ACOUSTIC CHARACTERISTICS OF PERFORATED DISSIPATIVE AND HYBRID SILENCERS DISSERTATION. the Degree Doctor of Philosophy in the Graduate

Transcription

1 ACOUSTIC CHARACTERISTICS OF PERFORATED DISSIPATIVE AND HYBRID SILENCERS DISSERTATION Presented in Partial Fulillment o the Requirements or the Degree Doctor o Philosophy in the Graduate School o The Ohio State University By Iljae Lee, M.S. * * * * * The Ohio State University 25 Dissertation Committee: Proessor Ahmet Selamet, Adviser Proessor Rajendra Singh Proessor Mohammad Samimy Dr. Norman T. Hu Approved by Adviser Graduate Program in Mechanical Engineering

2 Copyright by Iljae Lee 25

3 ABSTRACT Acoustic characteristics o silencers illed with ibrous material (hence dissipative) are investigated. Following a theoretical and experimental analysis o a single-pass, perorated, dissipative concentric silencer, the study is extended to a hybrid silencer designed by combining dissipative and relective (Helmholtz resonator) components. The ability to model these silencers relies heavily on the understanding o the acoustic behavior o the ibrous material and the perorations. Thereore, the present study has developed two experimental setups to measure: (a) the complex characteristic impedance and the wavenumber o the ibrous material with varying illing density and texturization conditions, and (b) the acoustic impedance o perorations in contact with the ibrous material, and with and without the mean low. New empirical expressions are then provided or the acoustic impedance o perorations with varying porosity, hole diameter, wall thickness, mean low rate, and the iber characteristics. The experimental results illustrate that the presence o absorbent signiicantly increases both the resistance and reactance o the peroration impedance. The addition o mean low is also shown in general to increase the resistance, while decreasing the reactance particularly at low porosities. ii

4 The empirical expressions or the iber acoustic properties and the peroration impedance are then integrated into the predictions o transmission loss. These predictions are based primarily on a three-dimensional boundary element method (BEM) developed in the present study, due to its ability to treat silencers with complex internals, in addition to one- and two-dimensional analytical approaches also introduced. Comparisons o predictions with the acoustic attenuation experiments support the proposed relationships or the properties o ibrous material and the peroration impedance. The inluence o the internal geometry modiications o dissipative silencers, such as bales and extended inlet/outlet, and the impact o connecting duct length between a pair o silencers, are investigated with BEM. Hence, the contributions o the present study include the development o methodologies or the measurement o acoustic properties o the ibrous material and the impedance o perorations, particularly in contact with the absorbent along with the resulting empirical expressions, thereby assisting towards analytical and computational design tools or the dissipative and hybrid silencers. iii

5 Dedicated to my parents and wie iv

6 ACKNOWLEDGMENTS I truly thank my adviser, Proessor Ahmet Selamet, or his guidance and support during my Ph.D. studies, as well as his patience in correcting this dissertation. I hope that I relect, in this work, his enthusiasm or and dedication to scientiic truth. I am also in debt to Proessor Ahmet Selamet or giving me opportunities to disseminate the indings through publications and presentations. I would like to thank Owens Corning and Dr. Norman T. Hu or providing the inancial support or this work. I am grateul to Dr. Norman T. Hu or many productive discussions and his gracious personality. I also wish to express thanks to my committee members, Proessors Rajendra Singh and Mohammad Samimy, and Dr. Norman T. Hu, or their valuable eedback and suggestions on this dissertation. I am grateul to the current and ormer members o the research group who helped me: Dr. Nolan Dickey, Dr. Emel Selamet, Dr. Mubing Xu, Adam Christian, Don Williams, Cam Giang, Lauren Lecuru, Yuesheng He, Jacques Paul, Vincent Mariucci, and Yale Jones. I particularly thank Dr. Nolan Dickey or discussing with me various aspects o this investigation. Finally, this dissertation is dedicated to my parents and wie, Jungmi. Thank you or your unconditional love, support, and understanding. v

7 VITA March 19, Born Seoul, Korea M.S.M.E., The Ohio State University B.S.M.E., Yonsei University, Korea 1997 present...graduate Research Associate, The Ohio State University JOURNAL PUBLICATIONS 1. Selamet, A., Xu, M. B., Lee, I.-J., and Hu, N. T., 25, Analytical approach or sound attenuation in perorated dissipative silencers with inlet/outlet extensions, Journal o the Acoustical Society o America 117, Selamet, A., Xu, M. B., Lee, I.-J., and Hu, N. T., 25, Helmholtz resonator lined with absorbing material, Journal o the Acoustical Society o America 117, Selamet, A., Xu, M. B., Lee, I.-J., and Hu, N. T., 24, Analytical approach or sound attenuation in perorated dissipative silencers, Journal o the Acoustical Society o America 115, Xu, M. B., Selamet, A., Lee, I.-J., and Hu, N. T., 24, Sound attenuation in dissipative expansion chambers, Journal o Sound and Vibration 272, Selamet, A. and Lee, I.-J., 23, Helmholtz resonator with extended neck, Journal o the Acoustical Society o America 113, Selamet, A., Lee, I.-J., and Hu, N. T., 23, Acoustic attenuation o hybrid silencers, Journal o Sound and Vibration 262, FIELD OF STUDY Mechanical Engineering vi

8 TABLE OF CONTENTS Page Abstract... ii Dedication... iv Acknowledgments...v Vita... vi List o Tables...x List o Figures... xi Chapters: 1. Introduction Background Objective Outline Literature review Hybrid silencers Dissipative silencers Analytical models...8 One-dimensional decoupled models...8 Multi-dimensional analytical models Numerical models Acoustic properties o absorbing material...13 Surace impedance and absorption coeicient...14 Characteristic impedance and wavenumber...16 Flow resistivity Acoustic impedance o perorations in contact with absorbing material Helmholtz resonators Theoretical models...23 vii

9 3.1 Lumped model Simple Helmholtz resonator Perorated silencers Dissipative perorated silencers One-dimensional decoupled model Two-dimensional analytical model Three-dimensional boundary element method Physical and acoustic properties o absorbing materials Physical properties o absorbing material Acoustic properties o absorbing material Peroration impedance Experimental setups Impedance tube setup in the absence o mean low Characteristic impedance and wavenumber o absorbing material Acoustic impedance o perorations Transmission loss Impedance tube setup in the presence o mean low Acoustic impedance o perorations Acoustic properties o absorbing material Selected existing ormulations Flow resistivity Characteristic impedance and wavenumber o ibrous material Fiber samples Good texturization Normal texturization Speed o sound in the ibrous material Acoustic impedance o perorations Peroration samples Peroration impedance in the absence o mean low Acoustic impedance o perorations in contact with air-air...13 Available ormulations...13 New Expressions...17 viii

10 6.2.2 Acoustic impedance o perorations in contact with air-ibrous material..11 Available ormulations New expressions Peroration impedance in the presence o grazing mean low Resistance o peroration impedance with mean low Reactance o peroration impedance with mean low Transmission loss o silencers Simple expansion chamber Perorated reactive silencers Comparisons o predictions and experiments Eect o peroration geometry on the transmission loss Single pass dissipative silencers Comparisons o predictions and experiments Eect o acoustic properties o ibrous material on the transmission loss Eect o acoustic impedance o perorations on the transmission loss Eect o mean low on the transmission loss Eect o bales and extended inlet/outlet on the transmission loss Hybrid silencers Conclusions...17 Appendix A: Nomenclature Reerences ix

11 LIST OF TABLES Table Page 5.1 Flow resistivity measured by Owens Corning Peroration samples R and α o the peroration in contact with air-air in the absence o mean low R and α o the peroration in contact with air-ibrous material in the absence o mean low ( Good texturization) R and α o the peroration in contact with air-ibrous material in the absence o mean low ( Normal texturization) x

12 LIST OF FIGURES Figure Page 3.1 The schematic o a simple Helmholtz resonator The schematic o a perorated reactive single-pass straight silencer The schematic o a perorated dissipative single-pass straight silencer Helmholtz resonator model o the dissipative silencer in Fig Wave propagation in a perorated dissipative silencer in axisymmetric twodimension Wave propagation in a perorated dissipative silencer in three-dimension Wave propagation through a perorated plate in a duct The schematic o the impedance tube setup in the absence o mean low The schematic o the experimental setup or obtaining the acoustic properties o absorbing material in the absence o mean low The schematic o the experimental setup or obtaining the acoustic impedance o perorations in the absence o mean low The schematic o the impedance tube setup in the presence o mean low The schematic o the experimental setup or obtaining acoustic impedance o perorations in the presence o mean low Measurement o distance using quarter wave resonance; min is the requency at which the acoustic pressure o p m is minimum xi

13 5.2 Pictures o iber samples ( ρ = 1 kg/m 3 ); (a) good texturization and (b) normal texturization Individual and averaged experimental results o characteristic impedance o ibrous 3 material with good texturization ( ρ = 1 kg/m ); (a) real part and (b) imaginary part Individual and averaged experimental results o wavenumber o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part Characteristic impedance o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part Wavenumber o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part Characteristic impedance o ibrous material with good texturization ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part Wavenumber o ibrous material with good texturization ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part Comparison o characteristic impedance o ibrous material or both texturization conditions ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part Comparison o wavenumber o ibrous material or both texturization conditions ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part Comparison o characteristic impedance o ibrous material or both texturization conditions ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part Comparison o wavenumber o ibrous material or both texturization conditions ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part...93 xii

14 5.13 Estimated speed o sound through absorbing materials Perorations in contact with air-air and air-ibrous material; (a) air-air and (b) airibrous material Dimensions o perorated plate samples (unit: cm [inch]) Pictures o perorated plate samples End correction o the perorations in contact with air-air (Melling, 1973); (a) single hole and (b) double holes Zone area o a hole Fok unction Comparison o measurement and Fok unction or the acoustic impedance o a hole in contact with air-air; (a) resistance and (b) end correction coeicient End correction o the perorations acing air-air and air-ibrous material; (a) air-air and (b) air-ibrous material Comparison o Eqs. (6.2) and (6.6) or the acoustic impedance o the perorated plate with φ = 8.4 % and d =.249 cm; (a) real part and (b) imaginary part h 6.1 Acoustic impedance o the perorated plate A2; (a) real part and (b) imaginary part Acoustic impedance o a hole, Group A; (a) resistance and (b) end correction coeicient Acoustic impedance o a hole, Group B; (a) resistance and (b) end correction coeicient Acoustic impedance o a hole, Group C; (a) resistance and (b) end correction coeicient xiii

15 6.14 The measured resistance o a hole in contact with air-ibrous material in the presence o mean low; (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m The measured end correction coeicients o a hole in contact with air-ibrous material in the presence o mean low; (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m The schematic o a reactive expansion chamber The transmission loss o the reactive expansion chamber without perorations; experiments vs. predictions Sample mesh o the expansion chamber using IDEAS; 2 2 cm mesh size The schematics o perorated ducts; (a) φ = 8.4 %, d h =.249 cm, (b) φ = 8.4 %, d h =.498 cm, (c) φ = 25.7 %, d h =.249 cm, and (d) φ = 25.7 %, d h =.498 cm The pictures o perorated ducts; (a) φ = 8.4 %, d h =.249 cm, (b) φ = 8.4 %, d h =.498 cm, (c) φ = 25.7 %, d h =.249 cm, and (d) φ = 25.7 %, d h =.498 cm A perorated reactive silencer; (a) picture and (b) schematic The transmission loss o the perorated reactive silencer ( φ = 8.4 %, d h =.249 cm) The transmission loss o the perorated reactive silencer ( φ = 8.4 %, d h =.498 cm) The transmission loss o the perorated reactive silencer ( φ = 25.7 %, d h =.249 cm) The transmission loss o the perorated reactive silencer ( φ = 25.7 %, d h =.498 cm) xiv

16 7.11 The eect o peroration impedance on the transmission loss o the perorated reactive silencers; experiments The eect o peroration impedance on the transmission loss o the perorated reactive silencer ( φ = 8.4 %, d h =.249 cm), BEM predictions; (a) resistance and (b) end correction coeicient The eect o hole spacing on the transmission loss o a perorated reactive silencer ( φ = 8.4 %, d =.249 cm); experiments h 7.14 The distances between holes or two dierent perorated ducts ( φ = 8.4 %, d =.249 cm); (a) equal spacing and (b) unequal spacing h 7.15 The pictures and schematics o a dissipative perorated reactive silencer; (a) picture and (b) schematic The transmission loss o the perorated dissipative silencer; Experiments; (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m The transmission loss o the perorated dissipative silencer ( φ = 8.4 %, d h =.249 cm); (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m Transmission loss o the perorated dissipative silencer ( φ = 25.7 %, d h =.249 cm); (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m The eect o acoustic properties o ibrous material on the transmission loss o the perorated dissipative silencer ( φ = 25.7 %, d h =.249 cm, ρ = 1 kg/m 3 ), BEM predictions; (a) characteristic impedance and (b) wavenumber The eect o acoustic properties o ibrous material on the transmission loss o the perorated dissipative silencer ( φ = 25.7 %, d h =.249 cm, ρ = 2 kg/m 3 ), BEM predictions; (a) characteristic impedance and (b) wavenumber xv

17 7.21 The eect o iber texturization condition on transmission loss o the perorated dissipative silencer ( φ = 25.7 %, d =.498 cm, ρ = 1 kg/m 3 ) h 7.22 The eect o peroration impedance on the transmission loss o the perorated dissipative silencer ( φ = 8.4 %, d h =.249 cm); (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m The eect o peroration impedance on the transmission loss o the perorated dissipative silencer ( φ = 8.4 %, d h =.249 cm, ρ = 1 kg/m 3 ); BEM predictions; (a) real part and (b) imaginary part The eect o peroration impedance on the transmission loss o the perorated dissipative silencer ( φ = 8.4 %, d h =.249 cm, ρ = 2 kg/m 3 ), BEM predictions; (a) real part and (b) imaginary part The eect o mean low on the transmission loss o the dissipative silencer ( d h =.249 cm, ρ = 1 kg/m 3 ) The schematic o the dissipative silencer with a bale ( φ = 8.4 %, d h =.249, and ρ = 1 kg/m 3 ) The eect o a bale on the transmission loss o the dissipative silencer, BEM predictions The schematic o the dissipative silencer with extended inlet and outlet ( φ = 8.4 %, d =.249, and ρ = 1 kg/m 3 ) h 7.29 The eects o extended inlet and outlet on the transmission loss o the dissipative silencer, BEM predictions The schematic o a hybrid silencer Transmission loss hybrid silencers, BEM predictions; (a) overall requency range and (b) low requency range xvi

18 CHAPTER 1 INTRODUCTION 1.1 Background Due to its signiicant acoustic attenuation characteristics at high requencies, ibrous material has been used in interior sound absorption and silencers o machinery that generates high requency noise while operating at low temperature and low velocities, or example, heating, ventilation and air-conditioning (HVAC) systems, or ans. In spite o their desirable acoustic characteristics, the use o traditional absorbing material in the intake and exhaust systems o internal combustion engines has been limited, due to environmental concerns such as blowout o ibers due to low and mechanical/chemical ailure o the ibers at high temperatures. However, recent developments in materials and the demand or improved silencer perormance in industry have led to a renewed interest in dissipative silencers. The blowout has been practically eliminated by the use o continuous strand iber, such as the Advantex iber developed by Owens Corning, coupled with the perorated tubes. In addition to desirable acoustic characteristics, dissipative silencers may oer additional advantages such as improved low eiciency and thermal insulation. For example, in internal combustion engines, lower backpressure rom higher low eiciency will improve the engine perormance, 1

19 and lower muler skin temperatures due to thermal insulation may reduce the thermal atigue. A simple dissipative silencer consists o a perorated main duct and an outer chamber illed with absorbing material. Attenuation o acoustic waves in the absorbing material is mainly due to viscous and thermal dissipation. Thereore, the acoustic properties o absorbing material are essential to understand the noise control by dissipative silencers. In general, the complex numbers o characteristic impedance and wavenumber are employed to account or the dissipation o wave through the absorbing material. Due to the complex structure o absorbing material, these acoustic properties are oten determined experimentally (Delany and Bazley, 197). Perorated ducts are used in reactive silencers to improve the acoustic perormance at low requencies, to reduce low losses, and to minimize generation o low noise rom abrupt cross-sectional area changes. In dissipative silencers, perorated ducts are also used to protect the absorbing material and to control the acoustic behavior o the silencers by adding mainly reactance to the impedance. While the acoustic characteristics o a perorated duct within a reactive silencer is relatively well established primarily by experiments (reer, or example, to Sullivan and Crocker [1978]), the eect o a perorated duct in contact with absorbing material remains to be investigated. The exhaust systems o internal combustion engines generate noise with a wide requency range, including particularly strong low requency components. In such applications, dissipative silencers can be enhanced at low requencies by reactive acoustic elements such as Helmholtz or quarter wave resonators. The combination o dissipative and reactive components deines the hybrid silencer, which is an eective noise 2

20 attenuator over a wide range o requencies. Hybrid silencers commonly have multiple chambers connected to each other by ducts. The interactions among these chambers may substantially aect the overall perormance o hybrid silencers. Thus, the design o hybrid silencers requires understanding o acoustic characteristics o individual dissipative and reactive elements, as well as the interactions among them. The signiicance o such interactions or reactive chambers and resonators was demonstrated by Davis et al. (1953). However, interactions between dissipative and reactive components remain to be investigated. 1.2 Objective The objective o the present study is to analytically, computationally and experimentally investigate the acoustic behavior o hybrid silencers that consist o dissipative and reactive components including mean low eect. First, the acoustic perormance o uniormly perorated cylindrical dissipative silencers is explored with varying duct porosity, hole diameter, and illing density. The transmission loss predictions rom analytical and computational approaches or these conigurations are compared with the experiments. Finally, the analytical and numerical approaches are utilized to illustrate the eect o both individual chamber coniguration and the connecting tubes (or the relative location o the chambers) on the acoustic perormance o the hybrid silencers. One-dimensional decoupled and two-dimensional analytical approaches and three-dimensional boundary element method (BEM) are developed to predict the transmission loss o silencers. Two experimental setups are developed in the present 3

21 study to determine the acoustic properties o absorbing material, the peroration impedance, and the transmission loss. The irst setup is used to measure the acoustic properties o absorbing materials, the impedance o perorations acing absorbent or air, and the transmissions loss in the absence o mean low. The second setup is designed to measure, in the presence o mean low: (1) the acoustic impedance o perorations acing absorbing material, and (2) the transmission loss o silencers. Based on these experimental results, the characteristic impedance and the wavenumber o absorbing material are presented along with an empirical ormulation or the acoustic impedance o perorations in contact with the absorbent with and without the mean low. The study has assumed that (1) the absorbing material is homogeneous, isotropic, and rigid rame; (2) temperature is held constant at ambient conditions; (3) the thickness o the perorated duct is much smaller than the wavelength; and (4) the eect o inite amplitudes is negligible. 1.3 Outline Following this introduction, Chapter 2 presents a review o pertinent works rom literature. Chapter 3 describes the analytical methods and the boundary element method. Experimental setups or acquiring the acoustic characteristics o absorbing material, peroration impedance, and transmission loss with and without mean low are described in Chapter 4. Experimental results or the acoustic properties o absorbing materials, complex characteristic impedance and wavenumber, are presented in Chapter 5. Chapter 6 provides new empirical ormulations or the acoustic impedance o perorations in contact with absorbing material with and without mean low. Using the acoustic 4

22 properties o absorbing material and the peroration impedance developed in Chapters 5 and 6, Chapter 7 compares the predicted transmission loss o single-pass dissipative silencers with experiments ollowed by hybrid silencer. Based on the computational method, parametric studies are also provided in this chapter to illustrate the eect o variation in the acoustic properties o absorbing material and the peroration impedance. Finally, Chapter 8 presents a summary and concluding remarks along with suggestions or uture works. 5

23 CHAPTER 2 LITERATURE REVIEW This chapter gives an overview o existing literature on predominantly dissipative and hybrid silencers in the requency domain. First, examples o hybrid silencers are introduced, ollowed by the examination o studies related to dissipative silencers. Analytical and numerical approaches to predict the acoustic behavior o dissipative silencers are reviewed. Experimental methods to determine the acoustic properties o absorbing material and the impedance o perorations in contact with this material are also presented. Finally, only a brie review o literature on Helmholtz resonators is provided due to the act that the acoustic characteristics o Helmholtz resonators have been widely investigated. 2.1 Hybrid silencers The idea o combining dissipative and reactive chambers, hence comprising the hybrid silencers, dates back to 193s. An early example is a three-pass muler by Peik (1935) with a main chamber packed by absorbing material, and a recent one is a pair o hybrid silencer designs suggested by Sterrett et al. (1998), which involve a single dissipative chamber and multiple reactive chambers envisioned as Helmholtz resonators. 6

24 The use o absorbing material on silencers can be ound rom numerous patents (Nelson, 1937; Schmidt, 1937; Manning, 1939; Wolhugel, 1983; Fukuda, 1986; Tanaka et al., 1987; Hetherington, 1989; Udell, 199; Hisashige et al., 1991; Kraai et al., 1994). Yet, no analytical or experimental guidance has been provided to determine the acoustic characteristics o particularly hybrid conigurations. Recently, Selamet et al. (23) predicted the transmission loss o a hybrid silencer consisting o two dissipative chambers and a reactive resonator using BEM. They demonstrated computationally the eectiveness o hybrid silencers over a wide requency range, while not detailing the interaction among the elements or providing validation. 2.2 Dissipative silencers One- and two-dimensional analytical approaches and three-dimensional numerical methods have been widely used to predict the acoustic behavior o dissipative silencers. Analytical approaches may be suitable or simple geometries, while numerical methods need to be used or silencers with complex conigurations or inhomogeneous absorbing material properties. The acoustic properties o absorbing material are usually obtained by experiments due to their complex structures. Thus, this section summarizes the available analytical and numerical approaches or dissipative silencers, as well as the experimental methods to acquire the acoustic properties o absorbing material. In addition, studies on the acoustic impedance o perorations backed by absorbing material are reviewed. 7

25 2.2.1 Analytical models Depending on the dimensions o silencers and requencies o interest, a lumped model, one-dimensional decoupled, and two-dimensional analytical approaches can be utilized to predict the acoustic behavior o dissipative silencers in the requency domain. A simple lumped model may be useul to approximate the resonance requencies o perorated dissipative silencers and Helmholtz resonators with small dimensions compared to the wavelength o interest. One-dimensional decoupled approach may be used to predict the transmission loss o perorated reactive and dissipative silencers at low requencies. Two-dimensional analysis is used to account or higher order modes. One-dimensional decoupled models One-dimensional decoupling model or unilled perorated silencers was introduced by Jayaraman and Yam (1981) with the assumption o equal mean low in the main duct and outer chamber. The method was extended by Peat (1988) and Munjal (1987) to unilled perorated multiple duct silencers. Wang (1999) and Selamet et al. (21) applied the same method to dissipative perorated silencers. One-dimensional decoupling approach may be accurate at low requencies. Such a method, however, shows numerical instability or large duct porosities or long chambers (Peat, 1988). Multi-dimensional analytical models Absorbing material o dissipative silencers can be modeled as either locally or bulk reacting depending on the geometry o silencers and the acoustic properties o the material. For the locally reacting model, the wave is assumed to propagate only in the 8

26 normal direction o the material surace, and the normal surace impedance o the material is applied as boundary conditions to the main ducts o the silencers. Thereore, only the surace impedance o the absorbing material is important in this model. Such a model can be appropriate or the absorbing material with high low resistivity or small thickness, and or partitioned silencers (Mechel, 199a). Morse (1939b) used the locally reacting model to investigate wave propagation inside ininitely long ducts lined with absorbing material. Later, Mechel (199a, 199b) developed a theoretical analysis and provided the transmission loss o baled inite length o dissipative silencers. However, the locally reacting model may not be appropriate or low low resisitivity. On the other hand, the bulk reacting model considers wave propagation in multiple directions within the absorbing material, thus yielding more accurate predictions than the locally reacting approach. Typically, a modal analysis is used to describe the wave propagation within both the main duct and dissipative chamber. In this analysis, eigenvalues (wavenumbers) and eigenunctions (modes) or an ininite length o a silencer are calculated irst. Numerical schemes are generally used to calculate the common wavenumbers in two domains o the silencer. Then, boundary conditions at inlet and outlet endplates are applied to obtain overall acoustic perormance such as transmission loss. Transer matrix methods may be applied or multi-chamber silencers ater using the modal analysis or each chamber. Scott (1946c) provided a theoretical analysis in the requency domain or ininite lengths o rectangular and circular ducts lined with homogeneous and isotropic absorbing materials. He used wave equation with complex wavenumber and eective density to account or the wave dissipation through the absorbing material. Scott s analytical results were compared and veriied by Bokor 9

27 (1969), Wassilie (1987), and Kurze and Ver (1972). Bokor (1969) experimentally obtained the wavenumber and the characteristic impedance using a standing wave method or various sample thickness and compared them with the theoretical work o Scott (1946c). Kurze and Ver (1972) suggested a more generalized model o Scott s method including non-isotropic absorbing material, and compared their results with those rom Scott (1946c) and Bokor (1969). Wassilie (1987) also veriied the homogeneous isotropic model o Scott (1946c) and the non-isotropic model o Kurze and Ver (1972). While these studies ocused on constant rectangular or circular cross-sectional areas, Glav (1996a, 1996b) used point matching and null-ield methods to ind eigenvalues and eigenunctions o ininite length o dissipative silencers with arbitrary constant crosssectional areas. Aly and Badawy (21) developed a theoretical approach including the eect o variable duct cross-sectional area, as well as liner length, porosity, and low resistivity. The overall behavior o dissipative silencers with inite length is associated with inlet and outlet boundary conditions. Cummings and Chang (1988) used a match o sound ield at the interace between chamber and inlet/outlet pipe. Glav (2) extended his previous works (null-ield and mode-matching) to inite length o dissipative silencers with arbitrary cross-sectional areas using a transer matrix method. For many applications o dissipative silencers, mean low exists in the main duct, which may change the acoustic characteristics o the silencers. Thus many theoretical and experimental eorts have been made or dissipative silencers with mean low. The wave propagation in ininite length o dissipative silencers in the presence o mean low has been investigated using either locally or bulk reacting assumptions. Ko (1972) developed a theoretical method or the prediction o acoustic behavior o ininitely long 1

28 duct lined with absorbing material using the locally reacting model in the presence o mean low. He demonstrated the eect o Mach number, boundary layer reraction, and acoustic impedance; and concluded, in particular, that the undamental mode was not necessarily the least attenuated in the lined duct. Accordingly, a simple one-dimensional model that includes only the undamental mode may not provide accurate results even at low requencies. The eect o mean low on ininite length o bulk reacting materials has been investigated by Cummings and Chang (1987a, 1987b), Bies et al. (1991), Gogate and Munjal (1993), and Cummings and Sormaz (1993). Cummings and Chang (1987a, 1987b) investigated anisotropic bulk reacting liners including mean low within the material and concluded that the induced mean low inside the absorbing material can increase the eective low resisitivity. However, their assumption o uniorm mean low proile within the absorbing material may be valid or limited cases. Cummings and Chang (1988) extended their earlier works on ininite lengths to inite-length silencers by incorporating boundary conditions at the inlet and outlet ducts. They obtained then the transmission loss o dissipative silencers, again including the internal uniorm mean low within the absorbing material Numerical models Hybrid silencers may involve complex geometries due to a combination o reactive and dissipative elements. Multi-dimensional numerical methods such as inite element method (FEM) and BEM are eective or the silencers with such complicated conigurations, inhomogeneous acoustic ield, or acoustic-structure interactions. FEM is useul or the silencers with structure interaction or inhomogeneous acoustic ield. 11

29 However, FEM requires a large number o nodes and elements or the silencers with large dimensions. On the other hand, BEM is appropriate or complex silencers with homogeneous acoustic ield or unbounded radiation problems. FEM has been used or dissipative silencers, or example, by Craggs (1977), Kagawa et al. (1977), Astley and Cummings (1987), Cummings and Astley (1996), and Peat and Pathi (1995). Craggs (1977) used FEM and a locally reacting model to predict the behavior o a lined expansion chamber without experimental validation. Kagawa et al. (1977) compared their FEM results with the experiments or lined expansion chambers, including temperature gradient. Astley and Cummings (1987), Cummings and Astley (1996), and Peat and Pathi (1995) demonstrated the eect o mean low in the main duct on the overall behavior o dissipative silencers. BEM has been employed or reactive silencers, or example, by Wang et al. (1993), Ji et al. (1994, 1995), Wang and Liao (1998), and Wu and Zhang (1998). Wang et al. (1993) used BEM to demonstrate the higher order modes or simple expansion chambers with oset inlet and outlet. Ji et al. (1994) predicted the transmission loss o simple expansion chambers using the direct Green unction solution (that is, without the coordinate transormation) and the substructure method in the presence o mean low. In particular, a transer matrix can be acquired rom only one BEM run in their approach. Later, they also illustrated velocity and pressure distribution inside simple expansion chambers using BEM (Ji et al., 1995). Wang and Liao (1998) used a coordinate transormation to predict the transmission loss o straight-through perorated reactive silencers in the presence o mean low. Wu and Zhang (1998) also used BEM to obtain the transer matrix o a perorated silencer. They argued that their method was more 12

30 eicient than conventional transer matrix and three-point approaches. However, it still required two BEM runs. BEM was also applied to dissipative silencers by Cheng and Wu (1999), Selamet et al. (21), and Wu et al. (22, 23). Cheng and Wu (1999) and Wu et al. (22) developed a direct-mixed BEM, where multi-domains were integrated into a single domain. Recently, Wu et al. (23) incorporated perorated interace along with ibrous mats into BEM or bulk reacting dissipative silencers. Selamet et al. (21) employed BEM or the predictions o single-pass dissipative silencers, including the eect o absorbing material on the peroration impedance Acoustic properties o absorbing material Wave propagation through an absorbing material is dissipated into heat by the viscous boundary layer eect. Thus, in order to predict the acoustic behavior o dissipative silencers, accurate acoustic properties o absorbing material are necessary. Surace properties such as surace impedance and absorption coeicient are used as boundary conditions or locally reacting models. On the other hand, wave propagation through an absorbing material can be described by the complex characteristic impedance and the wavenumber, which are employed or a bulk reacting model. Due to their complexity, the acoustic properties o absorbing material are usually obtained by experiments. Standing wave and two-microphone methods are commonly used or the measurements o the wave pattern relected rom the absorbing material, rom which the acoustic properties can be acquired. Recently, multi-microphone techniques were examined to reduce the error (Jang and Ih, 1998). Jones and Stiede (1997) summarized 13

31 the experimental methods or dierent number o microphones and type o sound sources in terms o accuracy and time eiciency. Surace impedance and absorption coeicient The absorption coeicient or surace impedance represents the acoustic behavior at the surace o the absorbing material. While the absorption coeicient represents only the acoustic characteristics o the material itsel with a real number, the surace impedance can be incorporated into silencer analyses with a complex number. The measurements o these two surace acoustic properties can be perormed using an impedance tube setup. A classical standing wave method and a two-microphone approach are examples o such impedance tube setups. The classical standing wave method was used until 198s, or example by Beranek (194a, 194b, 1942, 1947a, 1947b) and Scott (1946b). The method locates pressure maximum and minimum (standing wave pattern) using a movable microphone at each requency in order to calculate the surace impedance. Though it gives a reasonable accuracy, the standing wave method may not be appropriate in the presence o mean low. In addition, it requires a measurement at each requency and needs a long pipe at low requencies. On the other hand, the two-microphone method (transer unction method), which was introduced by Seybert and Ross (1977) and standardized by ASTM in 1985, has been used extensively since it can obtain surace properties or a wide requency range with a single experiment using signal processing. Normal incident sound absorption coeicient and normal speciic impedance o absorbing materials can be determined by this method, which uses an impedance tube, two microphones, a digital requency analysis, and a 14

32 broadband random sound source. The wave inside the impedance tube is assumed planar. Two microphones are used to decompose the standing wave into two traveling waves. The standard two-microphone test method with broadband random excitation is described in ASTM E15-98, and the detailed theory, including the transmission loss calculation, were presented by Seybert and Ross (1977) and Chung and Blaser (198a, 198b). Error in the two-microphone method can be caused by inaccurate distances between microphones or sample, non-identical microphone characteristics, and the wave attenuation inside the impedance tube. Eorts have been made to reduce the error by Chung and Blaser (198a), Seybert and Soenarko (1981), Boden and Åbom (1986), Åbom and Boden (1988), and Katz (2). Chung and Blaser (198a) demonstrated the eectiveness o the use o a third microphone and a microphone switching technique or calibration. Seybert and Soenarko (1981) claimed that bias error could be reduced by locating microphones close to a sample. Boden and Åbom (1986) concluded that, in addition to a close microphone location to a sample, non-relective termination and high coherence between microphones reduced the error. Later, Åbom and Boden (1988) extended their previous error analysis to measurements with mean low and the attenuation o wave inside the ducts. Furthermore, they suggested the use o a high signal-to-noise ratio, large microphone spacing, and signal averaging in order to minimize the error due to mean low. Recently, a multi-microphone technique in the presence o mean low was presented by Jang and Ih (1998) using a least squares method. The method may provide accurate results, but requires more than two microphones or their movement. Katz (2) suggested a technique to determine the precise distance rom a sample and spacing between microphones using a third microphone. 15

33 Characteristic impedance and wavenumber The surace acoustic characteristics o absorbing material do not provide enough inormation to describe the wave propagation through the absorbing material. Thus the complex wavenumber and the characteristic impedance o absorbing materials are introduced to account or both propagation and dissipation o wave through the absorbing material. In general, these two values are requency-dependent complex numbers and are acquired by impedance tube experiments. The standing wave pattern inside the absorbing material was measured earlier by Scott (1946b) to determine the wavenumber. However, this direct measurement inevitably disturbs the acoustic ield resulting in inaccuracies. Alternatively, the surace impedance obtained by indirect methods can be used to determine the wavenumber and the characteristic impedance o the absorbing material. Two dierent measurements o the surace impedance at one side o a sample are required with a dierent end condition (two-cavity method) or material thickness (twothickness method). The two-cavity method with a ¼-wave cavity backing was adopted by Delany and Bazley (197) and Yaniv (1973). Two dierent cavity backings generate zero and ininite acoustic impedance at each requency. That is, this method requires two experiments at each requency. Two-thickness method proved to be more accurate and eicient than the two-cavity method with ¼-wave cavity backing depth (Smith and Parrott, 1983; Woodcock and Hodgson, 1992). However, the two-thickness method requires two identical samples, which may be diicult to produce. Furthermore, adding the second sample to the irst one may generate an interace between the two samples, 16

34 consequently leading to an error. Thus the improved two-cavity method by employing the transer unction approach and arbitrary backing space depth has been widely used, or example by Utsuno et al. (1989) and Nice and Godrey (2). Especially, Utsuno et al. (1989) discussed the selection o an appropriate set o backing space depth. The two-thickness and two-cavity methods necessitate two measurements on one side o a sample with two dierent conditions on the other, which may be impractical or a high density or long sample. Thereore using impedances at both sides o a sample in a single experiment becomes a simple and eective alternative or high as well as low material density. For example, Song and Bolton (2) provided a simple method that utilizes the reciprocal characteristics o transer matrix or homogeneous material with a single experiment. The method can be also used or low density material. Recently, Tao et al. (23) proposed a two-source method to measure the characteristic impedance and the wavenumber. The two-source method was shown to be advantageous compared to the two-cavity approach or the materials with absorption coeicients below.4. However, the two-source method exhibits, in their experiment, unstable characteristic impedance or high absorption coeicient at high requencies, and requires an exchange o speaker locations. Flow resistivity Flow resistivity, deined as the ratio o pressure dierence across a unit length o sample to low velocity, is one o the important acoustic properties o absorbing material. This quantity has been used, or example by Delany and Bazley (197), as a parameter in determining the characteristic impedance and wavenumber o the absorbing material. 17

35 Because o the complexity related to the material structure (size and shape), illing density, and types o construction (random or layered) o material, the low resistivity is usually determined by experiments. A steady-state low method uses an air-low measurement, which obtains the low resistivity irst with mean low and then applies an extrapolation or zero mean low. Since this method neglects the inertia eect, the low resistivity is a real number. Nichols (1947) presented a useul empirical ormulation o low resistance, which is closely related to the low resistivity, or the materials with high porosity (.9-1.). According to Nichols, perpendicular illing to low has higher low resistance than random illing or the same thickness. In addition, the low resistance is inversely proportional to the square o iber diameter. Recently, the anisotropic eect o low resistivity was explored by Tarnow (22), who claimed that the resisitivity perpendicular to the iber layers was doubled relative to that o the parallel direction Acoustic impedance o perorations in contact with absorbing material Perorated ducts in silencers without absorbing material are commonly used to enhance the acoustic perormance at low requencies, to reduce the low separation, thereby reducing the energy losses and the noise generated by low, and to redirect the low path. The peroration impedance without backing by absorbing material has been extensively investigated, or example, by Ingard (1953), Sullivan and Crocker (1978), Melling (1973), Dickey et al. (2, 21), Dean (1974), Rao and Munjal (1986), Cummings (1986), Jing et al. (21), Lee and Ih (23), Sun et al. (22), and Salikuddin et al. (1994). An empirical ormulation o the peroration impedance by Sullivan and Crocker (1978) has also been widely used in the linear regime in the 18

36 absence o mean low. Melling (1973) and Dickey et al. (2) considered the impact o high amplitudes on the peroration impedance. In the nonlinear regime, the peroration impedance is dependent on the acoustic velocity. The resistance and reactance increase and decrease, respectively, as the acoustic velocity increases in the nonlinear regime. The eect o grazing mean low on the impedance was included by Dean (1974), Rao and Munjal (1986), Cummings (1986), Dickey et al. (21), and Jing et al. (21). According to these studies, grazing low generally decreases the reactance and increases the resistance o the peroration impedance. A perorated duct or screen is used also in dissipative silencers to prevent absorbing material rom being blown out by low. The backing by absorbing material can alter the peroration impedance and consequently the overall characteristics o dissipative silencers. Peroration impedance with backing in the absence o mean low has been studied by Bolt (1947), Ingard and Bolt (1951a, 1951b), Ingard (1954a), Callaway and Ramer (1952), Davern (1977), Takahashi (1997), and Chen et al. (2). Bolt (1947) analytically considered the mass reactance o perorations acing the absorbing material. He concluded that the addition o perorations to the absorbing material generally increased the absorption coeicient at low requencies at the cost o a loss at high requencies. Later, Ingard and Bolt (1951b) also indicated the eectiveness o perorations at low requencies and discussed an increase o resistance o the impedance. They concluded that a combination o absorbing material and perorations established a Helmholtz resonator. Ingard (1954a) emphasized the increase o resistance by the addition o perorations and the eect o an air gap between the perorations and absorbing material. His analysis showed that the air gap between the perorations and 19

37 acing material narrowed the eective requency range o absorption coeicients. Callaway and Ramer (1952) also considered the perorations acing absorbing material as a Helmholtz resonator. They showed that the air gap between the perorations and absorbing material increased the absorption coeicients or a very low duct porosity (1.23%). Davern (1977) experimentally illustrated the eect o porosity and wall thickness o the perorations, absorbing material density, air space between perorations and absorbing material, and impervious layer between the perorations and absorbing material on the absorption coeicient. Chen et al. (2) used FEM to predict the absorption coeicients or perorations acing various surace shapes in the absence o mean low. Cummings (1976) suggested a theoretical peroration impedance acing absorbing material in the presence o mean low. He incorporated the wavenumber and characteristic impedance o the acing material into the peroration impedance. However, the mean low eect was simpliied by ignoring the end correction on the low side when the low speed was greater than 1 m/s. Kirby and Cummings (1998) presented a semiempirical ormulation or the peroration impedance backed by absorbing material in the presence o grazing mean low. They concluded that the absorbing material increased the peroration impedance and the results were strongly dependent on the density o absorbing material immediately adjacent to perorations. They also argued that the interaction among holes could be neglected and thus the ormulation or single hole could be used or a multi-hole peroration. This semi-empirical ormulation is employed in the literature (Kirby, 21, 23). However, Kirby (21) concluded that the use o semi- 2

38 empirical expression o Kirby and Cummings (1998) might overestimate the peroration impedance. 2.3 Helmholtz resonators Given its low-requency attenuation characteristics, Helmholtz resonators may be desirable reactive elements to use in a hybrid silencer. For simple geometries with small dimensions compared to wavelength o interest, lumped models can be applied to estimate the resonance requency. However, multi-dimensional analyses may be required or Helmholtz resonators in hybrid silencers since they may have relatively long necks or complex geometry as a result o the combination with dissipative chambers. Helmholtz resonators are well understood compared to dissipative silencers. Ingard (1953) investigated the eect o neck geometry such as cross-sectional area shape, location, and size on the resonance requency o such a resonator with circular or rectangular crosssectional area or the volume. He developed end corrections or both single and double holes to account or the higher order modes at the interace between neck and cavity. Chanaud (1994) examined the eect o both oriice and cavity geometry on the resonance requency o a Helmholtz resonator using the end corrections suggested by Ingard (1953). He presented the limitations o simple lumped and transcendental models based on the predictions, and concluded that or a ixed volume and oriice size, the oriice position changed the resonance requency substantially, while the oriice shape was not signiicant. Both reerences (Ingard, 1953; Chanaud, 1994) considered only very short neck length compared to wavelength. Tang and Sirignano (1973) studied a Helmholtz resonator with a neck comparable to wavelength as an application to reduce combustion 21

39 instability. They developed a general ormulation and applied to quarter wave and Helmholtz resonators with various neck lengths. Recently, Selamet and his coworkers (Dickey and Selamet, 1996; Selamet et al., 1997; Selamet and Ji, 2) have employed several approaches to examine the eect o cavity volumes and neck locations. They illustrated the eect o length-to-diameter ratio o the volume on the resonance requency and the transmission loss characteristics using lumped and one-dimensional approaches (Dickey and Selamet, 1996). Earlier works have been extended urther by studying a number o circular concentric conigurations with lumped, radial and axial one-dimensional models, two-dimensional analytical approaches, and a three-dimensional boundary element method (Selamet et al., 1997). They also developed a three-dimensional analytical approach to investigate the eect o neck oset on the behavior o circular asymmetric Helmholtz resonators (Selamet and Ji, 2). 22

40 Equation Chapter 3 Section 1 CHAPTER 3 THEORETICAL MODELS To predict the acoustic behavior o dissipative and hybrid silencers, dierent approaches can be used depending on their geometries and applications. Lumped model is suitable or the silencers with dimensions small compared to the wavelength o interest in order to approximate resonance requencies and transmission losses. One-dimensional analysis may be appropriate or the silencers with relatively small diameters leading to planar wave propagation. A decoupled one-dimensional analytical approach is applied in this study or both unilled and dissipative perorated silencers. However, the neglect o higher order mode propagation may lead to inaccuracies or the silencers with large diameters or at high requencies. The eect o higher order modes can be eectively accounted or by using multi-dimensional methods. In this study, a two-dimensional analytical approach is developed or axisymmetric, cylindrical, unilled and dissipative silencers to incorporate the higher order modes. Finally, the three-dimensional BEM is also employed or the acoustic prediction o silencers with generic geometries with and without absorbing material. The lumped, one-dimensional decoupled, and two-dimensional analytical approaches and the three-dimensional boundary element method used in the present study 23

41 are introduced in this chapter. In addition, the acoustic properties o absorbing material and the acoustic impedance o perorations acing absorbing material are also discussed. 3.1 Lumped model The lumped approach can be applied to resonators and silencers when their dimensions are so small ( k!" 1, k is the wavenumber and! is the characteristic dimension) that the spatial parameters can be ignored. The estimations o resonance requencies and transmission losses or Helmholtz resonators are typical examples o the lumped model applications. Unilled and dissipative perorated silencers are also can be modeled as undamped and damped Helmholtz resonators Simple Helmholtz resonator In terms o acoustic resistance R ac, acoustic inertance L ac, and acoustic compliance Fig 3.1 is deined as, where C ac, the acoustic impedance o the unilled Helmholtz resonator depicted in ph 1 ZH = = Rac + iω Lac +, (3.1) Su iωc k H ac L ac ρ = (! + δ ) k k S k, (3.2) C ac = V ρ, (3.3) c 2 c 24

42 Cavity V c δ 1 Neck p H S k, u p, u p, in in δ 2 H! k δ = δ + δ k 1 2 out u out d h Figure 3.1: The schematic o a simple Helmholtz resonator. 25

43 p H and u H are the acoustic pressure and particle velocity at the neck, respectively, S k the neck cross-sectional area, ω the angular velocity, ρ the air density, c the speed o sound,! k the neck length, V c the cavity volume, and δ k the end correction to account or higher modes excited at the discontinuities, which may be determined by the geometry and location o the neck relative to the volume and main duct. The acoustic resistance R ac due to viscous riction and radiation is generally neglected or simple Helmholtz resonators except or very small neck diameters. The resonance requency o the Helmholtz resonator can be calculated when the impedance in Eq. (3.1) is minimum, that is Z H = ignoring the acoustic resistance, thus 1 ωr = 2π r =. (3.4) L C Substitution o Eqs. (3.2) and (3.3) into Eq. (3.4) yields the resonance requency o the resonator as ac ac r c = 2π V S k (! + δ ) c k k. (3.5) The transmission loss o the Helmholtz resonator can be also estimated rom the lumped analysis. The transer matrix relating the acoustic pressure and particle velocity at the inlet and outlet o the main duct is given by 1 p in p out T11 T 12 p out = ρcs =, (3.6) k ρ 1 cu in ρcu out T21 T 22 ρcu out ZH S d 26

44 where S d is the cross-sectional area o the main duct, and subscripts in and out indicate the inlet and outlet o the silencer, respectively. The transmission loss o the resonator can be then calculated, assuming the same cross-sectional areas or the inlet and outlet, as and thus 1 1 ρcsk TL = 2log1 ( T11 + T12 + T21 + T22) = 2log1 + 1 (3.7) 2 2 Z S H d 1 ρ c S TL = 2 log + 1. (3.8) k ρ (! k + δk) ρc Sd Rac + iω + Sk iωvc Thus, the transmission loss is dependent on the impedance o the resonator and the crosssectional area ratio o the neck to the main duct Perorated silencers An empty (unilled) perorated silencer depicted in Fig. 3.2 may be simpliied as a combination o an expansion chamber and a Helmholtz resonator. The perorations and the outer chamber can be considered as a neck and a cavity, respectively. Sullivan and Crocker (1978) expressed the resonance requency o a perorated duct silencer as ollows: = c φd π ; k L <, (3.9) 4 1 rp, 2 2 π ( d2 d1 )( tw + δk) where φ is the duct porosity, t w the perorated duct wall thickness, L the silencer length, d 1 and d 2 the diameter o the main duct and the outer chamber, respectively. 27

45 d h p, u ρ, c, k 2 2 ρ, c, k p, u 1 1 d1 d2 x L Figure 3.2: The schematic o a perorated reactive single-pass straight silencer. 28

46 Equation (3.9) can be obtained by substituting the neck area, cavity volume, and neck length given by Sk = φπd L, (3.1) 1 Vc π 2 2 = ( d2 d1 ) L, (3.11) 4! = t (3.12) k w into Eq. (3.5). Equation (3.9) may be valid or low duct porosities since the unilled perorated silencer with high duct porosity may act as rather an expansion chamber than a combination o a Helmholtz resonator and an expansion chamber. Sullivan and Crocker (1978) argued that Helmholtz resonator model or the perorated silencer shows 27% error rom the experiments or the given dimension in the literature Dissipative perorated silencers The perorated dissipative silencer depicted in Fig. 3.3 also can be approximated as a damped Helmholtz resonator as shown in Fig The acoustic inertance and compliance o the damped Helmholtz resonator can be expressed, using complex density ( ρ# ) and speed o sound ( c# ), as C V = (3.13) ρc "" c ac, D 2 and L ac, D ρ = ( t + δ ) w S k " k, (3.14) 29

47 d h # ρ, c#, k# p2, u2 ρ, c, k p, u 1 1 d1 d2 x L Figure 3.3: The schematic o a perorated dissipative single-pass straight silencer. 3

48 Cavity V c δ # 1 Perorations p H S k, u H p, u p, in in δ 2! n # δ = # δ + δ k 1 2 out u out S d Figure 3.4: Helmholtz resonator model o the dissipative silencer in Fig

49 where # δ k is the total end correction o perorations acing air and absorbing material at each side as shown in Fig Since the acoustic inertance and compliance are complex numbers and requency-dependent, Eq. (3.4) may not be utilized to calculate the resonance requencies or dissipative silencers. However, the transmission loss can be obtained rom Eqs. (3.1), (3.7), (3.13), and (3.14), and then the requency at which the transmission loss is maximum can be viewed as the resonance requency o the resonator. 3.2 One-dimensional decoupled model Assuming harmonic planar wave propagation in both the main duct and the illed outer chamber (Fig. 3.3), the continuity and momentum equations yield, in the absence o mean low (Munjal, 1987; Wang, 1999), and 2 4 ik 4 ik + k p1 p2 dx d # 1 ζ + = p d # 1 ζ (3.15) p 2 d p1 2 2 d p2 2 dx 4d # 1 ρ ik 2 4d1 ρ ik p1+ k# # 2 2 p2 = d2 d1 ρ # ζ p d2 d1 ρ # ζ, (3.16) p where k denote the wavenumber in air, p 1 and p 2 is the acoustic pressure in the main duct (domain 1) and the outer chamber (domain 2), respectively, and # ζ p the acoustic impedance o the perorated duct, which can be deined as p p # 1 2 p =, (3.17) ρcu 1 ζ 32

50 where u 1 is the particle velocity in the main duct. Equation (3.17) is to be urther elaborated later in Section 3.6. Equations (3.15) and (3.16) may be rearranged as, p 1 1 p1 2 4 ik dp 1 4 ik dp k 1 dx d # 1 ζ p d # 1 ζ p dx =, (3.18) p 2 1 p2 4d1 ik 2 4d dp dp ρ 1 ρ ik 2 2 # k# # d d dx dx 2 d1 ρ ζ p 2 d1 ρ ζ # # p where () indicates derivative with respect to x. Using the linearized momentum equation, Eq. (3.18) can be rearranged as ik p 1 ik 1 p ρcu d # 1 1 ζ p d # 1 ζ p ρcu 1 = p 2 ik# p2 ρcu 2 4d1 ρ k 1 4d1 ρ k 1 ## # ρcu 2 ik# # ## d2 d1 ρ k# # ζ p d2 d1 ρ k# #. (3.19) ζ p p1 ρcu 1 = [ TA] p2 # ρcu # 2 The solution o Eq. (3.19) may be expressed in terms o eigenvalues and eigenvectors as λ1 x p1 ( x) ce 1 λ2x ρcu 1( ) x ce 2 = [ Ψ] λ3x, (3.2) p2 ( x) ce 3 λ4x # ρcu # 2 ( x) ce 4 where λ n is the eigenvalue o the matrix [TA] and [ Ψ ] the modal matrix whose columns are the eigenvectors. Equation (3.2) can be rewritten as 33

51 p1( x) c1 ρcu 1( ) c x 2 = [ Ψ ( x) ], (3.21) p2( x) c3 # ρcu # 2( x) c 4 which leads to a relationship between the acoustic pressure and the particle velocity at the inlet ( x = ) and outlet ( x = L ) as where p1( ) p1( L) ρcu( ) 1 ρcu 1( ) L = [ TB], (3.22) p2( ) p2( L) # ρcu # 2( ) # ρcu # 2( L) [ ][ ( )] 1 [ TB] = Ψ ( ) Ψ. (3.23) For the outer chamber, the boundary conditions at x = and x = L may be written as and L u 2 ( ) = (3.24) u2 ( L ) =. (3.25) Finally, combining Eqs. (3.22), (3.24) and (3.25) yields the transer matrix o the dissipative silencer as where ( ) ( ) ( ) ( ) p T T p L =, (3.26) ρcu 1 T21 T 22 ρcu 1 L T TB TB = TB11, (3.27) TB43 34

52 T T T TB TB = TB12, (3.28) TB43 TB TB = TB21, (3.29) TB43 TB TB = TB22. (3.3) TB43 Assuming the main duct with a constant cross-sectional area, the transmission loss o the silencer can be then calculated rom the transer matrix as ollows: 1 TL 2log = 1 T11 + T12 + T21 + T22 2. (3.31) 3.3 Two-dimensional analytical model A two-dimensional analytical approach is introduced next to determine the acoustic characteristics o the cylindrical, concentric dissipative silencer o length L, main duct radius r 1, and illed outer chamber radius r 2 (Fig. 3.5). For two-dimensional axisymmetric and harmonic wave propagation in a circular, the governing equation duct in the cylindrical coordinates ( rx, ) can be expressed by or ( ) k p( r x) 2 p r, x + 2, = (3.32) 2 2 p 1 p p k p =. (3.33) r r r x 35

53 IIb B + n B n # ρ, c#, k# r 2 r 1 I A + n A n IIa B + n B n ρ, c, k r III C + n C n r 1 x L Figure 3.5: Wave propagation in a perorated dissipative silencer in axisymmetric twodimension. 36

54 The solution o Eq. (3.33) in domain I or inlet duct (Fig. 3.5) can be written as + ikaxn (, ) ( ) ( ),, x ikaxn,, x p rx = Ae + Ae ψ r, (3.34) A n n A, n n= where subscript A denotes domain I, p A is the acoustic pressure, A + n and A n the modal amplitudes corresponding to components traveling in the positive and negative x directions in domain I, respectively, Axn,, k the axial wavenumber, and ( r) eigenunctions. For the circular duct, the eigenunctions are given by ( r) J ( karnr) An,,, ψ the An, ψ =, (3.35) where J is the Bessel unction o the irst kind o order zero, and k Arn,, (radial wavenumber) the roots satisying the rigid wall boundary condition at r = r1 o ( Arn ) ( Arn ) J k r = J k r =. (3.36),, 1 1,, 1 The relationship between axial and radial wavenumbers is given by k Axn,, k k ; k > k = k k ; k < k 2 2 Arn,, Arn,, 2 2 Arn,, Arn,,, (3.37) ikaxnx e where the negative sign in Eq. (3.37) is assigned so that,, decays exponentially in x direction. The particle velocity in the axial direction may then be written, in terms o the linearized momentum equation, as 1 + ikaxn,, x ikaxn,, x u ( r, x) = k A e A e ψ ( r). (3.38) Ax, Axn,, n n An, ρω n= The outlet acoustic pressure and particle velocity in the axial direction (domain III) are similar to those at the inlet duct and are expressed as 37

55 + ikcxn,,( x L) ikcxn,,( x L) C(, ) = ( n + n ) ψ C, n( ) n= p rx Ce Ce r (3.39) and 1 u r x k C e C e r + ikcxn,,( x L) ikcxn,,( x L) Cx, (, ) = Cxn,, n n ψ Cn, ρω n= ( ), (3.4) where subscript C denotes domain III, and C + n and C n the modal amplitudes corresponding to components traveling in the positive and negative x directions in domain III, respectively, and Cxn,, are given by k the axial wavenumber. The eigenunctions ψ ( r) ( r) J ( kcrnr) Cn,,, Cn, ψ =, (3.41) and k Crn,, (radial wavenumber) the roots satisying the rigid wall boundary condition at r = r 1 o ( Crn ) ( Crn ) J k r = J k r =. (3.42),, 1 1,, 1 The sound propagation in domain II is given by where p 1 p p r r r x p = 2 2 κ, (3.43) k; r r1 κ = k #. (3.44) ; r r r 1 2 The solutions to Eq. (3.43) are given by or domain IIa (air) + ikbxn,, x ikbxn,, x p ( rx, ) = ( Be + Be ) ψ ( r) ; r r; (3.45) Ba n n Ba, n 1 n= 38

56 and or domain IIb (absorbing material) + ikbxn,, x ikbxn,, x p ( rx, ) = ( Be + Be ) ψ ( r) ; r r r; (3.46) Bb n n Bb, n 1 2 n= subscript Ba and Bb denote domain IIa and IIb, respectively, B + n and B n the modal amplitudes corresponding to components traveling in the positive and negative x directions in domain II, respectively, k Bxn,, the common wavenumber in the axial direction or both air and absorbing material, and ψ Ba, n and ψ Ba, n the eigenunctions in domain IIa and IIb, respectively. The radial wavenumbers or air and absorbing material are dierent and is related by k = k k (3.47) 2 2 Brn,, Bxn,, and k# = k# k. (3.48) 2 2 Brn,, Bxn,, Using momentum equation, the acoustic velocities in the radial directions are expressed by and + Bxn Bxn ( ) ( r) 1 ψ u B e B e r r (3.49) ik,, x ik,, x Ba, n Ba, r = n + n iρω n= r ; 1 + Bxn Bxn ( ) ( r) 1 ψ u B e B e r r r ik,, x ik,, x Bb, n Bb, r = n + n ; 1 2 i # ρω n= r (3.5) or domain IIa and IIb, respectively. The transverse modal eigenunctions in Eqs. (3.45) and (3.46) can be expressed as ( r) ( kbrn,, r) ( kbrn,, r) ψ = B J + B Y ; r r ; (3.51) Ba, n 1, n 2, n 1 39

57 and ψ = # + # ; (3.52) ( r) ( kbrnr) ( kbrnr) Bb, n B3, nj B,, 4, ny ; r,, 1 r r2 where Y denotes Bessel unction o the second kind o order zero, B 1, n B 4, n the coeicients related by the ollowing our boundary conditions at r =, r 1, and r 2 : (1) At r =, the pressure is inite, thus Eq. (3.51) yields (2) At r r2 =, the rigid wall boundary condition, (, ) B 2, n =. (3.53) u xr =, gives Bbr,, 2 ( kbrnr ) n ( kbrnr ) B J # B Y # (3.54) 3, n 1 +,, 2 4, 1 =,, 2 where Y 1 and J 1 are Bessel unctions o the irst and second kind o order one, respectively. (3) At r r1 (3.53) yield =, the continuity o radial particle velocity, u ( x, r ) u ( x, r ) Brn,,,, 1, n 1 3, n 1 4, n 1 # =, and Eq. Ba, r 1 Bb, r 1 k k# B J Brn ( kbrn,, r1) = B J ( kbrn,, r1 ) B Y ( kbrn,, r1) ρ ρ # + #. (3.55) (4) At r = r1, the dierence o acoustic pressure across the perorated duct yields and thus ( x, r ) ( x, r ) ρ ζ ( x, r ) p p = c # u (3.56) Ba 1 Bb 1 p Ba, r 1 # ζ pkb, r, n B J ( kbrn,, r1 ) B J ( k# r ) + B Y ( k# r ) = B J ( kbrn,, r 1). (3.57) 1, n 3, n Brn,, 1 4, n Brn,, 1 1, n 1 ik 4

58 The coeicients B 3,n and B 1,n in Eqs. (3.54) and (3.55) are expressed in terms o the coeicient B 4,n as, ( # Brnr ) ( # Brn ) Y k 1,, 2 B3, n = B4, n J 1 k,, r 2 (3.58) and ( # r ) ( # Brn ) k# ρ Y k B J Y B Brn,, 1 1 Brn,, 2 1, n = 1 ( k# Brn,, r1 ) + 1 ( k# Brn,, r1) 4, n k # Brn,, ρ J1( kbrn,, r1) J1 k,, r2 Substitution o Eqs. (3.58) and (3.59) into Eq. (3.57) gives the characteristic equation: ( kbrn,, r1) ( kbrn,, r1) ρ k# J Brn,, k Brn,, + i # ζ p # ρ kbrn,, J1 k Y J Y J = Y J Y J ( k# Brn,, r1 ) ( k# Brn,, r ) 2 ( k# Brn,, r2 ) ( k# Brn,, r1) ( k# Brn,, r1 ) ( k# Brn,, r2 ) ( k# Brn,, r2 ) ( k# Brn,, r1) Equation (3.6) can be expressed, using Eqs. (3.47) and (3.48), as ρ # ρ. (3.59). (3.6) k# k J ( k k,, Bxn,, r Bxn 1) k k Bxn,, + i # ζ 2 2 p 2 2 k k Bxn,, J1( k kbxn,, r1) k ( k# kbxn,, r1 ) ( k# kbxn,, r2 ) ( k# kbxn,, r2 ) ( k# kbxn,, r1) ( k# kbxn,, r1 ) ( k# kbxn,, r2 ) ( k# kbxn,, r2 ) k# 2 ( k r Bx,, n 1) Y J Y J = Y J Y J and the common axial wavenumber k Bxn,, can be obtained by solving Eq. (3.61). Equations (3.51) and (3.52) can be rewritten using Eqs. (3.53), (3.58), and (3.59) as ollows: Ba, n 1, n B, r, n 1, (3.61) ψ () r = B J ( k r); r r ; (3.62) 41

59 and ψ where ( ) kbrn,, # ρ J k 1 Brn,, r1 ( r) = B J ( k#,, r2 ) Y ( k#,, r) Y ( k#,, r2 ) J ( k#,, r) k# ρ D ; Bb, n 1, n 1 Brn Brn 1 Brn Brn Brn,, ( kbrn,, r2 ) ( kbrn,, r1 ) ( kbrn,, r2 ) ( kbrn,, r1) r1 r r2, (3.63) D= J # Y # Y # J #. (3.64) From the linearized momentum equation, the particle velocities in the axial direction are then obtained as and 1 + jkbxn,, x jkbxn,, x ( ) ( ) u r, x = k B e B e ψ ( r); r r (3.65) Ba, x B, x, n n n Ba, n 1 ρω n= 1 + jkbxn,, x jkbxn,, x ( ) ( ) u r, x = k B e B e ψ ( r); r r r # Bb, x B, x, n n n Bb, n 1 2 ρω n= The solution o unknown coeicients A + n, A n, B + n, B n, C + n and C n (3.66) can be determined rom the boundary conditions at the inlet ( x = ) and outlet ( x = L). Then the coeicients are used to calculate the transmission loss o the silencer. The boundary conditions at the interaces o the inlet and outlet are p = p r r x=, (3.67) ;, A B 1 u B ua; r r1, x= =, (3.68) ; r1 r r2, x= p = p r r x= L, (3.69) ;, C B 1 u B uc ; r r1, x= L =. (3.7) ; r1 r r2, x= L 42

60 In view o the expressions o the pressure and velocity as ininite series o unknown amplitudes in Eqs. (3.34), (3.38), (3.39), (3.4), (3.45), and (3.65), Eqs. (3.67) (3.7) give n= n= + + ( An + An ) ψ A, n( r) = ( Bn + Bn ) ψba, n( r) ; r r1 (3.71) n= n= + ( ) ψ ( ) k B B r + ( ) ψ ( r) k A A ; r r, (3.72) Axn,, n n An, 1 Bxn,, n n Ban, = n= ; r1 r r2 + + jkbxn,, L jkbxn,, L ( n n ) ψc n( r) ( n n ) ψba n( ) C + C = B e + B e r ; r r, (3.73),, 1 n= n= + jk ( ) ( ),,,,,,, ; BxnL jkbxnl kcxn Cn Cn ψ Cn r r + r 1 kbxn,, ( Bne Bne ) ψ Ban, ( r) = n=. (3.74) ; r1 r r2 In order to solve Eqs. (3.71) (3.74), the ininite series o unknown amplitudes need to be truncated to a suitable number, and then the same number o equations is solved or the amplitudes o the acoustic waves. An approach proposed by Xu et al. (1999) is adopted here to match the sound ield. Imposing the continuities o the integral o the pressure and axial velocity over discrete zones o the interaces at the inlet ( x = ) and outlet ( x = L), Eqs. (3.71) (3.74) yield the pressure and velocity matching conditions as N r N pm, r + + pm, ( An + An ) ψ A, n( r) dr = ( Bn + Bn ) ψba, n( r) dr, (3.75) n= n= 43

61 N N n= r + um, ( ) ψ ( r) k B B dr Bxn,, n n Ban, N r + um, kaxn,, ( An An ) ψ, ( ) ; An r dr rum, r, (3.76) n= = N r + 1 kaxn,, ( An An ) ψ, ( ) ; An r dr r1 rum, r2 n= r N pm, r + + jk, ( ) ( ) ( ),,,, pm BxnL jkbxnl n + n ψc, n r = n + n ψb, n( r), (3.77) C C dr B e B e dr n= n= N n= r + jk um, Bxn,, L jkbxn,, L ( ) ψ ( r) k B e B e dr Bxn,, n n Bn, N r + um, kcxn,, ( Cn Cn) ψ, ( ) ; Cn r dr rum, r1 n= = N r + 1 kcxn,, ( Cn Cn) ψ, ( ) ; Cn r dr r1 rum, r2 n= (3.78) with m r ; 1,, 1 pm, = r1 m= $ N + (3.79) N + 1 and m r ; 1,, 1 um, = r2 m= $ N +. (3.8) N + 1 In view o 4( N + 1) coeicients solved in Eqs. (3.75) (3.78) and the assumptions that: (1) the incoming wave is planar and A + termination is imposed at the exit by setting C n the unity, (2) an anechoic to zero, and (3) all transmitted waves in the outlet pipe are non-propagating modes except the irst mode C +, the transmission loss is determined as TL = 2log 1 C +. (3.81) 44

62 3.4 Three-dimensional boundary element method Dissipative silencers can be irst divided into unilled and illed acoustic domains. Impedance matrix or each domain is then obtained using boundary element method. These impedance matrices are then combined by applying the boundary conditions between the two domains. The matrix or illed acoustic domain can be evaluated by using complex acoustic characteristics o absorbing material, such as density and speed o sound. This section describes the BEM or perorated dissipative silencers depicted in Fig For the perorated main duct (domain 1), the three-dimensional wave equation with uniorm mean low in the x direction is expressed as ϕ 2 ϕ ϕ + kϕ 2ikMa Ma = 2 x x (3.82) or ϕ ϕ ϕ 2 2 k ikMa ϕ ϕ Ma ϕ =, (3.83) 2 x y z x x where ϕ is the acoustic velocity potential, Ma is the Mach number o the mean low. The acoustic pressure and particle velocity can be written respectively, in terms o the acoustic velocity potential, as ϕ p1 = ρ iωϕ + V x (3.84) and u1 = ϕ, (3.85) 45

63 d h # ρ, c#, k# p2, u2 ρ, c, k p, u 1 1 d1 d2 Domain 1 L Domain 2 Figure 3.6: Wave propagation in a perorated dissipative silencer in three-dimension. 46

64 where V is mean low velocity and the del operator. The adjoint o Eq. (3.83) is given by 2 * 2 * 2 * * ϕ ϕ ϕ 2 * ϕ k ϕ + 2ikMa x y z x 2 * 2 ϕ Ma + δ 2 ( x xi)( y yi)( z zi) = x (3.86) with its undamental solution (Green s unction) expressed as where ϕ ikma ik ( ) * 1 x x R 2 i 2 1 Ma 1 Ma = e e 2 4π 1 Ma R, (3.87) 1 R = x x 2 i + y yi + z zi 1 Ma ( ) ( ) ( ) (3.88) and δ is the Dirac delta unction. The weighted residual integral o Eq. (3.82) is written as 2 * 2 2 ϕ 2 ϕ ϕ ϕ + k ϕ 2ik dv Ma Ma = 2 x x, (3.89) V where V is the acoustic domain volume. Integrating the irst term by parts twice and the second term once, V ϕ ϕ ϕ + ϕ + x ϕ dv x * 2 * 2 * 2 * 2 k 2ikMa Ma 2 * * * ϕ ϕ *% 2 * ϕ ϕ % + ϕ % ϕ % 2ikMaϕϕ nx Ma ϕ ϕ nx dγ=, Γ n n x x (3.9) where Γ is the boundary surace o the acoustic domain, and n % is the normal vector to the surace and n % x is its component in the x direction. Rearranging Eq. (3.9), 47

65 where C ϕ % ϕ % d % ϕ + % + Γ= % n d Γ, (3.91) * * * 2 * 2 2 iϕi 2ik 1 Ma n Maϕ nx Ma n ϕ x ϕ x Γ n x Γ C i are the edge or corner point coeicients. Discretizing the boundary suraces into a number o elements Γ j, Eq. (3.91) yields ϕ % ϕ % % ϕ C + + dγ= % dγ(3.92) n N * * N * 2 * 2 2 iϕi % 2ik 1 Maϕ n Ma n x Ma n ϕ x ϕ x j= 1 Γ n x j= 1 j Γj which can be expressed in the matrix ormat as ϕ [ H ]{ ϕ} = [ G ]{ } n %. (3.93) Using Eqs. (3.84) and (3.85), Eq. (3.93) can be rewritten in terms o pressure and particle velocity as [ ]{ p } [ G]{ u } H =. (3.94) 1 1 The acoustic pressure and velocity on the boundary can be generally categorized as inlet, outlet, rigid wall, and perorations. Applying rigid boundary condition on the solid wall yields the impedance matrix o the main duct, in terms o inlet, outlet, and perorations, as i { p } O { } p { p } i { u } O { } p { u } 1 [ ] [ ] [ ] 1 TC11 TC12 TC13 p1 = [ TC21] [ TC22] [ TC23] u1, (3.95) [ TC31] [ TC32 ] [ TC33] 1 1 where superscripts i, o and p denote inlet, outlet, and perorations. Impedance matrix o the dissipative chamber can also be derived by the similar method as explained earlier with complex density and speed o sound. Zero mean low can be assumed in the dissipative chamber (domain 2), thus V = Ma= in Eqs. (3.83) 48

66 and (3.84). Applying rigid boundary condition at the wall, impedance matrix or the dissipative chamber becomes { p p p2} [ TD]{ u2} =. (3.96) The impedance matrices o inner duct ([ TC ]) and outer chamber ([ ]) TD may be coupled by the boundary conditions at the perorations interace. The acoustic velocity continuity at the interace yields, { p p u1 } { u2} =, (3.97) where the negative sign is assigned since u 1 and u 2 are normal outward acoustic velocities or each domain. Assuming the peroration thickness much smaller than the wavelength, pressure dierence at the perorations may be expressed as, { p p 1 } { p p 2} ρcζ { p p u1 } = #. (3.98) Combining Eqs. (3.95) (3.98) yields the impedance matrix [ TI ] o the silencer, deined by where i { p } O { p } i { u } O { u } 1 [ TI11] [ TI ] 12 1 =, (3.99) [ TI ] [ TI ] ( ) [ ] [ ] = [ ] + [ ] ζ p [ ] ([ ] + [ ]) TI TC TC # I TC TD TC, (3.1) ( ) [ ] [ ] = [ ] + [ ] ζ p [ ] ([ ] + [ ]) TI TC TC # I TC TD TC, (3.11) ( ) [ ] [ ] = [ ] + [ ] ζ p [ ] ([ ] + [ ]) TI TC TC # I TC TD TC, (3.12)

67 and [ ] I is the identity matrix. 1 ( ) [ ] [ ] = [ ] + [ ] ζ p [ ] ([ ] + [ ]) TI TC TC # I TC TD TC, (3.13) For a single-chamber silencer, the average o acoustic pressure and velocity rom Eq. (3.99) at nodes on the inlet and outlet planes determine irst the transer matrix o Eq. (3.26), ollowed by the transmission loss through Eq. (3.31). For a multiple-chamber silencer, the overall impedance matrix is obtained by connecting the impedance matrix o each chamber in terms o acoustic pressure and velocity continuities. The overall impedance matrix then yields the transer matrix and thus the transmission loss o the silencer in view o Eq. (3.31). 3.5 Physical and acoustic properties o absorbing materials There are a number o sound absorbing materials including glass iber, oam, and mineral wool. Understanding their physical and acoustic properties is essential in using such materials in silencers. In this study, glass iber is considered as an acoustic absorbing material and its physical and acoustical properties are introduced next Physical properties o absorbing material One o the important physical properties o ibrous materials is the low resistance (or resisitivity). The deinitions or and the standard test method to determine them are presented by ASTM C (1997) and Beranek (1988). The speciic low resistance is deined as 5

68 R p 1 = [mks rayl], (3.14) u where p and u is the pressure dierence and the linear low velocity across the homogeneous sample, respectively. The low resistivity is then deined as the speciic low resistance per unit thickness o the material as R R t p 1 = = [mks rayl/m] or [N s/m 4 ], (3.15) u t s s where t s is the sample thickness. The low resistance (hence resistivity) is mainly dependent on iber diameter, layer orientation, and material illing density. Nichols (1947) presented an empirical ormulation o low resistance or glass wool as R ( 1+ β ) ρ 1 γ β 2 ts r =, (3.16) where γ is a constant, ρ the illing or bulk density, r the iber radius, and the parameter β is a unction o the iber layer orientation. The low resistivity o iber perpendicular to low direction, in general, is higher than that o random illing. Thereore the value o b approaches.3 or ibers layered perpendicular to low direction and 1. or a random illing. The eect o iber orientation on the low resistivity was also theoretically and experimentally investigated by Tarnow (1997, 2). He suggests that the low resistivity or the iber layered perpendicular to low direction is twice the value o the one parallel to the low direction. Another physical property o absorbing material is the porosity, which is deined as the ratio o the volume o the voids or air (V a ) to the total volume (V t ) as 51

69 Y V V a = (3.17) t and can be calculated, using bulk or illing density ( ρ ) and the material density o ibers ( ρ m ), as Y ρ = 1. (3.18) ρ m According to Nichols (1947), Eq. (3.16) may be valid only or the materials with porosity higher than.9. In addition, thermal and chemical properties need to be considered according to the applications o materials. Such properties o various absorbing materials are described in Beranek (1988) and Hu (21) Acoustic properties o absorbing material The absorption o acoustic waves in ibrous material is mainly due to viscous and thermal dissipation, which may be expressed in terms o complex characteristic impedance and wavenumber. The imaginary part o the wavenumber, which is called the attenuation constant, accounts or the decay o the waves. Since the structure o absorbing material is rather complex, the acoustic properties are oten determined experimentally. For example, Delany and Bazley (197) presented empirical expressions or complex characteristic impedance and wavenumber o a ibrous material as ollows: Z# ρ c ( R ) i ( R ) = / / (3.19) and 52

70 k# k ( R ) i ( R ).7.59 = / /, (3.11) where denotes requency and R the resistivity which is in general experimentally determined. 3.6 Peroration impedance In Eqs. (3.15), (3.16), (3.56), and (3.98), the acoustic impedance o peroration relates the acoustic waves in the main duct and the outer chamber. The acoustic impedance is the ratio o the pressure dierence to the particle velocity. Two existing deinitions o impedance dier by the way particle velocity is speciied. One is the impedance o a hole using the particle velocity at the hole u h in Fig. 3.7, and the other is the impedance o a perorated plate applying the velocity in the duct u 1 or u 2. These deinitions and the relationship in between are introduced next. The speciic acoustic impedance o a hole is deined as p p Z = = R + jωl = R + jωρ! = R + jk ρ c!, (3.111) 1 2 h S S S e S e uh where p 1 and p 2 are the acoustic pressures beore and ater the perorated plate, respectively, as shown in Fig. 3.7, u h the particle velocity at the hole; R S the speciic resistance; L S the speciic inertance; and! e the eective length o the hole generally expressed as! = t +αd, (3.112) e w h where t w is the wall thickness, d h hole diameter, and α end correction coeicient. 53

71 P 1 P 2 u 1 u h u = u 2 1 A 1 porosity na h φ = A 1 h A h t w Figure 3.7: Wave propagation through a perorated plate in a duct. 54

72 The end correction coeicient α is dependent on hole geometry and the characteristics o medium contacting the hole. The non-dimensional acoustic impedance o a hole is then given by p p R + jk ρ c! ζ = = = R + jk t + d ( α ) 1 2 S e h w h ρcu h ρc, (3.113) where R is the non-dimensional resistance. The non-dimensional acoustic impedance o a perorated plate can be deined as ζ p p p ρ cu 1 2 =, (3.114) 1 where the particle velocity u 1 in the duct is related to u h through conservation o mass by na = = φ, (3.115) h h u1 uh uh A1 n h being the number o holes, A h the area o a hole, and φ the porosity. Equations (3.113) (3.115) yield the relationships between the impedance o a hole and a plate as ζ p ( α ) p p ζ R + jk t + d ρ cuφ φ φ 1 2 h w h = = =. (3.116) h While the impedance o a hole is representative o the acoustic characteristic o peroration itsel, the impedance o a plate is usually used in analyses, or example, transmission loss predictions. The impedance o a plate ζ p is usually obtained irst by experiments, and then the impedance o a hole ζ h are calculated using Eq. (3.116). 55

73 Equation Chapter 4 Section 1 CHAPTER 4 EXPERIMENTAL SETUPS An impedance tube setup was used to measure the acoustic properties o absorbing material, the acoustic impedance o perorations, and the transmission loss o silencers using the two-microphone method in the absence o mean low. In addition, two new setups were developed to acquire the acoustic impedance o perorations acing the absorbing material and the transmission loss o silencers in the presence o mean low. This chapter describes oregoing setups with and without mean low, along with the data reduction procedures to calculate the acoustic characteristics o ibrous material, perorations, and silencers. 4.1 Impedance tube setup in the absence o mean low Figure 4.1 shows the schematic o the impedance tube setup used in this study to obtain the characteristic impedance and wavenumber o the absorbing material, the acoustic impedance o perorations, and the transmission loss o silencers in the absence o mean low. Wide-band white random noise or single requency sine wave can be generated by a loudspeaker which is enclosed in a wooden box to reduce the radiation 56

74 Signal generator module B & K 317 Power ampliier Speaker Upstream Tube Multi-channel signal analyzer B & K 355 p p m1 m2 pm3 p m 4 Acoustic Element Downstream Tube Anechoic or relective termination Speaker housing Figure 4.1: The schematic o the impedance tube setup in the absence o mean low. 57

75 noise. The loudspeaker is driven by a signal generator module (B&K 317) and ampliied by a 1 W ampliier. A conical transition o 48 cm long is used to connect the speaker to an upstream PVC with 4.9 cm inner and 5.8 cm nominal outer diameters. The downstream tube is also made o PVC. A relective or anechoic termination at the end o the downstream tube is employed depending on the measurements. To accomplish the anechoic termination, a 4.5 m long PVC tube illed with absorbing iber is attached to the downstream o the setup. The relective termination may be achieved by either placing a rigid block inside the downstream tube or opening the tube. Each pair o ¼ inch condenser microphones (B&K 4135) are mounted at the upstream and downstream ends o the acoustic element. The signals rom these our microphones are processed by a multi-channel analysis system (B&K 355). Auto- and cross-spectra rom microphones are post-processed depending on the purpose o measurement. Ensemble averaging and microphone calibration are employed to reduce the experimental error. Averaging 1 ensembles is applied or each measurement to reduce uncertainties mainly due to signal noise. The mismatches o magnitude and phase among microphones are compensated using a microphone switching technique established by Chung and Blaser (198a). Next, the methodology is described to determine the characteristic impedance and wavenumber o the absorbing material, acoustic impedance o perorations, and transmission loss o silencers in the absence o mean low utilizing the impedance tube shown in Fig

76 4.1.1 Characteristic impedance and wavenumber o absorbing material A setup depicted in Fig. 4.2 in conjunction with the impedance tube was developed in this study to obtain the complex characteristic impedance and wavenumber o the absorbing material in the absence o mean low. A sample o absorbing material is mounted inside the main duct bounded by wire screens with a porosity higher than 75 % deining the locations a and b as shown in Fig Transer unctions among microphones and two measurements with distinctly dierent terminations are required in the present method to determine the complex characteristic impedance and wavenumber o the absorbing material in the requency domain. Anechoic and ully relective boundaries are applied as two dierent termination conditions. The transer matrix, which expresses the relationships between acoustic pressure and velocities at the suraces a and b o the sample, is given or the irst termination in ( ) terms o complex characteristic impedance & c sample (! ) as, ρ, wavenumber ( ) ( k#! ) i & ρcsin ( k#! ) ( k ) cos & ( k ) k #, and the length o the cos pa pb u = sin #! a i u, (4.1) #! b ρc and or the second termination (designated by prime) ( k#! ) i & ρcsin ( k#! ) ( k ) cos & ( k ) cos p a p b u = sin #! a i u. (4.2) #! b ρc 59

77 sound source p p m1 m2 p p m3! 1!! m 4 2! 3! 4 p u a a a ρc &, k# screens b p u b b two dierent termination conditions Figure 4.2: The schematic o the experimental setup or obtaining the acoustic properties o absorbing material in the absence o mean low. 6

78 The combination o Eqs. (4.1) and (4.2) yields cos( k#! ) & ρ sin ( k#! ) sin ( k#! ) pa pb ub i c u a pb u b = p a p b u b i & ua p c b u ρ b cos( k#! ) (4.3) or cos( k#! ) & ρ sin ( k#! ) sin ( k#! ) 1 pb ub pa T i c 1 pb u b u a T 2 = =. (4.4) i p b u b p a T 3 & ρc p b u b u a T4 cos( k#! ) In Eq. (4.4), acoustic pressure ( p, p, p, p ) and velocities ( u, u, u, u ) a b a b a b a b at the sample suraces a and b can be calculated rom the pressure measurements at our microphones ( p, p, p, p, p, p, p, p ) m1 m2 m3 m4 m1 m2 m3 m4. The transer matrix between the sample surace a and microphone #2 is given by, or the irst termination, ( k! ) iρ c sin ( k! ) 1 cos 2 2 pa pm2 1 u = i a sin ( k 2) cos ( k 2) u!! m2 ρc. (4.5) The acoustic velocity at microphone #2 ( u m2 ) can also be expressed in terms o pressure measurements at microphones #1 and #2 (, ) p p as m1 m2 u m2 = (! ) (! ) p p cos k m1 m2 1 iρ c sin k 1. (4.6) 61

79 Substitution o Eq. (4.6) into Eq. (4.5) yields 1 cos( k! 2) iρcsin ( k! 2) pm2 1 p 1 2cos sin ( 1) ( 2) cos m pm k!! (! 2) c iρ c sin ( k! ) pa u = i k k, (4.7) a ρ 1 which expresses the acoustic pressure and velocity at the sample surace a in terms o only pressure measurements at microphones #1 and #2. Similarly, the acoustic pressure and velocity at the sample surace a or the second termination is given by 1 cos( k! 2) iρcsin ( k! 2) p m2 1 p 1 2cos sin ( 1) ( 2 ) cos m p m k! k! ( k! 2) c iρ c sin ( k! ) p a = u i. (4.8) a ρ 1 The acoustic pressure and velocity at sample surace b can be similarly expressed in terms o pressure measurements at microphones #3 and #4 or the irst termination as, cos( k! 3) iρcsin ( k! 3) pm3 1 p cos sin ( ) ( 3) cos m k p ( 3)!!! c iρ c sin ( k! ) p =, (4.9) b 3 4 m4 u i k k b ρ 4 and or the second termination, cos( k! 3) iρcsin ( k! 3) p m3 1 p cos sin ( ) ( 3) cos m k p ( 3)!!! c iρ c sin ( k! ) p =. (4.1) b 3 4 m4 u i k k b ρ 4 The complex characteristic impedance and wavenumber o the sample can then be obtained rom Eq. (4.4) and Eqs. (4.7) (4.1). The complex characteristic impedance (& ρ c) is given, in terms o T 2 and T 3 o Eq. (4.4), by & T2 ρ c = (4.11) T 3 62

80 and the complex wavenumber, in terms o T 1 or T 4 o Eq. (4.4), by 1 k# = cos! 1 ( T ) 1 (4.12) or 1 k# = cos! 1 ( T ) 4. (4.13) Measured T 1 and T 4 o Eq. (4.4) are expected to be the same in ideal experiments. However, they may not be the case in experiments due, or example, to inhomogeneity in the illing and imperect sample suraces. Thus, the complex wavenumber may be obtained using an average o T 1 and T 4 as, 1 1 T cos 1 T4 k# + =! 2 (4.14) instead o either Eq. (4.12) or (4.13) Acoustic impedance o perorations Once the acoustic properties o the absorbing material are obtained, the acoustic impedance o perorations contacting the absorbent on one side can be also determined experimentally. Unlike the measurement o acoustic properties o the absorbing material, only a single experiment with one termination is necessary or the peroration impedance measurement. An anechoic termination is applied in this study to reduce the intererences at microphones by the relected wave rom the termination. The acoustic impedance o perorations shown in Fig. 4.3 is deined as 63

81 sound source pm1 p p m2 m3 p! 1!! m 4 2! 3! 4 p u c c p u d d ρc &, k# p u b b anechoic termination perorations Figure 4.3: The schematic o the experimental setup or obtaining the acoustic impedance o perorations in the absence o mean low. 64

82 ζ p p p ρ cu c d =, (4.15) c where p c and p d are the acoustic pressures at the peroration suraces contacting air and absorbing material, respectively, and u ( u ) c = is the acoustic velocity at the perorations. d The acoustic pressure and velocity at the peroration surace acing air are obtained in terms o pressures measured at microphones #1 and #2 by using Eqs. (4.5) and (4.6) as 1 cos( k! 2) iρcsin ( k! 2) pm2 1 p 1 2cos sin ( 1) ( 2) cos m pm k!! (! 2) c iρ c sin ( k! ) pc u = i k k. (4.16) c ρ 1 The acoustic pressure and velocity at the peroration surace in contact with iber are expressed in terms o a series o transer matrices and the pressures measured at microphones #3 and 4 as where m3 pd [ Tab ][ Tb 3] pm3cos( k 4) pm4 u =! d iρcsin ( k! 4) [ T ] ab p ( k#! ) & iρcsin ( k#! ) ( k ) cos & ( k ) is the transer matrix between suraces a and b, and, (4.17) cos = sin #! (4.18) i #! ρc [ T ] b3 ( k! ) iρ c sin ( k! ) cos 3 3 = 1 i sin ( k! 3) cos ( k! 3) ρc (4.19) 65

83 is the transer matrix between surace b and microphone #3. Finally, the peroration impedance can be determined rom Eqs. (4.15) (4.19) and sound pressure measurements at our microphones. In Eq. (4.15), the experimentally determined acoustic velocities at the perorations may not be exactly identical in experiments. Thus, instead o Eq. (4.15), the averaged value o the two velocities can be used or the calculation o peroration impedance as ζ p = pc pd 1 ρc uc u 2 ( + ) d. (4.2) Transmission loss Transmission loss is used to characterize the acoustic behavior o a silencer, since it is representative o the silencer itsel and independent o the source and termination impedances. The transmission loss is deined as the ratio o incident and transmitted power ( and ) L L o a silencer and is given under the assumption o anechoic wi wt termination and plane wave propagation inside upstream and downstream tubes by 2 Sin Ain 2 TL = L wi L = wt 1log 2 S A, (4.21) out 2 tr where S in and S out are the cross-sectional areas o inlet and outlet ducts, and A in and A tr are the magnitudes o incident and transmitted planar waves. When the inlet and outlet pipe diameters are the same, Eq. (4.21) is reduced to A TL = 2log in. (4.22) A tr 66

84 The magnitudes o incident and transmitted waves needed in Eq. (4.22) can be expressed in terms o relection coeicients at the inlet and outlet suraces o the silencer ( and ) R R as a b A in p 1 R m2 = (4.23) + a and A tr p 1 R m3 =. (4.24) + b As described in ASTM E15-98, the relection coeicients are given by R a H = e e ik! ik (! 1+! 2) e (4.25) ik! 1 H12 and R b H = e e ik! ik(! 3+! 4) e, (4.26) ik! 3 H34 where H ij is the transer unction between microphones i and j, which is deined as the ratio o cross-spectrum between microphones i and j ( G ij ) and auto-spectrum at microphone i ( G ii ): H ij G ij =. (4.27) G ii Combination o Eqs. (4.22) (4.27) yields the transmission loss o a silencer. 67

85 4.2 Impedance tube setup in the presence o mean low An impedance tube setup in the presence o mean low was designed and abricated as depicted in Fig The sound source consists o two 8 in speakers with 3 W each, which are connected to the main duct by conical sections. Numerical simulations have shown that the junction where the main duct and conical sections rom speakers merge needs to be small to reduce the internal resonances. Such resonances could interere with the transer o acoustical energy generated by the speakers to the main duct. In order to suppress the low noise generated at the inlet, a perorated dissipative silencer o cm is introduced upstream o the speakers. A perorated duct o 25% porosity is used and the outer chamber is illed with ibrous material. Anechoic termination is necessary to measure the transmission loss, which may also be beneicial or the peroration impedance measurements since it can reduce the wave relected rom the termination to the acoustic element under consideration. The approach in Fig. 4.1 that places the absorbing material directly within the duct is not suitable when low is considered. Thus, a termination with two steps o diverging conical sections and a cylindrical chamber is designed based on the numerical simulations o relection coeicients or various conigurations. While an exponential expansion would be ideal to accomplish lower relection coeicients, the high cost o manuacturing o such an expansion led to the design o a two step (4.5 and 13. o hal angle) conical sections and a cylindrical chamber o 3.2 cm diameter and 5.8 cm length. The conical sections and the downstream cylindrical chamber, which are separated rom the main duct by a perorated tube o 4.9 cm diameter and 25 % duct porosity, are illed with ibrous material o 15 kg/m 3 to suppress o the low noise and minimize the relection 68

86 Power ampliier Signal generator module B & K 317 Multi-channel signal analyzer B & K 355 low Perorated dissipative silencer p p m1 m2 pm3 p m 4 Upstream Tube Acoustic Element Downstream Tube Anechoic termination To low bench Speaker Figure 4.4: The schematic o the impedance tube setup in the presence o mean low. 69

87 coeicient. The downstream end is connected to a low bench which can pull a maximum air low rate o.63 kg/s at standard conditions with a pressure drop o up to 8.72 kpa. For transmission loss measurements, two pairs o ¼ inch condenser microphones (B&K 4135) are mounted at upstream and downstream ends o the acoustic element. While random noise can be used as a sound source or low mean low rates (thus low background noise), a single sine wave is applied at high low rate due to elevated background noise to ensure at least 1 db o signal-to-noise ratio as recommended by ASTM E This setup can be modiied or the measurements o acoustic impedance o perorations in the presence o grazing mean low, as described next Acoustic impedance o perorations The impedance tube setup o Fig. 4.4 is now modiied by introducing a sidebranch as shown in Fig. 4.5 to measure the acoustic impedance o perorations, particularly in contact with the absorbing material, and in the presence o grazing mean low. The circular sidebranch with an inner diameter o 4.9 cm is perpendicular to the rectangular main duct, with the circular perorated sample plate being placed at the interace o the two. The perorated plate is located 4 m rom the inlet o main duct to ensure a ully developed low at the plate, and its surace is lush with the inner wall o the main duct. The internal dimensions o the main duct is cm and the wall thickness is 1.27 cm. An! -long section o the sidebranch starting at the perorations is illed with the ibrous material. Two microphones, one in the main duct and the other 7

88 Power ampliier Signal generator module B & K 317 Multi-channel signal analyzer B & K 355 low Perorated dissipative silencer Perorated plate p, u a p m1 a Anechoic termination To low bench Speaker!! 5 & ρ ck, # p, u b b p m2 Figure 4.5: The schematic o the experimental setup or obtaining acoustic impedance o perorations in the presence o mean low. 71

89 at the end o sidebranch are mounted, as shown in Fig The pressure measurements p and p m2 ) rom these two microphones are used to calculate the acoustic impedance. ( m1 The acoustic impedance o the perorated plate in contact with the absorbing material depicted in Fig. 4.5 is deined as ζ p p p ρ cu m1 a =, (4.28) a where p a and u a are respectively the acoustic pressure and velocity at the peroration surace contacting the material, speciied by ( # ) & iρc ( # ) 1 sin & (! ) cos (! ) ( 5) ρ ( 5) sin ( ) cos( ) cos k! sin k! cos k! i c sin k! p p =, (4.29) a m2 1 u a i k k i k 5 k 5 u # #!! m2 ρc ρc where & ρ c and k # are known complex characteristic and wavenumber o the absorbing material. Equation (4.29) can be rewritten, with the rigid wall condition ( u m2 = ) as pa T11 T12 pm2 u = a T21 T 22 (4.3) Substitution o Eq. (4.3) into Eq. (4.28) gives the acoustic impedance o perorations as ζ p T m1 11 pm2 p =. (4.31) ρct 21 72

90 Equation Chapter 5 Section 1 CHAPTER 5 ACOUSTIC PROPERTIES OF ABSORBING MATERIAL The absorption o acoustic wave in ibrous material may be expressed in terms o complex characteristic impedance and wavenumber as described in Chapter 3. Thus, accurate estimation o such acoustic properties is essential in the use o theoretical approaches and in understanding how the acoustic properties aect transmission loss (to be covered later in Chapter 7.) In addition, the measurement o peroration impedance in contact with absorbing material (to be presented in Chapter 6) requires accurate acoustic properties o the material. Since the structure o absorbing material is rather complex, the acoustic properties are usually determined by experiments as in the present study, which obtained the characteristic impedance and wavenumber o a ibrous material using the impedance tube setup depicted in Figs. 4.1 and 4.2. Advantex strand glass iber developed by Owens Corning is used in the experiments. The texturization process separates ilament roving strands o iber glass into individual ilaments by turbulent air low (Silentex process). The average diameter o the individual ilaments in the roving strand is 24 µm. The degree to which the strands are separated into individual ilaments (texturization condition) aects the acoustic properties o the absorbing material. In 73

91 addition to the acoustic properties, high thermal and corrosion resistance o the Advantex glass iber may also be desirable or particularly exhaust systems. The physical and chemical properties o various absorptive materials and their relative durability in automotive silencer operations have been described by Beranek (1988) and Hu (21). This chapter irst introduces the selected existing empirical ormulations, given by Delany and Bazley (197) and Mechel (1992), o the complex characteristic impedance and wavenumber in terms o requency and low resistivity. Flow resistivity o the ibrous material used in this study is also provided based on the work by Nice (1999). Then, complex characteristic impedance and wavenumber or two dierent texturization conditions and two illing densities are experimentally determined in the present study and presented along with their comparisons to existing expressions. Finally, the estimated speed o sound in the absorbing material rom the measured complex wavenumber is briely discussed. 5.1 Selected existing ormulations The normalized empirical complex characteristic impedance Z # and wavenumber k # o Delany and Bazley (197) or ibrous materials are given in MKS units by Z# ρ c ( R ) i ( R ) = / / (5.1) k# k ( R ) i ( R ).7.59 = / /, (5.2) 74

92 where ρ is the air density, the requency, and R the low resistivity which is mainly dependent on iber diameter, layer orientation, and illing density. The data points used or the curve-its o Eqs. (5.1) and (5.2) are obtained using a two-cavity method within the range.1 < R < 1 in MKS units or the homogeneous material with its porosity (Y) close to one. Mechel (1992) also presented similar empirical ormulations o characteristic impedance and wavenumber or the glass iber in MKS units with ρ / R >.25 as Z# ρ c ( ρ R ) i ( ρ R ) = / / (5.3) k# k ( ρ R ) i ( ρ R ) = / /. (5.4) He also provided the coeicients or ρ / R <.25 and other material, such as mineral and basalt wool. Equations (5.1) (5.4) are unctions o requency and low resistivity. While the negative imaginary part o the wavenumber particularly ensures the dissipation o sound, the real part greater than unity indicates a lower speed o sound through the absorbing material. When the low resistivity approaches zero, both characteristic impedance and wavenumber approach the values o air. 75

93 5.2 Flow resistivity The low resistivity or the material used in the present study is experimentally determined by Owens Corning (Nice, 1999) using the standard test method described in the ASTM C (1997), and given in Table 5.1. The characteristic impedance and wavenumber as proposed by o Delany and Bazley (197) and Mechel (1992) can then be obtained by substituting the low resistivity in Table 5.1 into Eqs. (5.1) (5.4). Bulk density ρ (kg/m 3 ) Flow resistivity R (mks rayl/m) 1 4, ,378 Table 5.1 Flow resistivity measured by Owens Corning. 5.3 Characteristic impedance and wavenumber o ibrous material A transer unction method with two dierent termination conditions in the impedance tube setup depicted in Fig. 4.2 is developed in this study to acquire the complex characteristic impedance and wavenumber o the ibrous material in the absence o mean low. The mathematical process or this method is described in Section The precise measurements o distances between the sample suraces and microphones or among microphones are critical in the experiments. The accuracy within.1 mm or the materials with well deined suraces is recommended by ASTM (1998). 76

94 However, a precise physical measurement may not be possible due, or example, to the diiculty in inding the centers o the microphones with their diameter o 6.35 mm (1/4 inch). Thus, the distances,! 1 through! 4 and!, are acoustically estimated using quarter wave resonances as shown in Fig The estimated distances by this method in Fig. 4.2 are as ollows:! 1 =! 4 = 3.55 cm,! 2 =12.81 cm,! 3 = cm, and! =1.16 cm. The sample length! is chosen considering two aspects. When the sample length is too short, the impact o inhomogeneous illing may be magniied. On the other hand, too long sample may not transer suicient sound energy to the downstream microphones due to wave dissipation through the material, particularly at high requencies. An accurate estimation o speed o sound is also important or the measurements, which is calculated here by c = TC [m/s], (5.5) as suggested by ASTM (1998). All experiments were perormed at the room temperature, T C = 19±1 C. Existing ormulations o Eqs. (5.1) (5.4) were determined rom numerous experiments or various materials and illing densities, and then normalized in terms o low resistivity and requency. The present study rather ocuses on two illing densities ( ρ = 1 and 2 kg/m 3 ; approximate bounds in practical applications), and experimentally determines their acoustic properties. The properties obtained or both illing densities are then directly applied to the measurements o acoustic impedance o peroration in contact with ibrous material as described in Section 4.1.2, and also used or the transmission loss predictions or dissipative silencers. 77

95 p m c! = 4 min sound source rigid block Figure 5.1: Measurement o distance using quarter wave resonance; min is the requency at which the acoustic pressure o p m is minimum. 78

96 5.3.1 Fiber samples Absorbent samples, 1 cm long and 4.9 cm in diameter and with two dierent illing densities ρ = 1 and 2 kg/m 3, are placed in the test tube. Densities signiicantly lower than ρ = 1 kg/m 3 may cause the movement o iber inside silencers and higher than ρ = 2 kg/m 3 may be undesirable in terms o cost, weight, and in some cases acoustic attenuation, hence the justiication or the choices in this study. The material density o the iber ( ρ ) is kg/m 3 and thus the porosity o m the material (Y) or illing or bulk density ρ = 1 and 2 kg/m 3, with the ideal texturization being calculated as.962 and.924 using Eq. (3.18), respectively. The iber orientation and the texturization condition o samples need to be considered due to their inluence on the acoustic properties. Random illing is applied to achieve overall isotropic and homogeneous conditions. However, partially or locally non-isotropic or inhomogeneous raction may exist due, or example, to the hand illing o the iber into the test tube. Thus, experiments with ive dierent samples or each illing density are perormed to examine the eect o variations o illing conditions. The averaged values rom ive individual experiments are then used as the inal result. The texturization condition o the iber, the degree to which the strands are separated into individual ilaments, signiicantly aects the acoustic properties o the material. To illustrate the eect o texturization condition on the acoustic properties o the ibrous material, two set o samples with dierent texturization conditions were applied in the experiments. Since the texturization condition is diicult to deine 79

97 quantitavely or the given ibrous material, two qualitative conditions, good and normal, are considered in this study. The iber structures o the two texturization conditions or ρ = 1 kg/m 3 are shown in Fig While the good texturization reveals mostly individual ilaments, the normal texturization exhibits partially bonded strands. The ront suraces o the iber samples also need to be well deined in the experiments. The suraces o the samples perpendicular to the direction o wave propagation are identiied by the two screens. The screens have high porosity not to inluence the surace properties o the iber and suiciently high tension to hold the iber lat at the surace Good texturization Figures 5.3 and 5.4 show the individual measurements o ive dierent samples and their averaged values or the characteristic impedance and wavenumber at ρ = 1 kg/m 3 with good texturization condition. Although some scatter is displayed or the characteristic impedance, high repeatability is clearly observed, particularly or the wavenumber. Thus, the eect o locally non-isotropic or inhomogeneous illing is negligible in these experiments. The curve-itted complex characteristic impedance and wavenumber based on the averaged values o ive experiments or good texturization condition are then expressed by Z# ρ c ( ) i ( ) = (5.6) 8

98 (a) (b) Figure 5.2: Pictures o iber samples ( 1 ρ = kg/m 3 ); (a) good texturization and (b) normal texturization. 81

99 Real(Z/Z) #1 #2 #3 #4 #5 Average (a) Frequency (Hz) -.5 Imag(Z/Z) (b) #1 #2 #3 #4 #5 Average Frequency (Hz) Figure 5.3: Individual and averaged experimental results o characteristic impedance o 3 ibrous material with good texturization ( ρ = 1 kg/m ); (a) real part and (b) imaginary part. 82

100 2 1.5 #1 #2 #3 #4 #5 Average Real(K/K) Frequency (Hz) (a) -.5 Imag(K/K) -1 (b) #1 #2 #3 #4 #5 Average Frequency (Hz) Figure 5.4: Individual and averaged experimental results o wavenumber o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part. 83

101 k# = ( ) + i 38.39( ) k or ρ = 1 kg/m 3 and (5.7) Z# ρ c ( ) i ( ) = (5.8) k# k ( ) i ( ) = (5.9) or ρ = 2 kg/m 3. The unction it o the MATLAB is applied or all curve its in this chapter in the requency range o 1 32 Hz. At lower requencies, the anechoic termination condition may not be achieved and thus the two distinct termination conditions, which are necessary or accurate results, may not be applied. Figures show measured and curve-itted real and imaginary parts o characteristic impedance and wavenumber or both ρ = 1 [Eqs. (5.6) and (5.7)] and 2 [Eqs. (5.8) and (5.9)] kg/m 3 illings, along with comparisons to the ormulations o Delany and Bazley (197) and Mechel (1992). The low resistivity rom Table 5.1 is used or the existing ormulations [Eqs. (5.1) (5.4)]. The curve-its represent most o the data points well, particularly or the wavenumber at both illing densities. The experimental results obtained in this study show trends similar to those o Delany and Bazley, and Mechel, though some deviations exist. For example, the real part o the characteristic impedance o Mechel has lower magnitude compared to the results o Delany and Bazley, and the present study. The impact o each parameter on the transmission loss will be illustrated later in Chapter 7. 84

102 1.5 Experiment Curve it Delany & Bazley (R= 4896) Mechel (R= 4896) 1 Real(Z/Z) Frequency (Hz) (a) -.5 Imag(Z/Z) -1 (b) -1.5 Experiment Curve it Delany & Bazley (R= 4896) Mechel (R= 4896) Frequency (Hz) Figure 5.5: Characteristic impedance o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part. 85

103 2 1.5 Experiment Curve it Delany & Bazley (R= 4896) Mechel (R= 4896) Real(K/K) Frequency (Hz) (a) -.5 Imag(K/K) -1 (b) Frequency (Hz) Experiment Curve it Delany & Bazley (R= 4896) Mechel (R= 4896) Figure 5.6: Wavenumber o ibrous material with good texturization ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part. 86

104 2.5 2 Experiment Curve it Delany & Bazley (R= 17378) Mechel (R= 17378) Real(Z/Z) Frequency (Hz) (a) -.5 Imag(Z/Z) (b) Experiment Curve it Delany & Bazley (R= 17378) Mechel (R= 17378) Frequency (Hz) Figure 5.7: Characteristic impedance o ibrous material with good texturization ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part. 87

105 4 3 Experiment Curve it Delany & Bazley (R= 17378) Mechel (R= 17378) Real(K/K) Frequency (Hz) (a) -1 Imag(K/K) -2 (b) Frequency (Hz) Experiment Curve it Delany & Bazley (R= 17378) Mechel (R= 17378) Figure 5.8: Wavenumber o ibrous material with good texturization ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part. 88

106 5.3.3 Normal texturization The complex characteristic impedance and wavenumber or normal texturization with illing densities o ρ = 1 and 2 kg/m 3 are also experimentally determined using the same setup and methodology utilized or the good texturization. The curveitted properties are given by Z# ρ c k# k ( ) i ( ) = ( ) i ( ) = (5.1) (5.11) or ρ = 1 kg/m 3 and Z# ρ c ( ) i ( ) = (5.12) k# k ( ) i ( ) = (5.13) or ρ = 2 kg/m 3. Figures show the comparisons o acoustic properties o two dierent texturization conditions. The good texturization shows higher magnitudes o the properties compared to normal texturization. These dierences in acoustic properties o material due to the texturization conditions may cause substantial deviations in predicted transmission loss as will be illustrated later in Chapter 7. 89

107 1.5 Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) 1 Real(Z/Z) Frequency (Hz) (a) -.5 Imag(Z/Z) -1 (b) Frequency (Hz) Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Figure 5.9: Comparison o characteristic impedance o ibrous material or both texturization conditions ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part. 9

108 2 1.5 Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Real(K/K) Frequency (Hz) (a) -.5 Imag(K/K) -1 (b) Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Frequency (Hz) Figure 5.1: Comparison o wavenumber o ibrous material or both texturization conditions ( ρ = 1 kg/m 3 ); (a) real part and (b) imaginary part. 91

109 2 1.5 Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Real(Z/Z) Frequency (Hz) (a) -.5 Imag(Z/Z) -1 (b) Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Frequency (Hz) Figure 5.11: Comparison o characteristic impedance o ibrous material or both texturization conditions ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part. 92

110 3 2.5 Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) 2 Real(K/K) (a) Frequency (Hz) -.5 Imag(K/K) Experiment (Good texturization) Curve it (Good texturization) Experiment (Normal texturization) Curve it (Normal texturization) Frequency (Hz) (b) Figure 5.12: Comparison o wavenumber o ibrous material or both texturization conditions ( ρ = 2 kg/m 3 ); (a) real part and (b) imaginary part. 93

111 5.3.4 Speed o sound in the ibrous material The knowledge o speed o sound through the absorbing material may be useul, or example, in calculating the cut-o requencies o higher modes. The phase speed o sound c# ph through the absorbing material can be estimated rom the measured complex wavenumbers as, c# ph c =. (5.14) k# Real k Figure 5.13 shows the requency-dependent speed o sound estimated in this study using the real part o wavenumbers or both ρ = 1 and 2 kg/m 3 illing densities. The speed o sound or ρ = 2 kg/m 3 is lower than that o ρ = 1 kg/m 3. Scott (1946c) also presented the phase velocity in a Rock-wool in the range o 1 3 m/s or 1 32 Hz. Song and Bolton (2) also illustrated the low phase speed o sound through the absorbing material,.5.8 times the speed o sound in air. The lower phase speed o sound in the ibrous material suggests that that higher order modes can propagate at lower requency in absorbing material than in air. Thereore, the predictions o acoustic behavior by one-dimensional analysis or dissipative silencers may ail at requencies lower than or reactive silencers with similar dimensions. 94

112 35 3 air 1 kg/m^3 2 kg/m^3 Speed o Sound (m/s) Frequency (Hz) Figure 5.13: Estimated speed o sound through absorbing materials. 95

113 Equation Chapter 6 Section 6 CHAPTER 6 ACOUSTIC IMPEDANCE OF PERFORATIONS Acoustic impedance o perorations is essential or the prediction o both reactive and dissipative silencers, because perorations orm an interace, or example, connecting a main duct and an outer chamber. This impedance is dependent on the peroration geometry (porosity, shape and size o holes, wall thickness, and stagger pattern), the low ield (glazing and through low), sound pressure level, and the medium in contact with the peroration. The complexity o the peroration impedance conines the analytical approaches only to simple perorations. Thus, experimental methods are generally applied to obtain the impedance, such as those by Sullivan and Crocker (1978), Melling (1973), Dickey et al. (2, 21), and Lee and Ih (23). However, these results are limited to perorations in contact with air, learning the peroration impedance acing ibrous to be explored. The present study experimentally determines the acoustic impedance o perorations acing air-air (Fig. 6.1a) and air-ibrous material (Fig. 6.1b) in the absence and presence o mean low. First, the measurement methodology developed in this study (see Chapter 4) is applied to the perorations acing air-air in the absence o 96

114 (a) Fibrous material (b) Figure 6.1: Perorations in contact with air-air and air-ibrous material; (a) air-air and (b) air-ibrous material. 97

115 mean low, and the results are compared with the available literature. Then, peroration impedance in contact with air-ibrous material o ρ = 1 and 2 kg/m 3 are acquired. Two dierent texturization conditions, normal and good (See Chapter 5), are considered or each illing density. Finally, the eect o mean low on the peroration impedance acing air-ibrous material is presented. 6.1 Peroration samples Table 6.1 presents 11 samples o circular plates with dierent porosity φ, wall thickness t w, and hole diameter d h that are considered in this study to investigate the eect o such parameters on the peroration impedance. The dimensions and pictures o the sample plates are shown in Figs. 6.2 and 6.3. Peroration acing air-air with higher than 1 % porosity has not been, in general, considered in the literature since the eect o varying perorations at high porosities on the transmission loss is insigniicant. However, with the ibrous material, the transmission loss may be aected by the peroration impedance even at higher porosities. Thus, our duct porosities ( φ = 2.1, 8.4, 13.6, and 25.2 %) are considered here, deined by the area ratio o all holes to the plate. The samples are categorized into three groups; (1) group A is the base with d h =.249 cm and t w =.8 cm; (2) group B has the same hole diameter as group A and doubles the wall thickness t w =.16 ; and (3) group C has the same wall thickness as group A, while doubling the hole diameter d h =.498 cm. The numbers (1 4) ater the group identiication by (A C) in Table 6.1 indicate dierent duct porosities. The holes on the plates are distributed to achieve similar distances between the holes in the vertical 98

116 Sample No. Hole diameter, d h (cm) Wall thickness, t w (cm) Porosity, φ (%) A1 2.1 A A A B1 2.1 B B B C2 8.4 C C Table 6.1 Peroration samples. 99

117 (a) A1 and B1 (b) A2 and B2 (c) A3 and B3 (d) A4 and B4 (e) C2 () C3 (g) C4 Figure 6.2: Dimensions o perorated plate samples (unit: cm [inch]). 1

118 (a) A1 and B1 (b) A2 and B2 (c) A3 and B3 (d) A4 and B4 (e) C2 () C3 (g) C4 Figure 6.3: Pictures o perorated plate samples. 11

119 and horizontal directions, as shown in Fig For impedance measurements, the circular sample plates are installed in the impedance tube o 4.9 cm inner diameter, as shown in Fig The distance rom the tube wall to the nearest hole is chosen to be longer than the distance between holes in order to minimize the eect o the interaction between holes and wall. 6.2 Peroration impedance in the absence o mean low Acoustic impedance o a perorate plate can be expressed, using Eq. (3.116), as ζ p ( α ) R + ik t + d w h =, (6.1) φ where R is the resistance, k the wavenumber in air, and α an end correction coeicient o reactance which is associated with the interaction among holes. The present study experimentally determines R and α or the perorations acing air-air and air-ibrous material. Although the resistance has been considered as requency-dependent by some researchers, such as Dickey et al. (21) and Melling (1973), the present study assumes requency-independent resistance consistent with Sullivan and Crocker (1978) and Rao and Munjal (1986). The resulting single value o resistance provides an eective comparison o perorations without and with the ibrous material. Furthermore, the eect o variation o resistance on transmission loss will be shown later in Chapter 7 not to be signiicant. 12

120 6.2.1 Acoustic impedance o perorations in contact with air-air Available ormulations The empirical relationship o Sullivan and Crocker (1978) given by ζ p ( ).6 + ik t +.75d w h = (6.2) φ has been widely used or the peroration impedance with air-air contact. This expression was obtained using a peroration with φ = 4.2 %. In Eq (6.2), the coeicient.75 is associated with the end correction o the peroration. While the theoretical maximum value is.85 or a single hole, the coeicient may be smaller or multi-holes due to the interaction among holes. Melling (1973) graphically illustrated the interaction between two holes as shown in Fig He also introduced the Fok unction ψ (See also Rschevkin, 1963), which analytically considers the eect o interaction among holes on the end correction as ( ξ ) ψ = ξ ξ ξ.2287ξ +.315ξ.1641ξ ξ.1248ξ +.125ξ.985ξ (6.3) and d h ξ =, (6.4) 2 S / π where S is the zone area o each hole and d h is the hole diameter as shown in Fig The end correction coeicient or multi-holes can be expressed in term o the end correction or single hole and the Fok unction as.85/ψ ( ξ ). As the porosity increases, the Fok unction shown in Fig. 6.6 increases, hence the end correction coeicient α decreases. 13

121 Actual attached mass distribution Idealized total mass distribute (a) Interaction region (b) Figure 6.4: End correction o the perorations in contact with air-air (Melling, 1973); (a) single hole and (b) double holes. 14

122 S d h Figure 6.5: Zone area o a hole. 15

123 ψ' ξ Figure 6.6: Fok unction. 16

124 New Expressions The resistance R and the end correction coeicient α o the peroration samples [reer to expression Eq. (6.1)] with air-air contact are experimentally determined using the impedance tube setup depicted in Fig This setup is originally designed to measure the peroration impedance acing ibrous material (to be elaborated in Section 6.2.2), but it can also be utilized or the peroration in contact with air only. Five experiments or each sample are perormed and their averaged values are post-processed. The end correction coeicients are calculated rom the curve-itted reactance in the requency range o 1 32 Hz or the most samples. The only exception is A1 and B1 with φ = 2.1 % which are curve-itted in the range o 1 26 Hz due to the unstable signals at requencies higher than 26 Hz. The mean values o resistance in the same requency range are also obtained. Table 6.2 and Fig. 6.7 present the measured R and α or the peroration samples o Table 6.1, acing air-air. Figure 6.7 shows that the resistance R decreases as the porosity increases, especially rom φ = 2.1 to 8.4 %. The group B ( t =.16 cm) has higher resistance than group A ( t w =.8 cm), due to the doubled wall thickness. The resistance.6 in Eq. (6.2) obtained by Sullivan and Crocker (1978) or φ = 4.2 % is in the range between the resistance o φ = 2.1 and 8.4 % in Fig. 6.7(a). Figure 6.7(b) illustrates the eect o porosity, wall thickness, and hole diameter on α, thus reactance o the peroration impedance. As the porosity increases, the end correction coeicient α decreases because o the shorter distance between holes, hence w 17

125 Sample No. R α A A A A B B B B C C C Table 6.2 R and α o the peroration in contact with air-air in the absence o mean low. 18

126 Group A, Experiment Group B, Experiment Group C, Experiment Sullivan and Crocker Resistance R (a) Porosity End Correction Coeicient α Group A, Experiment Group B, Experiment Group C, Experiment Groups A and B, Fok unction Group C, Fok unction Sullivan and Crocker.1 (b) Porosity Figure 6.7: Comparison o measurement and Fok unction or the acoustic impedance o a hole in contact with air-air; (a) resistance and (b) end correction coeicient. 19

127 the stronger interaction among holes. The groups A and B have similar values o α indicating that the end correction coeicient is independent o wall thickness. The group C has higher values o α than the counterparts o group A since the larger hole diameter results in a longer distance among holes, thus reducing the interactions. The end corrections obtained in the present study are lower than.75 o Eq. (6.2) or all porosities. Figure 6.7(b) also shows that the predicted end correction coeicients using Fok unction, Eqs. (6.3) and (6.4), have a trend similar to the results o the present study, although some deviations in magnitude are observed Acoustic impedance o perorations in contact with air-ibrous material The acoustic impedance o perorations in contact with air-ibrous material is dierent than the one acing air-air due to the inluence o the material as shown in Fig The iber illing density and texturization condition in contact with the peroration are particularly important. Ingard (1954a) claims that the iber properties only within a distance o one hole diameter is signiicant. Thus the air gap between the peroration and the ibrous material may also alter the peroration impedance. The present study considers only the perorations directly in contact with the ibrous material without the air gap. 11

128 (a) Fibrous material (b) Figure 6.8: End correction o the perorations acing air-air and air-ibrous material; (a) air-air and (b) air-ibrous material. 111

129 Available ormulations Kirby and Cummings (1998) suggested a semi-empirical acoustic impedance # ζ p or perorated plates acing air-ibrous material by modiying the one that aces air-air as ζ p ' ζ p Z' ζφ p i.425kdh + i.425 kd # h ρc =. (6.5) φ In Eq. (6.5), hal o the end correction or a peroration acing air-air [the second term in Eq. (6.5)] is replaced by a third term to account or the eect o the ibrous material. However, the use o Eq. (6.5) or dissipative silencers lead to erroneous transmission loss predictions possibly due, according to Kirby (21), to an overestimated peroration impedance. Such overestimation may be attributed to the ignored interaction among holes. Selamet et al. (21) improved the concept by applying Eq. (6.2) to the perorations acing air-ibrous material, accounting or the eect o interaction among holes, as ollows: w h ρck # p =. (6.6) ζ.75 Z# k#.6 + ik t + 1+ d 2 φ In Eq. (6.6), the eect o absorbent is considered with the complex characteristic impedance Z # and wavenumber k # o the material, thus both real and imaginary parts o the impedance can be modiied in Eq. (6.2). Figure 6.9 shows the comparison o real and imaginary parts o # ζ p between Eqs. (6.2) and (6.6) or the peroration with φ = 8.4 %, d =.249 cm, and t =.9 cm. The acoustic properties o the ibrous material given h w 112

130 .4.35 Eq. (6.2) Eq. (6.6).3.25 Real (ζp) (a) Frequency (Hz) Eq. (6.2) Eq. (6.6) Imag (ζp) (b) Frequency (Hz) Figure 6.9: Comparison o Eqs. (6.2) and (6.6) or the acoustic impedance o the perorated plate with φ = 8.4 % and d =.249 cm; (a) real part and (b) imaginary part. h 113

131 by Eqs. (5.6) and (5.7) are used in this calculation. Figure 6.9 shows increased resistance and reactance due to Eq. (6.6) relative to Eq. (6.2). Equation (6.6) may be valid only near φ = 4.2 %, since it is obtained by modiying Eq. (6.2) which is based on this particular porosity. New expressions Acoustic impedance o perorations acing air-ibrous material is measured in the present study using the impedance tube presented in Fig The averaged results rom ive experiments with dierent ibrous material samples are utilized in the post-process. For example, the averaged values o the resistance and reactance or the sample plate A2 with and without ibrous material, along with their constant and curve its, are shown in Fig The relative range o variation in individual experiments or the resistance is broader than that or the end correction coeicient, particularly at high illing densities. Since the resistance and reactance orm the impedance o a plate given by Eq. (3.114), the resistance R and end correction coeicient α or a hole are obtained by using the relations given in Eqs. (3.113) and (3.116). The experimental results o R and α o the perorations (groups A, B, and C) acing an absorbent with ρ = 1 and 2 kg/m 3 as well as air only are shown in Figs , and Tables 6.3 and 6.4. In order to investigate the eect o texturization conditions on the peroration impedance, two texturization conditions, good and normal, are also applied in the experiments. Figures show that ibrous material substantially increases both resistance and end correction coeicients or all three groups. The resistance R o ρ = 2 kg/m 3 are 114

132 1.8 No Filling, Experiments No Filling, Constant-it 1 kg/m^3, Experiments 1 kg/m^3, Constant-it 2 kg/m^3, Experiments 2 kg/m^3, Constant-it Real (ζp) (a) Frequency (Hz) No Filling, Experiments No Filling, Curve-it 1 kg/m^3, Experiments 1 kg/m^3, Curve-it 2 kg/m^3, Experiments 2 kg/m^3, Curve-it Imag (ζp) 1.5 (b) Frequency (Hz) Figure 6.1: Acoustic impedance o the perorated plate A2; (a) real part and (b) imaginary part. 115

133 No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization Resistance R (a) Porosity End Correction Coeicient α No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization.1 (b) Porosity Figure 6.11: Acoustic impedance o a hole, Group A; (a) resistance and (b) end correction coeicient. 116

134 No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization Resistance R (a) Porosity End Correction Coeicient α No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization.1 (b) Porosity Figure 6.12: Acoustic impedance o a hole, Group B; (a) resistance and (b) end correction coeicient. 117

135 No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization Resistance R (a) Porosity.9.8 (b) End Correction Coeicient α No iber 1 kg/m^3, Good Texturization 1 kg/m^3, Normal Texturization 2 kg/m^3, Good Texturization 2 kg/m^3, Normal Texturization Porosity Figure 6.13: Acoustic impedance o a hole, Group C; (a) resistance and (b) end correction coeicient. 118

136 Sample No. ρ = 1 kg/m 3 ρ = 2 kg/m 3 R α R α A A A A B B B B C C C Table 6.3 R and α o the peroration in contact with air-ibrous material in the absence o mean low ( Good texturization). Sample No. ρ = 1 kg/m 3 ρ = 2 kg/m 3 R α R α A A A A B B B B C C C Table 6.4 R and α o the peroration in contact with air-ibrous material in the absence o mean low ( Normal texturization). 119

137 higher than those o lower illing density ρ = 1 kg/m 3 or all samples. However, the eect o dierent illing density on α diminishes or the samples A4 and B4 with good texturization conditions. For groups A and B, as the porosity increases, both R and α decrease or both illing densities except or the samples A4 and B4. Particularly, the R o φ = 25.2 % is higher than that o φ = 13.6 % or the samples A4 and B4. The relative increase o resistance by the illing material in group C ( d h =.498 cm) is considerably higher than in groups A and B ( d h =.249 cm). Unlike the resistance, the α o group C does not increase more rapidly by the illing material than groups A and B. Figures also show that while the impact o texturization condition o the ibrous material on R and α is insigniicant or ρ = 1 kg/m 3, the normal texturization or ρ = 2 kg/m 3 exhibits higher R and α or all samples particularly at high porosities, possibly due to the presence o more iber locally in contact with the perorations. Such trends with normal texturization should be treated, however, with caution, since there is a substantial experimental spread among the results rom corresponding samples. 6.3 Peroration impedance in the presence o grazing mean low The present study explores the eect o the grazing mean low on the peroration impedance in contact with the air-ibrous material using the setup described in Section and depicted in Fig Two low speeds o Ma =.5 and.1 at room temperature (19±1 C) and 1 atm are applied in the main duct. The low is ully developed at the perorate samples which are installed lush with the main duct. Two 12

138 dierent densities o ρ = 1 and 2 kg/m 3 with good texturization condition are illed in the side branch as shown in Fig Three iber samples are used or each peroration plate and their averaged values are presented here. The plane wave is assumed in the main duct and the cut-on requency o the irst higher order mode is calculated as 225 Hz based on the dimensions o the rectangular duct ( cm). Thus, the requency range o the experiment is limited up to 2 Hz. The resistance and end correction coeicient in the presence o mean low are presented next as normalized relative to their no-mean-low values Resistance o peroration impedance with mean low Figure 6.14 shows the measured resistance o peroration impedance in contact with air-ibrous material with ρ = 1 and 2 kg/m 3 or Ma =.5 and.1. The resistance with mean low is normalized relative to that without mean low, R Ma =., thus lower than unity, or example, indicates a decrease in resistance due to mean low. The resistance obtained in the experiments with φ = 25.2 % (samples A4, B4, and C4) exhibit negative values in the presence o mean low, thus those results are excluded rom the present study. Such negative values o resistance have also been measured by Cummings (1986) especially or the peroration with a large hole diameter, or example, o 1.38 cm. Cummings attributes the negative resistance to low-interactions, which was discussed by Howe (1979). Figure 6.14 shows that the normalized resistance or Ma =.5 is lower than unity, suggesting that the resistance decreases due to the mean low. Such decrease o resistance at low low rate has also been observed by Dickey et al. (21). 121

139 Resistance, R/R Ma= Group A, Ma=.5 Group A, Ma=.1 Group B, Ma=.5 Group B, Ma=.1 Group C, Ma=.5 Group C, Ma=.1.2 (a) Porosity Resistance, R/R Ma= Group A, Ma=.5 Group A, Ma=.1 Group B, Ma=.5 Group B, Ma=.1 Group C, Ma=.5 Group C, Ma=.1.2 (b) Porosity Figure 6.14: The measured resistance o a hole in contact with air-ibrous material in the presence o mean low; (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m

140 For Ma =.1, groups A and B have higher resistance than those without low, due possibly to the shedding and diusion o acoustically induced vorticity (Cummings, 1986; Dickey et al., 21). Noting that the present study has used a representative single value over the entire requency range o interest, the reduced resistance or group C may be attributed to the dominating contribution o lower values at high requencies. It is conceivable that such an averaging approach may not be suitable or a strong requencydependent behavior Reactance o peroration impedance with mean low Figure 6.15 shows the measured end correction coeicient or perorations in contact with air-ibrous material o ρ = 1 and 2 kg/m 3 or Ma =.5 and.1. The end correction coeicient with mean low is normalized relative to that without mean low, α Ma =., thus lower than unity indicates a decrease in the end correction due to mean low. The end correction coeicient or two illing densities show trends similar to each other in Fig. 6.15, even though the magnitude is dierent. Mean low is generally observed to reduce α, since the low tends to blow away the end correction, consistent with earlier studies (Dickey et al., 21). While the eect o mean low with Ma =.5 is not drastic (within 1 % variation rom α Ma =. ) or both ρ = 1 and 2 kg/m 3, the low with Ma =.1 shows a substantial eect or groups A and B, especially or a low porosity. For group C, the low inluence is not as substantial. Thus, the eect o mean 123

141 low needs to be considered particularly or a small hole diameter and with a high low speed. On the contrary, the peroration high porosity or large hole diameter may not be signiicantly inluenced by the mean low. 124

142 1 (a) End Correction Coeicient, α/αma= Porosity Group A, Ma=.5 Group A, Ma=.1 Group B, Ma=.5 Group B, Ma=.1 Group C, Ma=.5 Group C, Ma= (b) End Correction Coeicient, α/αma= Porosity Group A, Ma=.5 Group A, Ma=.1 Group B, Ma=.5 Group B, Ma=.1 Group C, Ma=.5 Group C, Ma=.1 Figure 6.15: The measured end correction coeicients o a hole in contact with airibrous material in the presence o mean low; (a) ρ = 1 kg/m 3 and (b) ρ = 2 kg/m

143 CHAPTER 7 TRANSMISSION LOSS OF SILENCERS The acoustic properties o ibrous material and the peroration impedance developed in the study are integrated into the predictions o silencers in the present chapter. Transmission loss is used to assess the acoustic perormance since it is independent o the input and termination impedances, thereore representative o the silencer itsel. The predictions are compared with the experimental results o reactive and dissipative silencers with various porosities and dierent hole diameters. For the dissipative silencers, dierent illing densities and texturization conditions are also applied or both experiments and predictions. The inluence o variation in each property o the ibrous material and the peroration impedance is illustrated using the BEM predictions. The eects o grazing mean low and the variation o internal geometry, such as bales and extended inlet/outlet, are also examined. Finally, the transmission loss o multi-chamber silencers are examined to illustrate the eect o connecting tube length between the two chambers. 126

144 7.1 Simple expansion chamber Beore the investigation o perorated silencers, the measured and predicted transmission loss o a simple concentric expansion chamber depicted in Fig. 7.1 are presented. The diameters o the chamber and the inlet/outlet ducts are 16.44, and 4.9 cm, respectively, and the length o the chamber is cm. These dimensions are adopted or all reactive and dissipative silencers investigated in the present study. The predicted transmission loss rom the BEM and 1D analytical approach are compared with the experiments in Fig The BEM predictions show good agreement with the experimental results, while the 1D analytical method deviates at requencies above 21 Hz due to its inability to account or the higher order modes o wave propagation. Thus, the 1D approach would not be appropriate at high requencies, even or a simple expansion chamber. A sample mesh o the expansion chamber used or the BEM predictions is displayed in Fig This mesh is generated by IDEAS and has 8 nodes on each element o 2 2 cm. This mesh type and size are applied or all BEM predictions o silencers hereater. 7.2 Perorated reactive silencers The acoustic characteristics o perorated reactive silencers are investigated next experimentally and numerically. These reactive silencers are then used in the ollowing section as baselines or the dissipative silencers. By comparing the transmission loss rom experiments and predictions, this investigation also provides an assessment o the acoustic impedance o perorations measured in the present study in contact with air only. 127

145 unit: cm Figure 7.1: The schematic o a reactive expansion chamber. 128

146 4 Experiment BEM 1-D Analytical Transmission Loss (db) Frequency (Hz) Figure 7.2: The transmission loss o the reactive expansion chamber without perorations; experiments vs. predictions. 129

147 Figure 7.3: Sample mesh o the expansion chamber using IDEAS; 2 2 cm mesh size. 13

148 Four perorated brass tubes with dierent porosities ( φ = 8.4 and 25.7 %) and hole diameters ( d h =.249 and.498 cm) are abricated and applied to the expansion chamber o Fig Such porosities and hole diameters are the same or close to those o the lat circular peroration samples used or the measurement o peroration impedance (see Table 6.1). The inner diameter o the tube is 4.9 cm and the wall thickness is.9 cm. The schematics and pictures o the abricated perorated tubes are shown in Figs. 7.4 and 7.5. The tubes are designed to have comparable distance between holes in axial and circumerential directions. For example, the perorated duct o Fig. 7.4(a) has.76 and.79 cm between the holes in two directions. The abricated reactive silencer with the perorated tube on the impedance tube setup is shown in Fig The transmission loss measured using the impedance tube setup are compared next with the predictions using the peroration impedance o available literature vs. the present study. Using the BEM predictions, the eect o real and imaginary parts o peroration impedance on the transmission loss is evaluated and presented in this section. 131

149 (a) (b) (c) (d) Figure 7.4: The schematics o perorated ducts; (a) φ = 8.4 %, d =.249 cm, (b) φ = 8.4 %, d =.498 cm, (c) φ = 25.7 %, d =.249 cm, and (d) φ = 25.7 %, d =.498 cm. h h h h 132

150 (a) (b) (c) (d) Figure 7.5: The pictures o perorated ducts; (a) φ = 8.4 %, d h =.249 cm, (b) φ = 8.4 %, d h =.498 cm, (c) φ = 25.7 %, d h =.249 cm, and (d) φ = 25.7 %, d h =.498 cm. 133

151 (a) perorations Unit: cm (b) Figure 7.6: A perorated reactive silencer; (a) picture and (b) schematic. 134

152 7.2.1 Comparisons o predictions and experiments The measured and predicted transmission loss o perorated reactive silencers with hole diameters ( d h =.249 and.498 cm) and φ = 8.4 % are presented in Figs. 7.7 and 7.8. The peroration impedance developed in the present study [Table 6.2 or Fig. 6.7] and Sullivan and Crocker s expression [Eq. (6.2)] are used or the predictions. The BEM predictions using the new expression o the present study show better agreement with experiments compared to those using Eq. (6.2). The results suggest that Sullivan and Crocker s expression (based on φ = 4.2 %) would not be appropriate or high porosities. Thus, Figs. 7.9 and 7.1 present the comparison o experiments with the predictions or two dierent hole diameters ( d h =.249 and.498 cm) and φ = 25.7 % using only the new expression o the present study. The BEM predictions show good agreement with the experiments or both hole diameters Eect o peroration geometry on the transmission loss Figure 7.11 shows the measured transmission loss or the simple expansion chamber, and the perorated reactive silencer with dierent porosities and hole diameters. The presence o perorated ducts o φ = 8.4 % substantially increases the transmission loss at the third attenuation dome, and reduces the width o the ourth dome along with a shit to lower requencies. The perorated duct with φ = 25.7 % also changes the transmission loss to some degree, but not as signiicantly as the one with φ = 8.4 %. 135

153 4 Experiment BEM, New expression BEM, Sullivan & Crocker's ormulation Transmission Loss (db) Frequency (Hz) Figure 7.7: The transmission loss o the perorated reactive silencer ( φ = 8.4 %, d =.249 cm). h 136

154 4 Experiment BEM, New expression BEM, Sullivan & Crocker's ormulation Transmission Loss (db) Frequency (Hz) Figure 7.8: The transmission loss o the perorated reactive silencer ( φ = 8.4 %, d =.498 cm). h 137

155 4 Experiment BEM, New expression Transmission Loss (db) Frequency (Hz) Figure 7.9: The transmission loss o the perorated reactive silencer ( φ = 25.7 %, d =.249 cm). h 138

156 4 Experiment BEM, New expression Transmission Loss (db) Frequency (Hz) Figure 7.1: The transmission loss o the perorated reactive silencer ( φ = 25.7 %, d =.498 cm). h 139

157 Transmission Loss (db) No Peroration porosity= 8.4 %, hole diameter=.249 cm porosity= 8.4 %, hole diameter=.498 cm porosity= 25.7 %, hole diameter=.249 cm porosity= 25.7 %, hole diameter=.498 cm Frequency (Hz) Figure 7.11: The eect o peroration impedance on the transmission loss o the perorated reactive silencers; experiments. 14

158 The variation o hole diameters ( d h =.249 and.498 cm) exhibits a substantial eect on the transmission loss or φ = 8.4 %, whereas this eect diminishes or φ = 25.7 %. Figures illustrate the importance o peroration impedance on the transmission loss predictions. Thus, the eect o resistance and end correction coeicient on the predicted transmission loss is shown in Fig or the silencer with φ = 8.4 % and d =.249 cm. While the variation in the end correction substantially h inluences the transmission loss at mid requencies, Hz, the eect o resistance is negligible except or high requencies. Thereore, accurate measurements o end correction are critical or improved transmission loss predictions. Unlike the similar hole spacing o Figs , Fig shows the experimental results o transmission loss with dierent hole spacing, as depicted in Fig. 7.14, while retaining the same porosity ( φ = 8.4 %) and the hole diameter ( d =.249 cm). The unequal spacing between holes [Fig (a)] tends to increase the peak attenuation compared to the one with nearly equal spacing [Fig. 7.14(b)] since the ormer may increase the end correction by reducing the interaction among holes. The considerable eect o the spacing illustrates the diiculty in obtaining a generalized peroration impedance or various porosity, and hole size and shape. h 141

159 Transmission Loss (db) Base.5R 1.5R Frequency (Hz) (a) Transmission Loss (db) Base.5alpha 1.5alpha Frequency (Hz) (b) Figure 7.12: The eect o peroration impedance on the transmission loss o the perorated reactive silencer ( φ = 8.4 %, d h =.249 cm), BEM predictions; (a) resistance and (b) end correction coeicient. 142

160 Transmission Loss (db) Experiment, Unequal space Experiment, Equal space BEM Prediction, Equal space Frequency (Hz) Figure 7.13: The eect o hole spacing on the transmission loss o a perorated reactive silencer ( φ = 8.4 %, d =.249 cm); experiments. h 143

161 unit; cm Circumerential direction Axial direction (a) Circumerential direction Axial direction (b) Figure 7.14: The distances between holes or two dierent perorated ducts ( φ = 8.4 %, d =.249 cm); (a) equal spacing and (b) unequal spacing. h 144

162 7.3 Single pass dissipative silencers The perorated reactive chambers o the preceding sections are now illed with ibrous material (Fig. 7.15) and the transmission loss is investigated experimentally and numerically. The measured transmission loss with dierent porosities, hole diameters, and illing densities are shown in Fig Filling the chamber with ibrous material dramatically changes the transmission loss rom a multi-dome behavior to a single peak with substantially higher attenuation, illustrating the eectiveness o iber. Figure 7.16 also demonstrates the dependence o the transmission loss o dissipative silencers on the porosity (φ ), iber illing density ( ρ ), and hole diameter ( d h ). Thus the understanding o the eect o such parameters on the transmission loss is important or the predictions o dissipative silencers. This section irst compares the experiments and the predictions using the complex characteristic impedance and wavenumber o ibrous material, and the peroration impedance presented in Chapters 5 and 6, respectively. Then the eect o variations in such parameters on the transmission loss is explored using the BEM predictions. The inluence o mean low and variation o internal geometry o dissipative silencers is also illustrated. 145

163 (a) Perorations Fibrous material 4.9 cm cm cm (b) Figure 7.15: The pictures and schematics o a dissipative perorated reactive silencer; (a) picture and (b) schematic. 146