and M, describe quantities associated with the reference duct. For pipe with Uniform Cross Section the acoustical pressure and mass velocity fields p i, ρ S i u i in a pipe with uniform cross section S i corresponding fields p i+, ρ S i u i+ we have [,4,5] pi+ ρ Siu i+ pi ρ Siu where the transmission matrix pipe is given by pipe e jmkcli cos kcli j sin kcl Yi pipe i i jyi sin kcli cos kcli () (4) For Cross Sectional Discontinuities case, the transition elements used in modeling the crosssectional discontinuities are shown in the first column of able. Using decreasing elementsubscript values with distance from the noise source, the cross-sectional areas upstream, at, and downstream of the transition (S, S, and S ) are related through [5] C S + C S + S (5) Element ype C C K S S S l S S S - - - S S S S l able. Parameter values of transition elements where the constants C and C are selected to satisfy the compatibility of the cross-sectional areas across the transition. ransfer matrices for cross-sectional discontinuities (csd) in the presence of mean flow that include terms proportional up to the fourth power (M 4 ) of the Mach number are presented in reference [5]. Components of the matrix csd is presented below [] where csd csd csd (6) csd csd csd csd csd csd Z km Y CS C S Z + S M Y CSZ MY ( CS + SK ) C S Z + S M Y j ( c S ) cot kli and l length of the extended inlet/outlet pipe, m Letting length l tend to zero yields the transfer matrix KM Y (7) he transfer matrix for a stationary medium is given by [,4] where Z j c st (8) Z ( S ) cot k l l length of the extended inlet/outlet pipe, m For l the approximate end correction δ inlet and outlet δ give by [4] δ δ inlet outlet 8ρSH π R 8ρS ( R R,kR ) ( R R,kR ) H π R (9) ISBN: 978-96-6766-74-9 96 ISSN 79-595
are used for inlet and outlet ducts, respectively, H being the Karal correction factor. A length correction is applied to both sides of the baffle hole. hese corrections take into account the excitation of higher modes at the area discontinuities. he transmission loss (L) is given by [,4,5] + / Y + Y + L log () Dimensions: l l 95 m, l l m,,, l 7, 95 m, l l4 l5 l6 l8 l9 l l di de d d,, m; D 4 m For a simple expansion chamber Fig. a, L l + l +. 7 l he L for a simple expansion chamber muffler configuration may be obtained by substituting into equation () the transmission matrices for these elements obtained from equations (4) and (6). In a simplified case Fig. a, where the flow is or is assumed to be insignificant, the product of these matrices is reduced to the expression [] L log +, 5 N sin kl () N where k is the wave number, L is the chamber length, and N is the area ratio given by N S S, where S is the area of cross section of the chamber and S that of the tail pipe or exhaust pipe (in this case is equal). a. 4 Case study he tests configurations for the expansion chamber muffler are shown in Fig., a, b. (case a and b) Sound source Sound source d i d i l l l 7 l l M M D a. l l l 7 l l M M D b. Fig.. Studied cases: a- single expansion chamber muffler; b- three expansion chambers muffler. d d d e Anechoic termination d e Anechoic termination b. Fig.. est stand for mufflers PULSE M 56B sound and vibration analysis system from Brüel&Kjær together with FF and CPB Analysis ype 77 software were used during the experimental tests for noise signal analysis. A pink noise generator and a controlled level sound source provided the acoustic excitation for the test configuration. High performance condenser microphones were placed in M and M measurement positions for sound pressure signals capture. est results are presented below as L spectral representation derived from the M and M signals spectra considering a low frequency range up to about khz. ransmission loss values are given in db and they are higher than the predicted ones in the frequency interval Hz khz in both configuration cases presented. ISBN: 978-96-6766-74-9 97 ISSN 79-595
5 5 occur at kl π n, while the peaks occur at kl π ( n ), where k ω c is the wave number and n is an integer. L [db] 5 5-5 5 5 5 65 8 95 5 4 55 7 85 5 - Frequency [Hz] Fig.4. FF - est results for the single expansion chamber muffler Fig.7. Predicted transmission loss for single expansion chambers muffler for area ratio N6 8 7 6 5 L [db] 4 - - 5 5 5 65 8 95 5 4 Frequency [Hz] Fig.5. FF - est results for the three expansion chambers muffler 55 7 85 5 Fig.8. Predicted transmission loss for three expansion chambers muffler (MM) for l l l l l l. m, l l. 4 m 4 6 8 5 9 Fig.6. Predicted transmission loss for single expansion chambers muffler (MM) he predicted L is presented for area ratio N6 and displayed versus the dimensionless quantity q kl π (Fig. 7). he troughs of the L curve Fig.9. Predicted transmission loss for three expansion chambers muffler (MM) for approximate end correction l l l l l l l l δ where [4] R δ baffle 4 5 6 8 9 +.54 (k R (.8898 +.58 k R ) +.875 (k R + ) baffle ) ISBN: 978-96-6766-74-9 98 ISSN 79-595
[4] Selamet A., Denia F.D., Besa A.J., Acoustic behavior of circular dual-chamber mufflers, Journal of Sound and Vibration, Vol. 65,, pp. 967-985. [5] Munjal, M.L., Acoustics of Ducts and Muffler, Wiley, New York, 987. Fig.. Predicted transmission loss for three expansion chambers muffler (MM) for approximate end correction l l l l l l l l δ 4 5 6 8 9 where δ. 5768 baffle 5 Conclusion baffle A major disadvantage of the simple expansion chamber Fig. a, is that in certain applications timevarying tones and their harmonics may align simultaneously with the periodic through and cause a severe deterioration in acoustical performance. he problem may be partly solved by using extended-tube (inlet and/or outlet) elements. he model for three expansion chambers Fig. b, computed without end corrections fails at considerably lower frequency while exhibiting a L close to that of a simple expansion chamber without baffle in its valid frequency range. ransmission loss values obtained for both single chamber and three chambers mufflers configurations under test are higher than the predicted ones over the frequency interval Hz khz. he three chambers muffler configuration provides considerably better L values than the single chamber muffler configuration. References: [] Randall F.B., Industrial Noise Control and Acoustics, Marcel Dekker, Inc.,. [] Gerges, S.N.Y, Jordan, R., hime, F.A., Bento Coelho, J.L., Arenas, J.P., Muffler Modeling by ransfer Matrix Method and Experimental Verification, J. Braz. Soc. Mech. Sci.& Eng., vol. 7, no., Rio de Janeiro, Apr./June 5, pp. -4. [] Ver, I.L., Beranek, L.L, Noise and vibration control engineering, Second Edition, John Wiley&Sons, Inc., 6. ISBN: 978-96-6766-74-9 99 ISSN 79-595