Design of Reactive Rectangular Expansion Chambers for Broadband Acoustic Attenuation Performance based on Optimal Port Location

Transcription

1 Design of Reactive Rectangular Expansion Chambers for Broadband Acoustic Attenuation Performance based on Optimal Port Location Akhilesh Mimani and M. L. Munjal chool of Mechanical Engineering The University of Adelaide outh Australia 5005 Australia (Corresponding author) Facility of Research in Technical Acoustics Department of Mechanical Engineering Indian Institute of cience Bangalore India munjal@mecheng.iisc.ernet.in Abstract This paper analyses the Transmission Loss (TL) performance of rectangular expansion chambers having a ingle- Inlet and ingle-outlet (IO) or ingle-inlet and Double-Outlet (IDO) by means of a 3-D semi-analytical formulation based on the modal expansion and the Green s function approach. To this end the acoustic field inside the rigid-wall rectangular chamber is obtained as the orthogonal modal solution of 3-D homogeneous Helmholtz equation. The IO/IDO rectangular chamber system is characterised using the uniform piston-driven model in terms of the impedance [] matrix parameters (equivalently the acoustic pressure response function) obtained by computing the average of the 3-D Green s function over the surface area of the inlet/outlet ports modelled as rigid pistons oscillating with uniform velocity. The TL graphs computed using the 3-D semi-analytical formulation are found to be in an excellent agreement with that obtained from the 3-D FEA for IO test-cases thereby validating the technique presented here. A parametric investigation is conducted to study the effect of arbitrary location of inlet/outlet ports (on the chamber surface) on the TL graph resulting in the formulation of guidelines towards designing an axially short and long IO/IDO rectangular chambers exhibiting a broadband TL performance in terms of optimal angular and axial location of ports (on the appropriate acoustic pressure nodes). In addition characteristic features of the TL spectrum of a general reciprocal and conservative ingle-inlet and Multiple-Outlet (IMO) muffler system such as () the effect of interchanging the position of inlet and outlet ports and () analysis of peaks and troughs are proved analytically by means of the scattering [] matrix parameters. These features are corroborated through the analysis of TL graph (obtained using the 3-D semi-analytical formulation) of IDO rectangular chambers. Keywords Rectangular plenum chambers Reciprocal and conservative systems 3-D Green s function Uniform piston-driven model ingle inlet and single/multiple outlet mufflers Optimal port location.. INTRODUCTION Rectangular expansion chambers are popularly used as plenum chambers in Heating Ventilation and Air-Conditioning (HVAC) duct systems and other industrial air-handling applications. The evaluation and analysis of the acoustical attenuation performance of rigid-wall (unlined) and dissipative (lined) plenum chambers as well as the problem of break-out/break-in noise and sound transmission through the thin-wall rectangular chambers has been a subject matter of several investigations see Refs. [-8] [8-0] and [ ] respectively. In particular previous papers present different analytical (modal summation approach) and numerical modelling techniques [-6 3 4] as well as experimental procedures [7 8] for the determination of the four-pole parameters and thence the Transmission Loss (TL) performance of a ingle-inlet and ingle-outlet (IO) and ingle-inlet and Double-Outlet (IDO) rigid-wall rectangular plenum. The [T] matrix multiplication approach (also known as cascading [3 5]) based on the -D plane wave theory is the simplest analytical technique to model wave propagation in axially long chambers; however it is valid only in the low-frequency range i.e. up to the cut-on frequency of the first transverse mode. Owing to the simplicity of the orthogonal modes (given by the trigonometric functions) inside a rigid-wall rectangular cavity that models its 3-D acoustic field the evaluation of the four-pole parameters and the TL performance of rectangular chambers using a 3-D analytical/semi-analytical approach based on modal summation is popular [-6]. Munjal [] devised a simple semi-analytical point-collocation approach to obtain the four-pole (or the transfer [T] matrix) parameters and the TL characteristics of a simple expansion chamber type rectangular (and circular) chamber by incorporating the effect of higher-order transverse modes. This approach made use of the compatibility conditions for acoustic pressure and acoustic particle velocity at a number of equally spaced points (or nodes) on the end face on which the port (sudden area discontinuity) is located. This generated the required number of algebraic equations for evaluation of the modal A preliminary version of this work has been published in the proceedings of the International Congress on ound and Vibration (ICV- 0) held in Bangkok Thailand during 7 th to th July 03. A part of this work was carried out when the corresponding author was pursuing his doctorate at the Facility of Research in Technical Acoustics Department of Mechanical Engineering Indian Institute of cience Bangalore India.

2 amplitudes associated with the 3-D modal functions the total number of which is proportional to the area ratio. One of the main limiting factors of this approach was the convergence of the modal solution which depends on the location of the nodal points requiring a constant track of the nodal mesh so that the collocated points do not fall on the pressure nodes. Furthermore this approach was not suitable for fractional area ratio besides being rather inconvenient to use for chambers having a side port. Regardless of these limitations the point-collocation approach was used by Chu et al. [4] to analyse a ingle-inlet and ingle- Outlet (IO) reverse-flow rectangular chamber configuration and more recently by Wu et al. [5] wherein a IDO as well as a Double-Inlet and ingle-outlet (DIO) straight-flow rectangular chamber configurations were analysed. Ih [] modelled the inlet/outlet ports (of a square or circular cross-section) as uniform velocity pistons and characterised a -port (IO) rigid-wall rectangular chamber (in terms of the four-pole parameters) based on the orthogonal modal summation and superposing the velocity potentials due to individual pistons. It is noted that the analytical expressions for the velocity potentials were obtained by solving the homogeneous 3-D Helmholtz equation subject to inhomogeneous boundary conditions at the face on which the piston-source (port) was located. This 3-D analytical approach does not suffer from the aforementioned limitations of the point-collocation approach [] and enabled him to readily conduct parametric studies on the effect of port location; the TL performance of different IO configurations such as end-inlet/end-outlet chambers (straightflow and reverse-flow configurations) end-inlet/side-outlet chambers (cross-flow configuration) Helmholtz resonators as well as 90 and 80 bends were analysed. A conceptually same analytical approach was followed by Venkatesham et al. [3] to characterise IO rigid-wall rectangular chambers wherein the inlet/outlet ports (of square cross-section) was modelled as uniform velocity pistons. They obtained the velocity potentials due to individual pistons in terms of the 3-D Green s function or point-source response (for a rigid-wall rectangular cavity) which was integrated over the port area to yield the average response. The total acoustic pressure at the ports was obtained by superposing their individual average velocity potentials following which the transfer [T] matrix of the -port system was obtained. The individual velocity potentials are in essence the different [] matrix parameters for a - port system. imilar to Ih [] Venkatesham et al. [3] also conducted a parametric investigation to study the TL characteristics of different IO configurations of rectangular chamber. However it is noted that Refs. [] and [3] do not explicitly formulate guidelines that recommends optimal location of ports towards designing rectangular chambers (of axially short or long length) for broadband attenuation pattern. It is important to note that Venkatesham et al. [3] obtain analytical expression for velocity potentials by solving the inhomogeneous 3-D Helmholtz equation subject to homogeneous boundary conditions. From a mathematically formal point-of-view this is the main difference from the analytical formulation adopted by Ih [] wherein the homogeneous 3-D Helmholtz equation was solved subject to inhomogeneous boundary conditions at the port-chamber interface (as indicated earlier). Although these two modelling techniques yield identical result (TL graph) their mathematical equivalence had not yet been shown for rectangular chambers. The present work demonstrates that both these analytical methods of obtaining the acoustic pressure response functions or equivalently the [] matrix parameters for a rectangular chamber are mathematically equivalent see Appendix A. Kadam and Kim [7] presented an experimental procedure to obtain the four-pole parameters of a rigid-wall 3-D rectangular cavity (made of hard wood) from the measured pressure response functions. Their experimental results was found to be in a good agreement with the analytical results [6] thereby validating the experimental approach for the first time. Li and Hansen [8] compared the experimental results of the TL performance of lined and unlined plenum chambers against several different and well-known prediction models such as the low- and high-frequency model proposed by Cummings [9] to analyse lined plenum chambers and the 3-D analytical model of Ih [] to analyse unlined (rigid-wall) plenum chambers. Pan et al. [6] analysed the low-frequency acoustic response in a damped rectangular enclosure using the modified method of weighted residual. Their model was able to successfully predict the general features of the acoustic response of both a helicopter passenger cabin and a laboratory enclosure. Ali et al. [7] presented a review of the different Boundary-Element Method (BEM) techniques used for solving the acoustic eigenvalue problem in a rigid cavity and compared the results of different techniques for a rectangular cavity (for which the analytical expression for resonance frequencies is well-known). While the Refs. [ 3] consider the problem of characterising a -port (IO) rigid-wall rectangular plenum chamber by means of the 3-D analytical uniform piston-driven model and evaluate its TL performance this approach has not yet been used for characterising and evaluating the TL performance of a multi-port rigid-wall rectangular plenum having a single inlet and multiple (in general M) outlet ports that may be located on the end or side face. This problem is important because a typical HVAC system may have multiple (two or more) outlet ports. Furthermore it is well-known through the previous papers by Ih [] Eriksson [8 9] and the monograph by Munjal [3] that certain mode(s) of a rectangular duct may be suppressed by appropriately locating the port centre on their respective pressure node(s). Indeed this technique of optimal port location has been exploited for the case of elliptical and circular cylindrical chambers to obtain broadband attenuation behaviour [3 0-3]. Despite its fundamental nature and previous parametric studies on the effect of port location on TL performance [ 3] the problem of determining optimal port location (with a view to obtain broadband attenuation) for the case of a IO and in general a ingle-inlet and Multiple-Outlet (IMO) rigid-wall rectangular plenum configuration has not been considered. In fact given the recent advancements presenting novel algorithms towards optimising a specific muffler configuration/geometry such as sub-chamber optimisation approach [4] the simulated annealing method for mufflers with perforated inlet extensions [5] and the differential evolution method for mufflers composed of multiple rectangular fin-shaped chambers [4] it becomes

3 imperative to solve the outstanding problem of optimal port locations for the relatively simpler IO/IMO rectangular plenum; these optimised port locations are intended to form the basis of design guidelines for such chambers. In parallel with the aforementioned studies the TL performance of IDO circular cylindrical chambers have been investigated using the 3-D analytical mode-matching approach [6] and the uniform piston-driven model via the 3-D Green s function method [7]. In a previous work [8] the authors had presented a general algorithm to characterise a network of multi-port elements (using the impedance [] matrix) and developed novel expressions for determining the TL (in terms of scattering [] matrix) for a general Multiple-Inlet and Multiple-Outlet (MIMO) muffler system. Their investigation was however confined to validate the proposed algorithm by comparing the TL graphs of arbitrary multi-port network configurations computed using the -D axial plane wave theory against the 3-D Finite-Element Analysis (FEA) results. Yang and Ji [9] also investigated the problem of characterising a network of multi-port (-port and 3-port) elements based on the [] matrix by using different numerical techniques such as point-collocation BEM and numerical mode-matching to compute the individual [] matrices of sub-systems. ubsequently the TL performance of the overall IDO muffler system was obtained. In these papers the characteristic features of a general IMO muffler system satisfying acoustic reciprocity and energy conservation were not explicitly derived or proved although Mimani and Munjal [3] later showed that interchanging the location of inlet and outlet ports of a IDO elliptical cylindrical chamber completely alters its TL characteristics. In light of the background provided above the objective of this work is therefore to first present a 3-D analytical approach based on the modal summation technique and the uniform piston-driven model for characterising a multi-port rectangular chamber having arbitrary number of ports that may be located on the end or side face. By virtue of this analytical formulation parametric studies are conducted with a view to examine the effect of axial and angular location of the inlet/outlet ports on the TL spectrum and subsequently identify specific IO/IDO rectangular chamber configurations (of short and long lengths) that yield a broadband attenuation performance based on optimal port location. Furthermore this paper analytically proves certain characteristic properties of a general reciprocal and conservative IMO muffler system which facilitate a better understanding or analysis of its TL spectrum. The parametric studies enable one to computationally corroborate these properties for IDO rectangular chambers. This paper is organised as follows. ection presents the theoretical formulation based on the uniform piston-driven model and the modal summation approach (via the 3-D Green s function) for characterising a multi-port rigid-wall rectangular chamber having ports located arbitrarily on the chamber surface. Towards the end of this section an expression for computing the TL performance of a general IMO/IO system is obtained. ection 3 analytically shows that the TL performance of a reciprocal and/or conservative IMO muffler is significantly altered by interchanging the location of inlet and outlet ports. This section also shows that the peaks and troughs in the TL graph of a conservative IMO and IO system can be readily predicted or analysed in terms of the scattering [] matrix parameters whilst for the special case of a IO system the [] matrix parameters are better suited for this purpose. ection 4 presents the TL performance of different IO and IDO rectangular chamber configurations obtained using the 3-D semi-analytical approach validates the method by comparing the results with 3-D FEA prediction (and a previous result []) and carries out parametric studies to analyse the effect of axial and angular location of inlet/outlet ports on the broadband acoustic attenuation range. The paper is concluded in ection 5 wherein different IO and IDO rectangular chamber configurations exhibiting a broadband TL performance are mentioned.. THEORETICAL FORMULATION A multi-port chamber having N M ports denoted by P P P N + M is characterised by means of an impedance [] matrix representation shown hereunder [3 8]. T p p p p v v v v () N N N M N M x N M N N N M T where N M x N M N N N M N NN N N N N M N N N N N N N M N M N M N N M N N M N M. () The acoustic pressure (Pa) and mass-velocity kg m 3 at the ports P P P N+M of the N+M port system are denoted by p... pn pn... pn M and v... vn vn... vn M respectively. It is noted that direction of the massvelocity is considered positive looking into the system and a harmonic time-dependence is assumed so that 3 p jωt i pie and

4 v j ω t i vie i = N+M. Furthermore j and is the excitation angular frequency radian s and is given by f where f is the frequency in Hz. In the ensuing subsection a multi-port rectangular expansion chamber having N M ports is characterised using the uniform piston-driven model via the 3-D Green s function approach in terms of the [] matrix representation. It is noted that the walls of the plenum chamber are considered rigid (with no absorptive/dissipative linings) therefore a conservative system is being considered. Furthermore a stationary medium i.e. a zero mean flow is assumed implying that the rectangular expansion chamber satisfies the principle of acoustic reciprocity [30].. Characterisation of a multi-port Rectangular chamber The acoustic field in a rigid-wall rectangular chamber is obtained by solution of the homogeneous 3-D Helmholtz equation in Cartesian coordinates x y z shown as follows [-7 6 3]. k0 p 0 x y z ( ) where p p( x y z) represents the acoustic pressure field k0 c0 is the excitation wavenumber (m ) and c 0 denotes the sound speed (ms ). The application of homogeneous rigid-wall boundary condition on the chamber faces yields the orthogonal modal solution given by [-7 6 3] l = m = n = n m l p x y z Anml cos x cos y cos z l 0 m 0 n = 0 B H L (3) where B H and L denote the breadth height and length of the rectangular chamber respectively A nml denotes the modal coefficient corresponding to the ( n m l ) mode. It is noted that the resonance frequency of the ( n m l ) mode of the rectangular chamber is given by [ 3 6] f nml 0. c n m l B H L The acoustic pressure response function for characterising the rectangular chamber is obtained by modal solution of the inhomogeneous 3-D Helmholtz equation (subject to homogeneous rigid-wall boundary conditions) shown as follows [3 6 3] k p j ρ Q x x y y z z δ δ δ (5) -3 where 0 is the ambient density kgm of the air Q 0 is the volume flow-rate m 3 s - due to the source port that is modelled as point-source δ denotes the Dirac Delta function (in Cartesian co-ordinates) whilst x y z denotes the location of the centre of the source port. The orthogonal modal solution of Eq. () expressing the 3-D acoustic field inside the rigid-wall rectangular chamber given by Eq. (3) is inserted into Eq. (5) whereby the modal coefficients A nml are evaluated using mode orthogonality of circular functions; these are back-substituted in the modal solution to obtain the 3-D Green s x y z function G xr yr zr x y z or the acoustic pressure response due to a point-source port located arbitrarily on the chamber surface [3 3]. jk c n m l R R R R R R p x y z x y z G x y z x y z ρ Q ρ Q m yr n xr l zr m y n x l z cos cos cos cos cos cos H B L H B L jt e B H L 0 0 n0... m0... l 0... n m l N n m l k0 (4) (6) where xb yh zl n x m y l z Nn m l cos dx cos dy cos dz ε nmlbhl B H L x0 y0 z0 (7) 4

5 are product of the integrals of the square of a particular set of mode shape functions ( n m l ) integrated over the volume of rectangular chamber whilst εnml for m n l 0 m n l m n l m n l m n l m n l m n l m n l 0.5 for for for and xr yr zr denotes the co-ordinates of the centre of the receiver port R. It is observed from Eq. (6) that on interchanging the location of source and receiver ports the Green s function remains unaltered thereby indicating that the 3-D acoustic field inside the rectangular chamber with zero mean flow satisfies the principle of acoustic reciprocity [30]. Furthermore it is also noted that the Green s function response is purely imaginary which analytically shows that the rigid-wall rectangular chamber (i.e. the absence of a dissipative lining) having zero mean flow is both reciprocal and conservative [30]. The point-source modelling is the least accurate method for obtaining the acoustic pressure response of a cavity due to excitation at a port (sudden-area discontinuity) because the ports of finite cross-section area are approximated as points on the chamber surface [3 33]. In fact the point-source model is known to have convergence issues at the source port; while the modal summation series in the cross-impedance parameters i j (i.e. at the far-field of the source port) exhibit good converge when computed using the point-source model the modal summation series in the self-impedance parameters ii converge very slowly (i.e. an extraordinarily large number of modal terms are required) when computed using the pointsource model see hou and Kim [6]. Therefore in view of this limitation the more realistic and accurate uniform pistondriven model is used to compute all the acoustic pressure response functions i.e. both self- and cross-impedance parameters (using the 3-D Green s function given by Eq. (6)) by modelling the source port as rigid oscillating piston having uniform velocity distribution which is equal to the normal acoustic particle velocity in the chamber over the cross-sectional area of the port. Furthermore the normal acoustic particle velocity in the chamber over the annular cross-section of the rigid face (on which the source port is located) is set to zero whilst the acoustic pressure field in the chamber and port are taken equal over the port-chamber interface. Therefore the uniform piston-driven model assumes planar wave propagation in the ports from the port-chamber interface i.e. it neglects the higher-order transverse evanescent modes in the ports. This assumption is justified because the diameter of ports is significantly smaller than the dimensions of the rectangular chamber therefore the transverse modes propagate (or become cut-on) only at high frequencies whilst throughout the frequency region of interest (for an automotive muffler) these modes are evanescent that decay within a short distance approximately less than two times the port diameter [3]. Incidentally hou and Kim [6] termed the uniform piston-driven model as the surface-source model and showed that this method is equivalent to computing the average of the Green s function over the source port area and it guarantees overcoming the slow convergence issue (of the ii parameters) arising out of the point-source representation. Nevertheless it is mentioned here that the accuracy of modelling the acoustic fields at the sudden-area discontinuity can be further improved by also considering the higher-order transverse modes in the ports by using the 3-D analytical modematching approach [34 35] which is the most accurate of all modelling techniques. However it is algebraically much more tedious as compared to the uniform piston-driven model and yields nearly identical results throughout the frequency range of interest. Therefore in view of its relative simplicity and comparable accuracy use of the 3-D uniform piston-driven model is popular [ ] and is indeed used here. The mathematical rendition of the uniform piston-driven model assuming the source port as rigid oscillating piston that is located on the X-Y face is shown hereunder [ ]. where ij k0 p x y z j ρ0u 0 f x y z z (8a-d) (9) f x y 0 X-Y (0a b) In Eq. (0a) and (b) and X-Y denote the cross-sectional area of the source port and the X-Y face of the rectangular chamber respectively whilst the symbol X-Y denotes the annular area (excluding the port). On making use of Eqs. (5) (6) (9) and (0) it can be shown that the acoustic pressure response based on the uniform piston-driven is given by p x y z x y z G x y z x y z dx d y. () R R R R R R 5

6 Equation () is further integrated over the cross-sectional area of the receiver port R and divided by it to obtain the average response whereby the matrix parameter R is obtained. The R parameter for different cases of location of receiver port on the X-Y Y- or -X face is given by R R R R R R p x y z x y z G x y z x y z R dxd y dxrd yr Port R located on X-Y plane ρ0q0 R R ρ 0Q0 G x y z x y z R ρ0q R 0 R R R dxd y dyrd zr Port R located on Y- plane G xr yr zr x y z dxd y dzrd xr Port R located on -X plan R R ρ 0Q0 e (a-c) where R denotes the cross-sectional area of the receiver port. The self-impedance parameter the X-Y face) is obtained by replacing the suffix R by in Eq. () to obtain (for source port located on d d d d. (3) p x y z x y z G x y z x y z x y x y ρ0q0 ρ0q0 The [] matrix parameters characterising a rectangular chamber having N M ports are obtained by using Eqs. () and (3). It is noted that Venkatesham et al. [3] characterised a -port rectangular chamber following the same approach.. Integration of the Green s function over the port cross-sectional area The integration of the 3-D Green s function over the cross-sectional area of the source/receiver ports shown in Eq. () is carried out using numerically using the impson s three-eighths rule [36] for ports having a circular cross-section although Ih [] had previously obtained analytical expressions for these integrals in terms of the first-order ordinary Bessel function of the first kind. It is for this reason that the present modelling approach is termed as semi-analytical. For a port of diameter d or equivalently radius r centred at ( x y ) on the X-Y face the integral expression is given by y0 r x x x 0 0 r x0 r n x m y n x 0 x0 r B H x0 r y B 0 r x x 0 cos cos dy d x r x x cos d x m 0 x0 r H m y m r 0 0 x x cos sin n x cos d x m 0. m H x0 r H B (4) It is noted for the plane wave mode i.e. m n 0 Eq. (4) is evaluates to r the port cross-sectional area. imilarly for ports located on the Y- and X- planes the integral expressions are given by cos cos d d and cos cos d d (5 6) y0 r z z 0 x0 r z z z 0 0 r z0 r l z m y l z n x y z x z z0 r L H z0 r L B y0 r z z 0 x0 r z z 0 respectively. The integrals for ports of square cross-section can be readily evaluated through closed-form analytical expressions see Refs. [ 3 3]..3 Influence of the port location on the mode propagation/suppression The location of ports can significantly influence the excitation or suppression of the higher-order transverse modes (along the x and y directions) or axial modes (along the z direction) of the rectangular chamber [8 9] which directly influences the [] matrix parameters. It is studied analytically in this subsection using the point-source model or the Green s function given by Eq. (6). To this end the test-case of a port centred at x 0.5 B y 0.5H on the X-Y end face is considered wherein it is observed that 6

7 m n cos 0 m cos 0 n (7 8) implying that a centred location of the port on the end face results in the modal coefficient associated with the corresponding odd mode amount to zero thence not allowing the odd modes to propagate although the frequency may be greater than the cuton frequency of that mode. It is for this reason that for a concentric rectangular expansion chamber only the even modes propagate which explains the repetitive nature of the domes and troughs for a frequency range beyond the cut-on frequency of the first order higher mode. The test-case of a port located at x 0.5 B or 0.75 B y 0.5 H or 0.75H on the X-Y end face is now considered wherein it is observed that m 3m n 3n cos cos 0 m cos cos 0 n (9 0) Equations (9) and (0) signify that even modes given by m = n = 6 0 do not propagate if the port is centred at the aforementioned co-ordinates. imilar comments on the suppression of even or odd axial modes also hold for a side port located on the Y- or X- faces. These results/comments on the influence of port location on propagation/suppression of certain rigid-wall modes will be used in the ensuing section to explain the nature of TL graph in particular the broadband attenuation characteristics of different -port (IO) rectangular chamber configurations..4 Computation of TL of a IMO/IO muffler system An expression for the TL performance for a muffler system having a single inlet (say port P) and multiple outlet (ports P P 3 P M + ) is obtained in terms of the matrix or matrix parameters. To this end the matrix is expressed in terms of the matrix of a M port muffler system shown as follows [ 3 8]. Y Ι Y I I Y Ι. () M M In Eq. () the number of inlet ports N = (port P) I is the identity matrix and [Y] is a diagonal matrix consisting of the characteristic impedances of the ports. It is assumed that the ports are acoustically compact i.e. their diameter is much smaller compared to the shortest wavelength of interest (to automotive muffler designer) so that only planar waves propagate. Anechoic termination is imposed at M outlet ports (P P 3 P M+ ) to yield B A B A B A... B A () 3 3 M M following which the TL expression is obtained for a IMO muffler system given by Y TL 0log 0. 3 M... Y Y3 Y M (3) In Eq. (3) A is the incident wave amplitude at port P whilst B B BM is the vector of reflected wave amplitudes at ports P P P3 P M+ respectively propagating downstream. It is noted that at ports P P3 P M+ the incident wave amplitudes A A3 AM 0 respectively due to the imposition of anechoic conditions. The TL for the IMO system for the different cases of inlet port located at port P P 3 P M + is similarly given by Y Y M TL 0log 0... TL 0log 0 M 3 M M M M M Y Y3 Y M Y Y Y M (4 5) 7

8 respectively. The TL for a IO muffler system (with ports P and P taken as the inlet and outlet respectively) is obtained by substituting M = in Eq. (3). On expressing the parameter in terms of the matrix parameters followed by further algebraic manipulation one obtains [] Y Y Y TL 0log0 0log 0. Y 4Y Y (6) It is noted that the Abom [37] obtained an identical expression for TL for a IO muffler system in terms of the [] matrix parameter wherein the convective effects of mean flow at the ports were also taken into account. 3. TRANMIION LO PROPERTIE OF A RECIPROCAL AND CONERVATIVE IMO MUFFLER YTEM 3. Interchanging the positions of inlet and outlet ports of a reciprocal IMO system An analytical proof is outlined in this subsection to show that the TL performance of a reciprocal IMO muffler system is significantly altered by interchanging the position of inlet and outlet ports. To this end the matrix of a M port muffler system is expressed in terms of the matrix shown as follows [8]. Ι I Y. M M I I Y (7) T For a muffler system satisfying the principle of acoustic reciprocity [30] condition in Eq. (7) one obtains the following relation between the matrix parameters. T i Y Y or i j M From Eq. (8) one obtains the relation Y Y ij and on use of this reciprocity Y ji.... (8) Y j that is substituted in TL to yield Y TL 0log 0. Y Y 3... M Y Y3Y YM Y It is noted that in general since the parameters 3 3 (M+) (M+) the TL TL for a general reciprocal IMO muffler system. Equation (8) is similarly used to simplify the expression for TL j where < j (M+) to obtain Y TL j 0 log 0. j Y j Y j Y j Y j j j j j j M j Y j YY Y jy Y jy YM Y (9) (30) Based on a similar reasoning it can be readily shown that TL TL j. Therefore it is concluded that TL i for a reciprocal IMO muffler system corresponding to excitation at the i th port (port P i being the inlet port) would in general be different from TL j of the same IMO system corresponding to excitation at the j th port (port P j being the inlet port) i.e. TL TL thereby proving that interchanging the inlet and outlet port locations may significantly alter the TL characteristics of a reciprocal IMO system. As a corollary of the foregoing derivation it is noted that for a -port muffler system satisfying Y Y is used to obtain acoustic reciprocity (not necessarily being conservative) the relation i j Y Y TL 0log0 0log 0 TL Y Y (3) thereby signifying that the TL graph of a reciprocal IO muffler remains unaltered on interchanging the position of inlet and 8

9 outlet ports see Refs. [3] and [7]. 3. Analysis/prediction of the observed peaks and troughs in the TL spectrum 3.. A reciprocal and conservative IO muffler system The advantage of expressing the TL of a reciprocal and conservative IO muffler system [30] explicitly in terms of the matrix parameters (given by Eq. (6)) is that it enables one to analyse/predict the characteristic features of the TL spectrum such as the frequency of occurrence of the attenuation peaks and the troughs in a rather straightforward manner. A peak in the TL graph of a general IO muffler occurs at a frequency f when either of the following conditions is satisfied by the matrix parameters [38]: (a) whilst (and ) are finite (b) whilst (and ) are finite and (c) and whilst (and ) are finite (d) 0 (or equivalently 0 ) regardless of the order of magnitudes of the self-impedance parameters and. It is noted that since a reciprocal system is considered the following conditions will hold good at a given frequency f : () if is finite it implies that is also finite and () 0 0. When conditions (a-c) are satisfied the numerator of the RH of Eq. (6) tends to infinity whilst when condition (d) is satisfied its denominator tends to zero implying a large attenuation or peak at f in the TL spectrum of a reciprocal and conservative system. However it is worth mentioning that the TL graph of a IO system which is not necessarily conservative or reciprocal exhibits an attenuation peak at a frequency f when conditions (a-d) are satisfied. A trough in the TL graph of a IO muffler occurs at a given frequency f when all the matrix parameters (that are purely imaginary for a reciprocal and conservative muffler system [30]) tend to infinity i.e. ij f for i j at the same rate (precisely as given by Eq. (6) simplifies to O 0 ) which implies 0. Hence under this condition the TL expression Y Y Y Y TL 0log 0 0log 0 4YY 4YY and for ports with equal diameters i.e. Y Y the TL is exactly equal to zero thereby indicating the occurrence of a trough. 3.. A conservative IMO muffler system The principle of energy conservation is used to formulate general conditions which when satisfied the TL graph of a conservative IMO muffler system (having M outlet ports) exhibits a peak or a trough at a given frequency f. To this end the inlet port P is excited with a time-harmonic piston velocity and anechoic termination is implemented at the remaining M outlet ports P P 3 P M +. The energy conservation statement is invoked according to which the incident acoustic power at the inlet port P is equated to the sum total of acoustic power transmitted downstream of ports P P 3 P M + thereby yielding (3) A A M... ρ0y ρ0 Y Y Y M which then is substituted in Eq. (3) to yield the following canonical form of TL expression TL 0log 0. (33) (34) Equation (34) may readily be used to analyse/predict the location of attenuation peaks and troughs in the TL spectrum of a conservative IMO system as may be understood from the following discussion. 9

10 . The physical implication of f is that almost all the acoustic power incident due to the inlet piston at the port P is reflected back into the system and negligible acoustic power is transmitted downstream to the anechoic termination at the outlet ports P P 3 P M +. This signifies that the TL graph exhibits a resonance peak at the frequency f.. On the other hand f 0 signifies that almost all the acoustic power incident due to the inlet piston at port P is transmitted to the anechoic outlet ports P P 3 P M + whilst negligible fraction of acoustic power is reflected back into the system through the inlet port P. This signifies that the IMO muffler is acoustically transparent at the frequency f thereby resulting in a trough in the TL graph. On the basis of this discussion it is evident that in the ii versus f graph of conservative IMO system with arbitrary M outlet ports the frequency locations corresponding to ii f 0 would indicate the occurrence of a trough whereas ii f would indicate the occurrence of a resonance peak when port P i is the inlet port. Here i... M. The attenuation peak and trough in the TL graph of a reciprocal and conservative IO system may also be predicted by outlining different conditions satisfied by the [] matrix parameters that occur in the expression of or parameters that are given by Y Y Y Y Y Y Y Y. (35 36) From Eqs. (35) and (36) it may be readily shown that an attenuation peak occurs at a frequency f if any one of the conditions (a-d) indicated in section 3.. are satisfied. imilarly a trough occurs at a frequency f when all the matrix parameters tend to infinity i.e. ij f for i j. It is worth mentioning that due to algebraically simple forms of and parameters the different conditions for occurrence of attenuation peak and troughs in the TL graph can be explained in terms of the [] matrix parameters in a rather straightforward manner. However for a conservative IDO system (i.e. M ) outlining such conditions in terms of the [] matrix parameters is difficult as may be appreciated from the algebraic tediousness of the parameter for a 3-port system given by Y Y Y Y Y Y Y Y Y Y Y Y For IMO muffler systems with more than outlet ports i.e. M 3 outlining different such conditions (when an attenuation peak or trough occurs a frequency f ) in terms of the [] matrix parameters is still more formidable due to the increasing algebraic tediousness of the expression for parameter. In view of the foregoing discussion it is concluded that ii analyzing ii versus f graph offers a convenient and a rather powerful method for analysing/explaining the salient features of TL spectrum of a conservative IMO system. The TL expressions for a conservative IMO system with ports P P 3 P M + taken as the inlet ports are similarly obtained and are shown as follows. TL 0log 0 TL 3 0log 0... TL M 0log M M It is noted from Eqs. (34) and (38) that in general for a conservative (not necessarily a reciprocal) IMO system TL TL TL3 TL M inasmuch as M M in general. Therefore in general interchanging the inlet and outlet port locations would also significantly alter the TL characteristics of a conservative IMO system.. (37) (38) 4. REULT ANALYI AND FORMULATION OF DEIGN GUIDELINE This section presents the TL performance of different configurations of the IO and IDO rectangular chambers computed using the 3-D semi-analytical model (discussed in section ) and validates the method by comparing the results with those obtained using the 3-D Finite Element Analysis (FEA) and also against a previous result (from literature []). Unless otherwise noted the breadth and height of the rectangular chamber are both taken equal and given by B H 00 mm the length L of the axially short and long chamber is taken as 50 mm and 300 mm respectively ports of circular cross-section are considered and their diameters are taken equal given by d 0 = 40 mm. Furthermore the ambient temperature field in the rectangular 0

11 chamber and ports is assumed to be uniform and is taken as T0 0 C implying that the sound speed c m s and a 3 uniform ambient density 0.0 kgm. It is noted that for the different IO and IDO configurations analysed in this work the infinite modal summation in the Green s function response (given by Eq. (6)) is truncated to the first 0 modes in each of the x y and z directions i.e. n m l The truncated Green s function response coupled with the uniform piston-driven model exhibits good convergence throughout the frequency range of interest. 4. Chambers having a single end-inlet and single/double end-outlet: traight-flow/reverse-flow configuration Figure (a) shows the three orthogonal views of a -port rectangular expansion chamber configuration having an end-inlet port and end-outlet port located on the opposite B-H end faces whilst Fig. (b) shows its 3-D view. This configuration is referred to as the IO straight-flow configuration. Figure A -port straight-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as ) and end-outlet port (marked as ) located on the opposite B-H end faces: (a) Orthogonal projections and (b) 3-D view. Figure (a) and (b) shows the orthogonal views and the 3-D view respectively of a -port rectangular expansion chamber configuration having an end-inlet port and end-outlet port located on the same B-H end face. This configuration is referred to as the IO reverse-flow configuration. Figure A -port reverse-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as ) and end-outlet port (marked as ) located on the same B-H end face: (a) Orthogonal projections and (b) 3-D view. For both straight-flow and reverse-flow configurations the end ports and are centred at ( x y) and ( x y ) respectively on the B-H end face. It is noted that in this subsection only the axially short rectangular chamber configurations are analysed because the TL graph of axially long chambers are well-known to have either a simple expansion chamber type

12 behaviour for straight-flow configuration (characterised by the regular occurrence of domes and troughs in the low-frequency range [ 3]) or a quarter-wave type behaviour for reverse-flow configuration [ ]. Figures 3(a) and (b) presents the TL performance obtained using the 3-D semi-analytical approach and 3-D FEA for axially short straight-flow and reverse-flow configurations respectively when the port is centred at x y 00 mm (the pressure node of odd transverse modes) and port is centred at x y 50 mm (the pressure node of even transverse modes). It is noted that the vertical lines in Fig. 3 denote the resonance frequency of n m l mode and the same convention is followed henceforth. An excellent agreement is observed between the 3-D semi-analytical approach and 3-D FEA in Figs. 3(a) and (b) which validates the present method. It is important to mention here that the 0-noded tetrahedral elements were used during 3-D FEA which are especially attractive to use because of the availability of automatic tetrahedral meshing programs exhibit good numerical convergence and thus are a good general purpose element. The element size Δl of the tetrahedron is so chosen that there is a minimum of eight nodes per wavelength and for the sound speed considered and the maximum frequency of interest fmax 3000 Hz this amount to Δl min mm where min 4.38 mm. A broadband attenuation performance is obtained throughout the frequency range for both straight-flow and reverseflow configurations as shown in Figs. 3(a) and (b) respectively. Indeed both these configurations yield an attenuation over f Hz where the lower and upper limits correspond to the resonance frequency of 0 db in the frequency range the ( 0 0) or (0 0) mode and (3 0) or ( 3 0) mode respectively. The broadband attenuation pattern is due to the occurrence of peak at the resonance frequencies of the transverse modes which is explained in terms of the matrix parameters as follows. At the resonance frequencies of ( 0 0) (0 0) ( 0) (3 0 0) (0 3 0) (3 0) and ( 3 0) modes i.e. the odd transverse modes the (and ) parameters are finite whilst which signifies that the TL graph exhibits a peak at these frequencies (see section 3..). imilarly at the resonance frequencies of ( 0 0) (0 0) and ( 0) modes the even transverse modes the (and ) parameters are finite whilst which implies that the TL graph exhibits a peak at these frequencies. It is noted that both straight-flow/reverse-flow configurations exhibit a trough at the resonance frequency of the (4 0 0) or (0 4 0) mode (coincident in this case because B H ) i.e. at f Hz (not shown in Fig. 3) thereby leading to the breakdown of the broadband attenuation pattern. This frequency corresponds to the maximum frequency limit up to which an axially short straight-flow/reverse-flow IO rectangular chamber can exhibit a broadband attenuation and is given by f400 fupper c0 B or f040 fupper c0 H whichever occurs earlier provided that the resonance frequency of the first axial mode given by f c 0 L occurs after or is coincident with f 400 or f 040 implying that maximum length L for the short chamber must satisfy either L B 0.5 or L H 0.5. Incidentally for the chamber dimensions considered L B L H 0.5 implying maximum attenuation obtained whilst having the largest possible broadband frequency range of attenuation. Figures 3(a) and (b) demonstrate that one may easily exploit the influence of higher-order transverse modes in an axially short rectangular chamber by optimally locating the end-inlet port at x = 0.5B y = 0.5H and end-outlet port at x = 0.75B y = 0.75H on the same/opposite end face to obtain a broadband attenuation pattern. The optimal location of end ports is indeed similar to the double-tuning of the length of concentric inlet and outlet pipes in an extended-inlet and extendedoutlet chamber muffler [40]. In order to mathematically demonstrate the dominance of higher-order transverse modes over the axial plane wave modes a model based on the -D transverse modes is considered by ignoring all those modes in Eq. (6) that has axial variation/dependence to obtain the impedance matrix parameter. R R m y n x m y n x cos cos dx dy cos cos dx dy. B H R R R R m n R R R H B R H B R jk0c 0 ρ0q0 m0... n0... n m εnm0hbl k0 p ( x y z x y z ) (39) It is important to mention that the -D transverse model considered in this work (Eq. (39)) is in essence the same as that developed in previous papers [4 4] for thin (or axially short) muffler elements of uniform rectangular or circular crosssection and is valid up to the frequency when the corresponding wavelength is sufficiently larger than axial length. Figures 3(a) and (b) indicate that the -D transverse model is nearly coincident with the 3-D predictions for almost throughout the frequency range in particular for straight-flow configuration. This proves that for axially short rectangular chambers the transverse modes completely dominate the acoustic field and in fact the axial modes may be altogether neglected up to the resonance frequency of the first axial mode i.e. f00 0.5c0 L without significantly affecting the accuracy. Furthermore from the point-of-view of -D transverse model the axially short straight-flow and reverse-flow configurations are acoustically identical or equivalent inasmuch as this model does not distinguish between the relative axial

13 location of end ports. In fact in previous papers by the authors a similar conclusion was also arrived at wherein axially short elliptical and circular end-chambers were analysed using the -D transverse plane wave approach [43]. Figure 3 Broadband TL performance of axially short IO rectangular chamber in (a) straight-flow configuration shown in Fig. and (b) reverse-flow configuration shown in Fig.. The transverse location of the end ports and (on the X-Y end faces) are identical and is given by x = 0.5B y = 0.5H x = 0.75B y = 0.75H. The TL graph obtained using the 3-D semianalytical approach is compared with those obtained using the 3-D FEA and -D transverse model. (c) Broadband TL performance of axially short IO straight-flow and reverse flow configurations (computed using the 3-D semi-analytical method) having the same transverse location of end ports and given by x = 0.5B y = 0.75H and x = 0.75B y = 0.5H respectively. The chamber and port dimensions in parts (a) to (c) are same and given by B = H = 00 mm L = 50 mm and d 0 = 40 mm. Figure 3(c) presents the TL performance (obtained using the 3-D semi-analytical approach) for axially short straightflow and reverse-flow configurations when the end port is centred at x 00 mm y 50 mm and end port centred at x 50 mm y 00 mm. It is observed from Fig. 3(c) that a broadband attenuation performance is obtained throughout the frequency range for both straight-flow and reverse-flow configurations. The TL graphs in Fig. 3(c) exhibit peak at the resonance frequencies of transverse modes and are nearly identical with those shown in Figs. 3(a) and (b) except the absence of peaks at f0 f0 f30 or f 30 (because all [] parameters are finite and non-zero at these frequencies) and the occurrence of an additional peak at f0 or f. 0 imilar to the configurations analysed in Figs. 3(a) and (b) the TL graphs shown in Fig. 3(c) also exhibit a trough at f400 f Hz thereby indicating the broadband attenuation range for this configuration. Another set of parametric study was carried out to investigate the effect of angular location of the ports and on the broadband attenuation range of the axially short end-inlet and end-outlet straight-flow/reverse-flow configurations. To this end the following test-configurations were considered: () x = 0.5B y = 0.5H and x = 0.5B y = 0.75H and () x = 0.5B y = 0.75H and x = 0.5B y = 0.5H. (The end ports may be located on the same B-H face or on opposite faces.) It was found that 3

14 the TL graph (not shown here) for the test-configurations () and () exhibit a trough at f f00 and f f respectively 00 which results in a breakdown of the broadband attenuation pattern significantly before f400 or f. 040 In view of the outcome of the foregoing parametric studies it is concluded that for an axially short straightflow/reverse-flow IO chamber the transverse location of the end ports and should be taken either as () x = 0.5B y = 0.5H x = 0.75B y = 0.75H or () x = 0.5B y = 0.75H x = 0.75B y = 0.5H to nullify the troughs at the resonance frequencies of the first few transverse modes and maximise the broadband attenuation range up to f f400 or f f. Indeed 040 use of these high-performance short chamber configurations are recommended for engineering application such as hermetically sealed refrigeration compressors [4 4] and for medical application such as Continuous Positive Airway Pressure (CPAP) devices [44] where severe space/volume constraint is often a limiting requirement. Figure 4(a) shows the three orthogonal views of a IDO (3-port) rectangular expansion chamber configuration having end ports and that are located on the same B-H end face and centred at ( x y) and ( x y ) respectively whilst the end port 3 is located on the opposite B-H end face and is centred at ( x3 y 3). Figure 4(b) shows its 3-D view. Figure 4 A 3-port rectangular expansion chamber having two end ports (marked as and ) located on a B-H end face and the third end port (marked as 3) located on the opposite B-H end face: (a) Orthogonal projections and (b) 3-D view. Figure 5(a) depicts the TL graphs for the axially short IDO configuration (shown in Fig. 4) for different cases when (a) end port is the inlet and ports and 3 are outlet (b) end port is the inlet and ports and 3 are outlet and (c) end port 3 is the inlet and ports and are outlet. It is noted that end port is centred at x y 00 mm whilst the end ports and 3 are centred at x y 50 mm and x3 y3 50 mm respectively. Figure 5(a) indicates that when port is the inlet port a broadband attenuation pattern is observed throughout the frequency range which is qualitatively similar to the TL graphs shown in Figs. 3(a) and (b) for axially short IO chambers. Incidentally the peaks in this TL graph at the resonance frequencies can be readily explained by analysing the versus frequency f graph shown in Fig. 6(a). It is observed from Fig. 6(a) that for the frequency range f Hz and in particular at resonance frequencies which signifies that no acoustic power is transmitted downstream of the anechoic ports and 3 thereby explaining the occurrence of peaks at these frequencies thence the broadband attenuation characteristics. Figure 5(a) indicates that the TL characteristics for the axially short IDO configuration when the port or port 3 is taken as inlet is significantly different in comparison to the test-case when the port is the inlet; in this case the TL graphs exhibits a trough at the resonance frequency of ( 0 0) or (0 0) mode leading to the breakdown of the broadband attenuation pattern at this frequency. This demonstrates that interchanging the location of the inlet and outlet ports of a IDO muffler completely alters the nature of TL graph thereby corroborating the theoretical developments in section 3.. The troughs in the TL graph when port or 3 is the inlet may be explained by analysing the spectral variation of or 33 parameters shown in Fig. 6(b). It is noted that near the resonance frequencies of ( 0 0) or (0 0) ( 0) (3 0 0) or (0 3 0) and (3 0) or ( 3 0) modes both 0 and 33 0 thereby signifying that almost all the acoustic power incident at the inlet port is transmitted downstream into the anechoic outlet ports which renders the IDO muffler acoustically transparent at these frequencies thereby explaining the breakdown of the broadband attenuation characteristics. Incidentally it is also observed from Fig. 6(b) that the spectrum of and 33 parameters is nearly co-incident thereby explaining the nearly identical nature of the TL graphs in Fig. 5(a) for the test-case when ports and 3 are taken as inlet. 4

15 Figure 5 (a) TL performance of axially short IDO rectangular chamber shown in Fig. 4 Effect of interchanging the location of the inlet and outlet ports. The chamber and port dimensions for the IDO configuration are identical with the axially short ( x y z ) 0.5 B0.5 H0 IO configuration analysed in Fig. 3 whilst the location of end ports and 3 is given by ( x y z ) 0.75 B0.75 H0 and ( x y z ) 0.75 B0.75 H L respectively. (b) Comparison of the TL performance of axially short IDO configuration (with port as the inlet) with its IO counterparts analyzed in Fig. 3 Effect of an additional outlet port. Figure 6 Variation of the chamber configuration analysed in Fig. 5(a). parameter ( i 3) versus frequency f for the axially short IDO rectangular expansion ii Figure 5(b) compares the TL performance of the axially short IDO chamber configuration analysed in Fig. 5(a) (with port taken as inlet) with its IO counterpart analysed in Fig. 3 with a view to study the effect of considering an additional outlet port on attenuation performance. The following points are observed from the TL graphs shown in Fig. 5(b): 5

16 . The TL performance of the straight-flow/ reverse-flow type axially short IO rectangular chamber is qualitatively similar to its equivalent IDO counterpart wherein the TL graphs of the IO and IDO configurations are almost parallel to each other.. The IDO configuration yields slightly lower acoustic attenuation performance than the IO configuration as the presence of an additional outlet port results in a smaller effective expansion ratio [3]. 4. Chambers having a single end-inlet and single side-outlet: Cross-flow configuration Figure 7(a) shows the three orthogonal views of a -port rectangular expansion chamber configuration having an end-inlet port located on the B-H face and a side-outlet port located on the B-L face whilst Fig. 7(b) shows its 3-D view. This configuration is referred to as the IO cross-flow configuration. It is noted that end and side ports of this configuration are of a square cross-section. Figure 7 A -port cross-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as ) located on the B-H end face and a side-outlet port (marked as ) located on the B-L face: (a) Orthogonal projections and (b) 3-D view. An axially long chamber is considered and the chamber dimensions are given by B = H = 50 mm L = 5 mm. Furthermore the breadth and height of both ports are taken to be equal and given by b 0 = h 0 = 50 mm. The end-inlet port and the side-outlet port is centred at x = 0.5B y = 0.5H z = 0 and x = 0.5B y = H z = 0.5L respectively. It is noted that Ih [] and Venkatesham et al. [3] had previously analysed this configuration with the same chamber and port dimensions and port locations as noted above. Figure 8 demonstrates that the TL graph computed using the 3-D semi-analytical model presented in this work the 3-D analytical prediction by Ih [] and the 3-D FEA results are in excellent agreement throughout the frequency range of interest thereby validating the present model. In particular it is observed that the TL graph exhibits peak at the resonance frequencies of the (0 0 ) (0 0) or ( 0 0) modes resulting in a broadband attenuation pattern up to the resonance frequency of the (0 0 ) or the second axial mode. Indeed the TL graph of this cross-flow configuration is similar to that of an axially long elliptical/circular chamber having an end-inlet/outlet and a side-outlet/inlet [ 3 45]. The peaks at the resonance frequencies in Fig. 0 are explained in terms of the matrix parameters as follows. At the resonance frequency of (0 0 ) or the first axial mode the (and ) parameters are finite whilst which signifies that the TL graph exhibits a peak. Incidentally the occurrence of this peak may also be explained by noting that the resonance frequency of the (0 0 ) chamber mode is coincident with the first resonance frequency of the quarter-wave resonator of axial length L formed due to the cavity between the side port and the rigid B-H end face opposite to the B-H end face on which the end port is located. On the other hand at resonance frequency of the (0 0) transverse mode the (and ) parameters are finite whilst thereby explaining the occurrence of peak at this frequency. imilar to the peak at resonance frequency of the (0 0 ) mode this peak may also be explained by the noting the first resonance frequency of a quarter-wave resonator of height H formed along the y direction of the rectangular chamber. At the resonance frequency of the (0 0 ) or the second axial mode all the matrix parameters tend to infinity (due to the location of port on the B-H end face) which by the virtue of Eq. (3) implies that TL 0 thereby leading to the breakdown of broadband attenuation pattern at this frequency. 6

17 Figure 8 TL performance of the cross-flow configuration shown in Fig. 7 having the following chamber and port dimensions: B = H = 50 mm L = 5 mm b 0 = h 0 = 50 mm and port location given by x = 0.5B y = 0.5H z = 0 x = 0.5B y = H z = 0.5L - Comparison of the 3-D semi-analytical approach used in this work the 3-D analytical approach used by Ih [99] 3-D FEA and the -D axial plane wave theory. The -D axial plane wave model [3 8] is also used to obtain the TL performance of this configuration and the -D results are compared against the 3-D predictions. To this end the [] matrix based on the -D axial plane wave is first obtained (given by Eq. (40)) that characterises this cross-flow configuration [8]. where Y c BH cosk0 L z sin k0l sin k L cot k0l p v jy0 p cos k0 L z cos k0z cos k v 0 L z sin k0l 0 is the characteristic impedance of the rectangular chamber along the z direction. It is observed from 0 0 Fig. 0 that the -D axial plane wave model is in a satisfactory agreement with the 3-D approaches in the low-frequency range i.e. up to a frequency slightly less than the resonance frequency of the ( 0 0) or (0 0) mode beyond which significant deviation is observed between the -D model and the more accurate 3-D predictions thereby indicating the breakdown of -D axial plane wave theory. 4.3 Chambers having a single side-inlet and single/double side-outlet: Cross-flow configuration (40) Figure 9(a) shows the three orthogonal views of a -port rectangular expansion chamber configuration having a side-inlet port that is centred at x y = H z and located on the B-L face and a side-outlet port centred at x = B y z and located on the H-L face whilst Fig. 9(b) shows its 3-D view. This configuration is also referred to as the IO cross-flow configuration. 7

18 Figure 9 A -port cross-flow configuration of a rectangular expansion chamber having a side-inlet port (marked as ) located on the B-L face and a side-outlet port (marked as ) located on the H-L face: (a) Orthogonal projections and (b) 3-D view. Figure 0 presents the TL performance of the axially long IO cross-flow configuration (shown in Fig. 9) obtained using the 3-D semi-analytical approach and 3-D FEA when the side port is centred at x = 0.5B y = H z = 0.5L (the pressure node of the first axial mode) and the side port is centred at x = B y = 0.5H z = 0.75L (one of the pressure nodes of the second axial mode). It is noted that this location of side ports and corresponds to an equivalent relative angle between their respective centres. An excellent agreement is observed between the results of 3-D semi-analytical approach and 3-D FEA as may be observed from Fig. 0 which validates the present method. Figure 0 Broadband TL performance of the cross-flow configuration shown in Fig. 9 having the following chamber and port dimensions: B = H = 00 mm L = 300 mm d 0 = 40 mm and side port location given by x = 0.5B y = H z = 0.5L x = B y = 0.5H z = 0.75L - Comparison of the 3-D semi-analytical approach used in this work 3-D FEA and the -D axial plane wave theory. It is observed from Fig. 0 that a broadband attenuation performance is obtained up to resonance frequency of the ( 0 0) or (0 0) transverse mode given by f 75.7 Hz in particular this configuration yields an attenuation over 0 db in the frequency range f Hz. The broadband attenuation pattern is due to the occurrence of peak at the resonance 8

19 frequencies of the (0 0 ) ( 0 0) or (0 0) (0 ) or ( 0 ) (0 0 ) (0 ) or ( 0 ) modes which is explained in terms of the matrix parameters as follows. At the resonance frequencies of the first axial mode denoted by (0 0 ) and the axial-transverse modes given by ( 0 ) or (0 ) the (and ) parameters are finite whilst which signifies that the TL graph exhibits a peak. On the other hand at the resonance frequencies of the second axial mode denoted by (0 0 ) and the axial-transverse modes given by ( 0 ) or (0 ) the (and ) parameters are finite whilst that again explains the occurrence of peak in the TL graph. Furthermore at resonance frequencies of (a) the ( 0 0) transverse mode the (and ) parameters are finite whilst and (b) the (0 0) transverse mode the (and ) parameters are finite whilst thereby explaining the occurrence of peak in the TL spectrum. It is noted that at the resonance frequency of the ( 0) transverse mode all matrix parameters are finite and non-zero hence the TL graphs does not exhibit either a peak or trough. The TL graph in Fig. 0 exhibits a trough at the resonance frequency of the ( 0 0) or (0 0) transverse modes leading to a breakdown of the TL performance (noted earlier). The occurrence of trough at this frequency can be readily explained by noting that at the resonance frequency of the ( 0 0) or (0 0) modes all matrix parameters tend to infinity which implies (by virtue of Eq. (3)) that TL 0. However it is important to mention here that based on the chamber dimensions considered the resonance frequency of (0 0 3) or the third axial mode given by f003.5c0 L is coincident with the resonance frequencies of the ( 0 0) or (0 0) mode given by f00 c0 B f00 c0 H. Although the TL graph of the IO configuration considered exhibits a small peak at f 003 (that may be explained on the basis of axial plane wave modes) this peak is not able to cancel or nullify the dominant trough due to the ( 0 0) or (0 0) modes thereby indicating that the broadband attenuation range cannot be increased by fine-tuning the chamber length as L.5H or L.5 B. In fact based on this observation guidelines can also be formulated for specifying the minimum chamber length for axially long chambers having a side-inlet and side-outlet and is briefly discussed as follows. For chamber length L.5 B or L.5 H the peak due to the third axial mode occurs before f00 or f 00 because f003 f00 or f003 f00 which signifies a maximum broadband attenuation range equal to f00 or f00 whichever occurs earlier. Therefore the minimum length of the axially long chamber for obtaining broadband attenuation over the maximum frequency range is equal to L.5H or L.5 B. The maximum chamber length which yields a broadband attenuation over the maximum possible frequency range can also be determined by first noting that a trough always occurs at the resonance frequency of the (0 0 4) or the fourth axial mode given by f004 c0 L (because all matrix parameters tend to infinity). On increasing the chamber length the trough at f 004 tends to shift towards the left or lower side of the frequency spectrum and when L H f004 f00= f00 thereby signifying the coincidence of troughs due to the onset of these modes. A further increase in the chamber length will result in the breakdown of the broadband attenuation pattern earlier than the resonance frequency of the ( 0 0) or (0 0 ) modes as the trough due to the fourth axial mode will occur before the trough due to these transverse modes. Therefore the maximum chamber length is taken as L H. In view of the foregoing discussion the recommended optimal length range for axially long chambers is given by.5h L H which yields a high attenuation in the low-frequency region and simultaneously also has the maximum broadband attenuation range. This axially long IO rectangular chamber configuration with side-inlet port and side-outlet port located at x = 0.5B y = H z = 0.5L and x = B y = 0.5H z = 0.75L respectively is indeed the counterpart of axially long elliptical/circular chamber configurations having a side-inlet and side-outlet port that are located at axial distances given by one-half and three-quarters of chamber length with relative angular location between their centres equal to see Refs. [0 3 45]. The -D axial plane wave model [3 8] is also used to compute the TL performance of the axially long cross-flow IO rectangular chamber and the -D results are compared against the 3-D predictions. To this end the [] matrix based on the -D axial plane wave theory is first obtained (given by Eq. (4)) that characterises this configuration [8]. k0z k0 L z k0z k0 L z p jy cos cos cos cos 0 v p sin k0l cos k0z cos k0 L z cos k0z cos k v 0 L z where Y0 c0 BH. Figure 0 demonstrates that the -D axial is in a good agreement with the 3-D prediction in the lowfrequency range i.e. up to a frequency slightly smaller than the resonance frequency of the ( 0 0) or (0 0) modes beyond which significant deviation is observed between the -D and 3-D approaches thereby indicating the breakdown of the -D model. A parametric study was carried out to investigate the effect of angular location of the side-outlet port on the broadband attenuation range of the axially long side-inlet and side-outlet configuration. To this end the location of side-inlet port was fixed at x = 0.5B y = H z = 0.5L the axial location of the side-port was also kept constant at z = 0.75L (same 9 (4)

20 parameters as those considered in Fig. 0) whilst its angular location was varied: () x = 0.75B y = H () x = B y = 0.75H (3) x = B y = 0.5H and (4) x = 0.75B y = 0. It is noted that the side port is located on the same and opposite B-L faces for test-configurations () and (4) respectively whilst for the test-configurations () and (3) the side-port is located on the H-L face. Indeed the test-configurations ()-(4) are the counterparts of axially long elliptical/circular chamber muffler having a side-inlet port (z = 0.5L) and a side-outlet port (z = 0.75L) with relative angular location between their centres equal to respectively. It was found that the TL graphs (not shown here) for each of the test-configurations ()- (4) exhibits a peak at f f00 and f f00 due to the axial location of side ports and on the pressure nodes of the first and second axial modes respectively. However a trough was observed at f f00 or f f 00 thereby leading to a breakdown in the broadband attenuation pattern significantly before f. 00 In particular the TL graph for test-configurations () and () exhibits a peak in the low-frequency region (near f f00 ) and resembles that of a side-branch resonator. In view of the outcome of this parametric study it is concluded that angular location of the side ports and given by x = 0.5B y = H and x = B y = 0.5H respectively is crucial to nullify the trough at f 00 or f 00 and thus significantly enhance the frequency range of broadband attenuation up to f f00 or f f. 00 Figure (a) and (b) shows the three orthogonal views and the 3-D view respectively of a IDO (3-port) rectangular expansion chamber configuration having side port located on B-L face and side ports and 3 located on opposite H-L face. Figure A 3-port cross-flow configuration of a rectangular expansion chamber having a side-outlet port (marked as ) located on the B-L face and two side ports (marked as and 3) located on opposite H-L faces: (a) Orthogonal projections and (b) 3-D view. Figure depicts the TL graphs for the axially long cross-flow IDO configuration (shown in Fig. ) for different cases when (a) side port is the inlet and side ports and 3 are outlet and (b) side port is the inlet and side ports and 3 are outlet. It is noted that the side port is centred at x = 0.5B y = H z = 0.5L whilst the side ports and 3 are centred at x = 0 x 3 = B y = y 3 = 0.5H z = z 3 = 0.75L respectively. Figure indicates that when port is the inlet port a broadband attenuation pattern is observed throughout the frequency range which is qualitatively similar to the TL graphs shown in Fig. 0 for axially long IO chamber. Incidentally the peaks in this TL graph at the resonance frequencies can again be readily explained by analysing the versus frequency f graph shown in Fig. 3(a). It is observed from Fig. 3(a) that for the frequency range Hz f and in particular at resonance frequencies which signifies that no acoustic power is transmitted downstream of the anechoic ports and 3 thereby explaining the occurrence of peaks at these frequencies thence the broadband attenuation characteristics. Figure indicates that the TL characteristics when the side port is taken as inlet is significantly different in comparison to the test-case when the side port is the inlet; in this case the TL graphs exhibits a trough at the resonance frequency of (0 0 ) or the first axial mode leading to the breakdown of the broadband attenuation pattern at this frequency. This demonstrates that interchanging the location of the inlet and outlet ports of a IDO muffler completely alters the nature of TL graph thereby again corroborating the theoretical developments in section 3.. The troughs in the TL graph when the side port is the inlet may also be explained by analysing the spectral variation of parameters shown in Fig. 3(b). It is noted 0

21 that near the resonance frequencies of (0 0 ) ( 0 0) (0 0) (0 ) ( 0 ) (0 0) ( 0 0) and (0 0 3) modes thereby signifying that almost all the acoustic power incident at the inlet port is transmitted downstream into the 0 anechoic outlet ports which renders the IDO muffler acoustically transparent at these frequencies thereby explaining the breakdown of the broadband attenuation characteristics. For the axially long IDO rectangular chamber considered the TL graph (and 33 spectrum) when side port 3 is taken as the inlet port was found to be coincident with the corresponding TL graph (and spectrum) when the side port was taken as the inlet. It is for this reason that the test-case of side port 3 as the inlet port is not presented in Figs. and 3(b). Figure TL performance of the axially long cross-flow IDO configuration shown in Fig. having the following chamber and port dimensions: B = H = 00 mm L = 300 mm d 0 = 40 mm and location of side ports given by x = 0.5B y = H z = 0.5L x = 0 x 3 = B y = y 3 = 0.5H z = z 3 = 0.75L - Effect of interchanging the location of the inlet and outlet ports. Figure 3 Variation of the chamber configuration analysed in Fig.. parameter ( i ) versus frequency f for the axially long IDO rectangular expansion ii