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Boundary element acoustic analysis of hybrid expansion chamber silencers with perforated facing
ARTICLE IN PRESS Engineering Analysis with Boundary Elements 34 (2010) 690–696
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Boundary element acoustic analysis of hybrid expansion chamber silencers with perforated facing Z.L. Ji School of Power and Energy Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 11 April 2009 Accepted 25 February 2010 Available online 2 April 2010
The substructure boundary element approach is developed to predict and analyze the acoustic attenuation characteristics of hybrid expansion chamber silencers with perforated facing. The silencers are divided into a number of acoustic domains with single medium (air or sound-absorbing material), and treating the sound-absorbing material as an equivalent fluid with complex-valued density and speed of sound (or complex-valued characteristic impedance and wavenumber), and then the boundary element method (BEM) may be applied to each domain leading to a system of equations in terms of acoustic pressure and particle velocity. Using the specific acoustic impedance of perforate, which takes into account the effect of sound-absorbing material, the relationship of acoustic pressures and particle velocities between the inlet and outlet of silencer may be obtained and then transmission loss is determined. For the straight-through perforated tube reactive and dissipative silencers, the predictions of transmission loss agree reasonably well with experimental measurements available in the literature, which demonstrated the applicability and accuracy of the present approach. The BEM is then used to investigate the effect of internal structure on the acoustic attenuation characteristics of hybrid expansion chamber silencers with perforated facing. The numerical results demonstrated that the hybrid expansion chambers may provide higher acoustic attenuation than the reactive expansion chamber in the mid to high frequency range. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method Substructure approach Hybrid silencer Perforated facing Acoustic attenuation performance Prediction Analysis
1. Introduction The sound-absorbing materials are widely used to improve the acoustic attenuation performance of silencers at mid to high frequencies. The studies demonstrated that the multi-dimensional approach is needed for the accurate prediction of acoustic attenuation performance of silencers at higher frequencies, while the simple onedimensional solution provides a reasonable accuracy at lower frequencies only. The two-dimensional axisymmetric and threedimensional analytical approaches based on the mode-matching techniques have been developed to determine the acoustic attenuation performance of the perforated dissipative silencers with regular geometry [1,2]; however, the analytical methods are difficult to be employed for the silencers with complex internal structures. The numerical methods such as finite element method (FEM) [3] and boundary element method (BEM) [4–12] are suitable to predict the acoustic attenuation performance of reactive and dissipative silencers and they are not confined to the geometry. BEM is an ideal tool for the acoustic analysis of silencers due to its surface-only meshing scheme, especially the use of substructure BEM approaches and
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impedance matrix synthesis can save a significant amount of computational time and memory space for the complex and largesize silencer analysis. In consideration of the fact that the coupling in the conventional multi-domain BEM is tedious, Wu and Wan [13], and Wu et al. [14–16] developed the direct mixed-body BEM for the analysis of reactive and dissipative silencers. In order to reduce the matrix size and the total computational time, Lou and Wu [17,18] have also developed the substructuring techniques based on the direct mixed-body BEM for the complex and large-size silencer analysis. The major advantage of the direct mixed-body BEM over the traditional multi-domain BEM approach is that only one side of the thin-wall components needs to be discretized; therefore may reduce the discretization effort and save computational time; however the thickness of these components is neglected. For the engine intake and exhaust silencers, the thickness of perforated facing is typically 1–2 mm, the effect of thickness of perforated facing on the acoustic attenuation prediction is marginal and may be neglected; therefore both the traditional multi-domain BEM approach and direct mixed-body BEM may be used to determine the acoustic attenuation performance of the silencers. However, for some special applications, the thickness of perforated pipe or plate may be not small enough and needs to be considered for the accurate prediction. In comparison with the direct mixed-body BEM, the multi-domain BEM approach needs to discretize the both sides of perforated components for the
ARTICLE IN PRESS Z.L. Ji / Engineering Analysis with Boundary Elements 34 (2010) 690–696
two neighboring substructures; therefore the thickness of perforated components may be considered in the numerical models. The expansion chamber silencers containing sound-absorbing material and perforated facing (tube and plate) are widely used in the intake and exhaust systems of internal combustion engines and fluid machines. The objective of the present study is then to develop the three-dimensional substructure boundary element approach to determine the transmission loss of typical hybrid expansion chamber silencers with perforated facing as shown in Fig. 1, and to investigate the effects of internal structure on the acoustic attenuation characteristics of the silencers.
2. Boundary element approach There are two media inside the hybrid expansion chamber silencers shown in Fig. 1: the air and sound-absorbing material. Assuming homogeneous sound-absorbing material and harmonic
L S2
S1
d
D
wave propagation in both media, and treating the soundabsorbing material as an equivalent fluid with complex-valued density and speed of sound, the continuity and momentum equations yield [1,9]
r2 p þ k2 p ¼ 0,
ð1Þ
where p is the acoustic pressure, k is the wavenumber and equals to k0 for the air and k~ for the sound-absorbing material, respectively. Eq. (1) may be represented in the form of the boundary integral equation [4,7] as Z @G jkzvðYÞGðX,YÞ þ pðYÞ ð2Þ ðX,YÞ dGðYÞ, CðXÞpðXÞ ¼ @n G where G is the boundary surface of the acoustic domain, n the unit normal vector on G directed away from the domain, j the imaginary unit, z the characteristic impedance of the medium (equals to z0 for the air and z~ for the sound-absorbing material, respectively), v the outward normal particle velocity and G(X,Y)¼exp( jkR)/4pR the Green’s function of free space: R being the distance between any two points X and Y in the domain or on the surface, and C(X) is a coefficient which depends on the position of point X. Numerical solution of the boundary integral Eq. (2) can be achieved by discretizing the boundary surface of the domain into a number of elements, and then a set of algebraic system of equations may be obtained by using numerical integration, which can be written in matrix form [7] as ½HfPg ¼ z½GfVg,
L S2 d
S3 S1
D
L
691
ð3Þ
where [H] and [G] are the coefficient matrices, and {P} and {V} are the vectors whose elements are the acoustic pressure p and outward normal particle velocity v on the boundary nodes, respectively. The detailed treatment of the BEM numerical solution procedure is provided elsewhere [4,7]. The substructure BEM is used to determine the acoustic attenuation performance of hybrid silencers. The silencer is divided into a number of substructures with single medium and then the BEM is applied to each one of these substructures leading to a system of equations. The objective of the following derivation is to develop the relationship between inlet variables (Pi,Vi) and outlet variables (Po,Vo) which then facilitates the calculation of four-pole parameters and transmission loss of the silencers.
S2 2.1. Expansion chamber with packed cylindrical shell
d
S1
D
The expansion chamber with packed cylindrical shell is shown in Fig. 1(a). The silencer is divided into two substructures: the center air domain S1 and the outer sound-absorbing material domain S2. Eq. (3) combined with the rigid wall boundary condition yields [7,10] 8 S 9 8 S 9 2 S1 S1 S1 3 T11 T12 T13 > > V 1> P 1> > > < i > < i > = = 7 6 S1 S1 S1 7 S1 6 ¼ z0 4 T21 T22 T23 5 VoS1 , ð4Þ Po > > > > > S1 S1 S1 > : P S1 > : V S1 > ; ; T31 T32 T33 p p
D
fPpS2 g ¼ z~ ½T S2 fVpS2 g,
L S3 S2 d
S1
Fig. 1. Configurations of hybrid silencers: (a) expansion chamber with packed cylindrical shell, (b) expansion chamber with packed end-plates, (c) expansion chamber with full-packed housing and (d) straight-through hybrid expansion chamber.
ð5Þ
where the subscripts i, o and p represent the inlet, outlet and perforation, respectively. Introducing the specific acoustic impedance zp for the perforated facing, the boundary conditions may be expressed as fVpS1 g ¼ fVpS2 g
and
fPpS1 gfPpS2 g ¼ z0 zp fVpS1 g:
ð6; 7Þ
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9 8 < PpS2 =
From Eqs. (4) to (7), one may have 2 3 ( S1 ) S1 S1 S1 S1 S1 S1 ( S1 ) T11 þT13 ZT31 T12 þ T13 ZT32 Pi Vi 5 ¼ z0 4 S1 , S1 S1 S1 S1 S1 S1 T21 þT23 ZT31 T22 þ T23 ZT32 Po VoS1
1
ð8Þ
matrix matrix between the inlet and outlet of the silencer. The four-pole parameters and transmission loss calculation procedure are identical to those used earlier [7] and will not be repeated here.
The expansion chamber with packed end-plates as shown in Fig. 1(b) is divided into three substructures: the center air domain S1, left sound-absorbing material domain S2 and right soundabsorbing material domain S3. Eq. (3) combined with the rigid wall boundary condition yields 8 S 9 8 S 9 2 S1 S1 S1 3 T11 T12 T13 > > V 1> P1> > > = = < i > < i > 6 7 S S S T211 T221 T231 7 ð9Þ PoS1 ¼ z0 6 VoS1 , 4 5 > > > S > > > S1 S1 S1 > 1 ; ; : PS1 > : V T T T 32
33
ð10Þ
fPpS3 g ¼ z~ ½T S3 fVpS3 g:
ð11Þ
Eqs. (10) and (11) may be rewritten as fPpS23 g ¼ z~ ½T S23 fVpS23 g,
ð12Þ
where 9 8 < VpS2 = fVpS23 g ¼ , : VpS3 ;
"
½T
S23
T S2 ¼ 0
# 0 : T S3
On the perforated facings between substructures S1 and S2, as well as S1 and S3, the boundary conditions may be expressed as fVpS1 g ¼ fVpS23 g
and
ð16Þ
ð17Þ
On the perforated facings between substructures S1 and S2, as well as S2 and S3, the boundary conditions may be expressed as fVpS11 g ¼ fVpS12 g
and
fPpS11 gfPpS21 g ¼ z0 zp fVpS11 g,
ð18; 19Þ
fVpS33 g ¼ fVpS32 g
and
fPpS33 gfPpS23 g ¼ z0 zp fVpS33 g:
ð20; 21Þ
fPpS21 g ¼ z~ ½T S23 fVpS12 g,
ð22Þ
S2 S2 S2 1 S2 ½T11 þ ðz~ =z0 ÞT12 ½zp IS3 T S3 ðz~ =z0 ÞT22 T21 , S3
S23
and ½IS3 where ½T ¼ being the unity matrix with same order as ½T . Therefore, the same expression as Eq. (8) may be obtained, here S1 ðz~ =z0 ÞT S23 1 . ½Z ¼ ½zp Ip T33 3. Acoustic properties of sound-absorbing material and perforated facing
p
fPpS2 g ¼ z~ ½T S2 fVpS2 g,
9 8 < PpS2 = , fPpS23 g ¼ : PpS3 ;
S2 T21
Combining Eqs. (16), (17), (20) and (21) leads to
2.2. Expansion chamber with packed end-plates
31
¼ z~ 4
9 38 S2 < S2 = Vp1 T12 5 S2 : S2 ;, T22 Vp3
S2 T11
fPpS33 g ¼ z0 ½T S3 fVpS33 g:
S1 where ½Z ¼ ½zp Ip T33 ðz~ =z0 ÞT S2 1 , and [Ip] being the unity S1 . Eq. (8) defines the impedance with same order as ½T33
p
: PpS23 ;
2
fPpS1 gfPpS23 g ¼ z0 zp fVpS1 g:
ð13; 14Þ
Eqs. (9), (12)–(14) may be combined to obtain the impedance matrix between the inlet and outlet of the silencer, which is same S1 ðz~ =z0 ÞT S23 1 . as Eq. (8) with ½Z ¼ ½zp Ip T33 2.3. Expansion chamber with full-packed housing The expansion chamber with full-packed housing (i.e. packed cylindrical shell and two end-plates) as shown in Fig. 1(c) is divided into two substructures: the center air domain S1 and the outer sound-absorbing material domain S2. Then, the same treatment as the expansion chamber with packed cylindrical shell may be used to obtain the impedance matrix between the inlet and outlet of the silencer.
The complex-valued acoustic impedance and wavenumber of sound-absorbing material may be calculated usually by empirical expressions [19] or determined by experiments [20]. The present study uses texturized fiberglass roving as the sound-absorbing material. The complex acoustic impedance z~ and the complex wavenumber k~ of the sound-absorbing material with the filling density of 100 kg/m3 are expressed as [21] z~ =z0 ¼ 1:0 þ 33:20f 0:7523 j28:32f 0:6512 ,
ð23Þ
0:6841 ~ k=k j38:39f 0:6285 , 0 ¼ 1:0 þ 39:20f
ð24Þ
where f is the frequency. In order to predict the acoustic attenuation performance of silencers with perforated facing, the specific acoustic impedance of perforated facing must be obtained first. The specific acoustic impedance of perforated plate is defined [22] as
zp ¼
pi po
r0 c0 ui
or
zp ¼
pi po
r0 c0 uo
,
ð25Þ
where pi, po, ui and uo are the average sound pressures and particle velocities on each side of the perforated plate, r0 the density of air and c0 the speed of sound. Combining the mass conservation equation, Eq. (25) may be rewritten as
zp ¼
pi po uh
r0 c0 uh ui
¼
zh Rh þjXh ¼ , f f
ð26Þ
where uh is the particle velocity within the hole, zh the specific acoustic impedance of a single hole, f the porosity, j the imaginary unit and Rh, Xh the specific acoustic resistance and reactance of a single hole, respectively, which may be expressed as [22,23] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ Rh ¼ ð1 þ tw =dh Þ 8k0 m=z0 ,
2.4. Straight-through hybrid expansion chamber Xh ¼ k0 ðtw þ adh Þ, The straight-through hybrid expansion chamber as shown in Fig. 1(d) is divided into three substructures: the center air passage S1, sound-absorbing material domain S2 and outer air cavity S3. Eq. (3) combined with the rigid wall boundary condition yields 8 S 9 2 S1 S1 S1 38 S1 9 T11 T12 T13 > > V > P1> > > < i > < i > = = 6 S1 S1 S1 7 S1 7 T T T , ð15Þ PoS1 ¼ z0 6 V o 21 22 23 5 4 > S > > S > > > > S1 S1 S1 > 1 ; :P 1 ; : V T31 T32 T33 p1 p1
ð28Þ
where m is the dynamic viscosity of air, tw the thickness of perforated plate, dh the diameter of hole and a the end correction coefficient of a single hole. Based on the piston-driven model, Ingard [24] derived the formulation for the end correction coefficient of a circular hole in a rectangular tube as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 X J12 p ðmxÞ2 þðnZÞ2 X 4 1 0 a¼ 2 e , ð29Þ p ðxZÞ1=2 m ¼ 0 n ¼ 0 mn ½m2 ðh=bÞ þ n2 ðb=hÞ3=2
ARTICLE IN PRESS Z.L. Ji / Engineering Analysis with Boundary Elements 34 (2010) 690–696
zp ¼
~ ~ =z0 Þdh g Rh þ jk0 ftw þ 0:5a½1þ ðk=k 0 Þðz
f
:
50
Transmission loss (dB)
where x ¼dh/b, Z ¼dh/h, b and h are the distances between two neighboring holes in two directions, the prime on the summation sign implies that the term representing the (0, 0) fundamental mode is excluded, emn ¼1 if ma0 and na0 or emn ¼ 1/2 if otherwise, and J1 is the Bessel function of the first kind of order 1. In the presence of sound-absorbing material, Eq. (26) should be modified accordingly. In view of the fact that one side of the perforated facing is the air and another side is the soundabsorbing material, expression (26) is modified as
693
Experiment BEM using Eqs. (26)-(29) BEM using Eq. (31)
40 30 20 10
ð30Þ
0 0
500
1000
1500 2000 Frequency (Hz)
4. Results and discussions
2500
3000
Fig. 3. Transmission loss of straight-through perforated tube reactive silencer (f ¼8.4%).
L
50
Transmission loss (dB)
In order to validate the present substructure boundary element approach as well as to assess the applicability and accuracy of expressions (26)–(30) for the acoustic attenuation predictions of silencers with perforated facing, the straightthrough perforated tube silencer as shown in Fig. 2 (which may be considered as a special case of the configuration 1(a), where the diameter of perforated tube equals to diameters of inlet and outlet tubes) is considered. The silencer has a cylindrical concentric configuration, and the dimensions are D¼164.4 mm and L¼257.2 mm for the inner diameter and length of expansion chamber, respectively, d¼ 49.0 mm and tw ¼0.9 mm for the inner diameter and wall thickness of perforated tube, respectively, dh ¼4.98 mm for the diameter of hole. Transmission loss of the straight-through perforated tube silencer without and with sound-absorbing material filling (100 kg/m3) is predicted by the present approach. In view of the axisymmetrical configuration of the silencer, the axisymmetrical BEM [7] is applied and the three-node quadratic elements are used to discretize the boundary generator. The maximum size of elements is less than 12.5 mm. Figs. 3–6 compare the transmission loss predictions from the present BEM using Eqs. (26)–(30) and experimental results in the literature [25] for two different porosities (f ¼ 8.4% and 25.7%). The BEM predictions for the straight-through perforated tube reactive and dissipative silencers show reasonable agreements with experiments in the entire frequency range of interest. The discrepancy between BEM predictions and experimental results may possibly be attributed to (1) the texturization condition and uniformity of soundabsorbing material filling inside the silencer in the experiments, and (2) the approximations involved in the expressions for the perforate specific acoustic impedance and sound-absorbing material properties. The following empirical expression for the specific acoustic impedance of perforate suggested by Sullivan and Crocker [26] is widely used to calculate the acoustic attenuation performance of
Experiment BEM using Eqs. (26)-(29) BEM using Eq. (31)
40 30 20 10 0 0
500
1000
1500 2000 Frequency (Hz)
2500
3000
Fig. 4. Transmission loss of straight-through perforated tube reactive silencer (f ¼25.7%).
50
Transmission loss (dB)
4.1. Validation
Experiment BEM using Eq. (30) with Eqs. (27) & (29) BEM using Eqs. (26)-(29)
40 30 20 10 0 0
500
1000
1500 2000 Frequency (Hz)
2500
3000
Fig. 5. Transmission loss of straight-through perforated tube dissipative silencer (f ¼8.4%).
silencers with perforated elements [27,28]
d
D
zp ¼
Fig. 2. Straight-through perforated tube silencer (dissipative expansion chamber).
0:006 þ jk0 ðtw þ0:75dh Þ
f
:
ð31Þ
This expression was obtained for a small (1600 mm2) sample of perforated plate. The test sample consisted of a 0.81 mm thick plate with 2.49 mm diameter holes drilled on a 9.65 mm 9.65 mm
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Z.L. Ji / Engineering Analysis with Boundary Elements 34 (2010) 690–696
60 Experiment BEM using Eq. (30) with Eqs. (27) & (29) BEM using Eqs. (26)-(29)
40
Reactive 2" packing 4" packing Dissipative
50 Transmission loss (dB)
Transmission loss (dB)
50
30 20 10
40 30 20 10 0
0 500
1000
1500 2000 Frequency (Hz)
2500
0
3000
Fig. 6. Transmission loss of straight-through perforated tube dissipative silencer (f ¼25.7%).
center-to-center pattern, resulting in a percentage open area of 4.2%. When the geometric parameters of perforates differ from their sample plate, the specific acoustic impedance of perforates may deviate from that of Sullivan and Crocker. In order to examine the accuracy of BEM predictions using the Sullivan and Crocker formula, the transmission loss predictions using Eq. (31) are included in Figs. 3 and 4. It may be seen that the transmission loss predictions using Eq. (31) deviate from the measurements in the mid to high frequency range. The comparisons demonstrated that Eqs. (26)–(29) are more suitable and accurate than Sullivan and Crocker formula for the acoustic attenuation prediction of perforated tube reactive silencers. The effect of sound-absorbing material correction on the acoustic attenuation predictions of the straight-through perforated tube dissipative silencer is illustrated in Figs. 5 and 6. It may be seen that the transmission loss predictions using Eq. (30) with Eqs. (27) and (29) show better agreements with the experiments than Eqs. (26)–(29) due to the consideration of sound-absorbing material correction in Eq. (30), especially for the lower porosity. Based on the above analyses and discussions, Eq. (30) will be used to calculate the acoustic attenuation performance of the hybrid silencers in the next section.
4.2. Acoustic attenuation analysis of hybrid silencers The hybrid expansion chambers are commonly used in the intake and exhaust systems of high power internal combustion engines as well as the intake and discharge systems of blowers in view of their acoustic and aerodynamic performance with economy consideration. This section will investigate numerically the acoustic attenuation characteristics of the typical hybrid expansion chamber silencers as shown in Fig. 1 and examine the effects of internal structure on the acoustic attenuation performance of the silencers. For all configurations, the present study in this section considers D¼457.2 mm (1800 ) and L¼609.6 mm (2400 ) for the inner diameter and length of expansion chamber, respectively, d ¼152.4 mm (600 ) for the inner diameter of inlet and outlet tubes, tw ¼1.0 mm for the wall thickness of perforated facing, dh ¼4.0 mm for the hole diameter, f ¼15% for the porosity of perforation, and the filling density of sound-absorbing material is 100 kg/m3 for the hybrid and dissipative silencers. Fig. 7 compares the transmission loss of the simple (reactive) expansion chamber, the expansion chamber with packed cylindrical shell and the dissipative expansion chamber. The
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 7. Transmission loss of expansion chamber silencer with packed cylindrical shell.
50
Transmission loss (dB)
0
Reactive Packed inlet end-plate Packed outlet end-plate Packed inlet & outlet end-plates
40 30 20 10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 8. Transmission loss of expansion chamber silencer with packed end-plate (s).
reactive expansion chamber exhibits the period attenuation behavior in the low frequency range and the very low attenuation in the high frequency range. The packed cylindrical shell changes the acoustic attenuation characteristics of the expansion chamber and improves the acoustic attenuation at most frequencies. Increasing the thickness of sound-absorbing material improves the acoustic attenuation performance of the expansion chamber and shifts the attenuation peak in transmission loss curve to lower frequency. Fig. 8 shows the effects of packed end-plate (s) on the transmission loss of expansion chamber. The packed end-plate (s) improves the acoustic attenuation of expansion chamber at most frequencies and raises the troughs in transmission loss curve. The expansion chamber with packed outlet end-plate reveals the similar behavior to the expansion chamber with packed inlet end-plate. Packed both inlet and outlet end-plates lead to higher acoustic attenuation than the packed single endplate only at most frequencies. The numerical results with varying thickness of sound-absorbing material for the expansion chamber with packed inlet end-plate are depicted in Fig. 9. Increasing the thickness of sound-absorbing material improves the acoustic attenuation at lower frequencies particularly at the first trough, and changes transmission loss in the middle frequency range, while the effect of thickness of sound-absorbing material on the high frequency acoustic attenuation is marginal. Transmission loss predictions of the simple reactive expansion chamber, the expansion chamber with the full-packed housing
ARTICLE IN PRESS Z.L. Ji / Engineering Analysis with Boundary Elements 34 (2010) 690–696
Transmission loss (dB)
50
expansion chamber provides a relative flat broad band acoustic attenuation, and deletes the pass frequencies of the reactive expansion chamber and improves the acoustic attenuation performance at most frequencies. For all BEM computations in Figs. 7–11, the axisymmetrical BEM is applied and the three-node quadratic elements are used to discretize the boundary generator. The maximum size of elements is less than 40 mm.
Reactive 2" packing 4" packing 6" packing
40 30 20
5. Conclusions
10 0 0
250
500
750
1000
1250
1500
Frequency (Hz) Fig. 9. Effect of thickness of sound-absorbing material on transmission loss of expansion chamber silencer with packed inlet end-plate.
60
Reactive 2" packing 4" packing Dissipative
Transmission loss (dB)
50 40
The substructure boundary element approach is developed to predict and analyze the acoustic attenuation characteristics of four typical hybrid expansion chamber silencers with perforated facing. The hybrid expansion chambers change the acoustic attenuation behavior of the simple reactive expansion chamber by switching to a broad-band acoustic attenuation, raise the troughs in transmission loss curves and improve the acoustic attenuation at most frequencies. Increasing the thickness of sound-absorbing material may improve the acoustic attenuation of the hybrid expansion chambers in the specific frequency range.
Acknowledgments
30
The author wishes to thank Professor A. Selamet at The Ohio State University for providing the experimental results.
20
References
10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 10. Transmission loss of expansion chamber silencer with full-packed housing.
60
Reactive 2" packing 4" packing Dissipative
50 Transmission loss (dB)
695
40 30 20 10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 11. Transmission loss of straight-through perforated tube hybrid silencer.
and the dissipative expansion chamber are presented in Fig. 10. Similarly, the full-packed housing changes transmission loss shape and improves the acoustic attenuation performance at most frequencies. Fig. 11 compares the transmission loss of the simple reactive expansion chamber, the straight-through hybrid expansion chamber and the dissipative expansion chamber. The hybrid
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[23] Bauer AB. Impedance theory and measurements on porous acoustic lines. J Aircraft 1977;14:720–8. [24] Ingard U. On the theory and design of acoustic resonators. J Acoust Soc Am 1953;25:1037–61. [25] Lee IJ, Selamet A, Huff NT. Impact of perforation impedance on the transmission loss of reactive and dissipative silencers. J Acoust Soc Am 2006;120:3706–13. [26] Sullivan JW, Crocker MJ. Analysis of concentric-tube resonators having unpartitioned cavities. J Acoust Soc Am 1978;64:207–15. [27] Sullivan JW. A method for modeling perforated tube muffler components, II. Applications. J Acoust Soc Am 1979;66:779–87. [28] Munjal ML. Acoustics of ducts and mufflers. New York: Wiley; 1987.
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Boundary element acoustic analysis of hybrid expansion chamber silencers with perforated facing Z.L. Ji School of Power and Energy Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 11 April 2009 Accepted 25 February 2010 Available online 2 April 2010
The substructure boundary element approach is developed to predict and analyze the acoustic attenuation characteristics of hybrid expansion chamber silencers with perforated facing. The silencers are divided into a number of acoustic domains with single medium (air or sound-absorbing material), and treating the sound-absorbing material as an equivalent fluid with complex-valued density and speed of sound (or complex-valued characteristic impedance and wavenumber), and then the boundary element method (BEM) may be applied to each domain leading to a system of equations in terms of acoustic pressure and particle velocity. Using the specific acoustic impedance of perforate, which takes into account the effect of sound-absorbing material, the relationship of acoustic pressures and particle velocities between the inlet and outlet of silencer may be obtained and then transmission loss is determined. For the straight-through perforated tube reactive and dissipative silencers, the predictions of transmission loss agree reasonably well with experimental measurements available in the literature, which demonstrated the applicability and accuracy of the present approach. The BEM is then used to investigate the effect of internal structure on the acoustic attenuation characteristics of hybrid expansion chamber silencers with perforated facing. The numerical results demonstrated that the hybrid expansion chambers may provide higher acoustic attenuation than the reactive expansion chamber in the mid to high frequency range. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method Substructure approach Hybrid silencer Perforated facing Acoustic attenuation performance Prediction Analysis
1. Introduction The sound-absorbing materials are widely used to improve the acoustic attenuation performance of silencers at mid to high frequencies. The studies demonstrated that the multi-dimensional approach is needed for the accurate prediction of acoustic attenuation performance of silencers at higher frequencies, while the simple onedimensional solution provides a reasonable accuracy at lower frequencies only. The two-dimensional axisymmetric and threedimensional analytical approaches based on the mode-matching techniques have been developed to determine the acoustic attenuation performance of the perforated dissipative silencers with regular geometry [1,2]; however, the analytical methods are difficult to be employed for the silencers with complex internal structures. The numerical methods such as finite element method (FEM) [3] and boundary element method (BEM) [4–12] are suitable to predict the acoustic attenuation performance of reactive and dissipative silencers and they are not confined to the geometry. BEM is an ideal tool for the acoustic analysis of silencers due to its surface-only meshing scheme, especially the use of substructure BEM approaches and
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impedance matrix synthesis can save a significant amount of computational time and memory space for the complex and largesize silencer analysis. In consideration of the fact that the coupling in the conventional multi-domain BEM is tedious, Wu and Wan [13], and Wu et al. [14–16] developed the direct mixed-body BEM for the analysis of reactive and dissipative silencers. In order to reduce the matrix size and the total computational time, Lou and Wu [17,18] have also developed the substructuring techniques based on the direct mixed-body BEM for the complex and large-size silencer analysis. The major advantage of the direct mixed-body BEM over the traditional multi-domain BEM approach is that only one side of the thin-wall components needs to be discretized; therefore may reduce the discretization effort and save computational time; however the thickness of these components is neglected. For the engine intake and exhaust silencers, the thickness of perforated facing is typically 1–2 mm, the effect of thickness of perforated facing on the acoustic attenuation prediction is marginal and may be neglected; therefore both the traditional multi-domain BEM approach and direct mixed-body BEM may be used to determine the acoustic attenuation performance of the silencers. However, for some special applications, the thickness of perforated pipe or plate may be not small enough and needs to be considered for the accurate prediction. In comparison with the direct mixed-body BEM, the multi-domain BEM approach needs to discretize the both sides of perforated components for the
ARTICLE IN PRESS Z.L. Ji / Engineering Analysis with Boundary Elements 34 (2010) 690–696
two neighboring substructures; therefore the thickness of perforated components may be considered in the numerical models. The expansion chamber silencers containing sound-absorbing material and perforated facing (tube and plate) are widely used in the intake and exhaust systems of internal combustion engines and fluid machines. The objective of the present study is then to develop the three-dimensional substructure boundary element approach to determine the transmission loss of typical hybrid expansion chamber silencers with perforated facing as shown in Fig. 1, and to investigate the effects of internal structure on the acoustic attenuation characteristics of the silencers.
2. Boundary element approach There are two media inside the hybrid expansion chamber silencers shown in Fig. 1: the air and sound-absorbing material. Assuming homogeneous sound-absorbing material and harmonic
L S2
S1
d
D
wave propagation in both media, and treating the soundabsorbing material as an equivalent fluid with complex-valued density and speed of sound, the continuity and momentum equations yield [1,9]
r2 p þ k2 p ¼ 0,
ð1Þ
where p is the acoustic pressure, k is the wavenumber and equals to k0 for the air and k~ for the sound-absorbing material, respectively. Eq. (1) may be represented in the form of the boundary integral equation [4,7] as Z @G jkzvðYÞGðX,YÞ þ pðYÞ ð2Þ ðX,YÞ dGðYÞ, CðXÞpðXÞ ¼ @n G where G is the boundary surface of the acoustic domain, n the unit normal vector on G directed away from the domain, j the imaginary unit, z the characteristic impedance of the medium (equals to z0 for the air and z~ for the sound-absorbing material, respectively), v the outward normal particle velocity and G(X,Y)¼exp( jkR)/4pR the Green’s function of free space: R being the distance between any two points X and Y in the domain or on the surface, and C(X) is a coefficient which depends on the position of point X. Numerical solution of the boundary integral Eq. (2) can be achieved by discretizing the boundary surface of the domain into a number of elements, and then a set of algebraic system of equations may be obtained by using numerical integration, which can be written in matrix form [7] as ½HfPg ¼ z½GfVg,
L S2 d
S3 S1
D
L
691
ð3Þ
where [H] and [G] are the coefficient matrices, and {P} and {V} are the vectors whose elements are the acoustic pressure p and outward normal particle velocity v on the boundary nodes, respectively. The detailed treatment of the BEM numerical solution procedure is provided elsewhere [4,7]. The substructure BEM is used to determine the acoustic attenuation performance of hybrid silencers. The silencer is divided into a number of substructures with single medium and then the BEM is applied to each one of these substructures leading to a system of equations. The objective of the following derivation is to develop the relationship between inlet variables (Pi,Vi) and outlet variables (Po,Vo) which then facilitates the calculation of four-pole parameters and transmission loss of the silencers.
S2 2.1. Expansion chamber with packed cylindrical shell
d
S1
D
The expansion chamber with packed cylindrical shell is shown in Fig. 1(a). The silencer is divided into two substructures: the center air domain S1 and the outer sound-absorbing material domain S2. Eq. (3) combined with the rigid wall boundary condition yields [7,10] 8 S 9 8 S 9 2 S1 S1 S1 3 T11 T12 T13 > > V 1> P 1> > > < i > < i > = = 7 6 S1 S1 S1 7 S1 6 ¼ z0 4 T21 T22 T23 5 VoS1 , ð4Þ Po > > > > > S1 S1 S1 > : P S1 > : V S1 > ; ; T31 T32 T33 p p
D
fPpS2 g ¼ z~ ½T S2 fVpS2 g,
L S3 S2 d
S1
Fig. 1. Configurations of hybrid silencers: (a) expansion chamber with packed cylindrical shell, (b) expansion chamber with packed end-plates, (c) expansion chamber with full-packed housing and (d) straight-through hybrid expansion chamber.
ð5Þ
where the subscripts i, o and p represent the inlet, outlet and perforation, respectively. Introducing the specific acoustic impedance zp for the perforated facing, the boundary conditions may be expressed as fVpS1 g ¼ fVpS2 g
and
fPpS1 gfPpS2 g ¼ z0 zp fVpS1 g:
ð6; 7Þ
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9 8 < PpS2 =
From Eqs. (4) to (7), one may have 2 3 ( S1 ) S1 S1 S1 S1 S1 S1 ( S1 ) T11 þT13 ZT31 T12 þ T13 ZT32 Pi Vi 5 ¼ z0 4 S1 , S1 S1 S1 S1 S1 S1 T21 þT23 ZT31 T22 þ T23 ZT32 Po VoS1
1
ð8Þ
matrix matrix between the inlet and outlet of the silencer. The four-pole parameters and transmission loss calculation procedure are identical to those used earlier [7] and will not be repeated here.
The expansion chamber with packed end-plates as shown in Fig. 1(b) is divided into three substructures: the center air domain S1, left sound-absorbing material domain S2 and right soundabsorbing material domain S3. Eq. (3) combined with the rigid wall boundary condition yields 8 S 9 8 S 9 2 S1 S1 S1 3 T11 T12 T13 > > V 1> P1> > > = = < i > < i > 6 7 S S S T211 T221 T231 7 ð9Þ PoS1 ¼ z0 6 VoS1 , 4 5 > > > S > > > S1 S1 S1 > 1 ; ; : PS1 > : V T T T 32
33
ð10Þ
fPpS3 g ¼ z~ ½T S3 fVpS3 g:
ð11Þ
Eqs. (10) and (11) may be rewritten as fPpS23 g ¼ z~ ½T S23 fVpS23 g,
ð12Þ
where 9 8 < VpS2 = fVpS23 g ¼ , : VpS3 ;
"
½T
S23
T S2 ¼ 0
# 0 : T S3
On the perforated facings between substructures S1 and S2, as well as S1 and S3, the boundary conditions may be expressed as fVpS1 g ¼ fVpS23 g
and
ð16Þ
ð17Þ
On the perforated facings between substructures S1 and S2, as well as S2 and S3, the boundary conditions may be expressed as fVpS11 g ¼ fVpS12 g
and
fPpS11 gfPpS21 g ¼ z0 zp fVpS11 g,
ð18; 19Þ
fVpS33 g ¼ fVpS32 g
and
fPpS33 gfPpS23 g ¼ z0 zp fVpS33 g:
ð20; 21Þ
fPpS21 g ¼ z~ ½T S23 fVpS12 g,
ð22Þ
S2 S2 S2 1 S2 ½T11 þ ðz~ =z0 ÞT12 ½zp IS3 T S3 ðz~ =z0 ÞT22 T21 , S3
S23
and ½IS3 where ½T ¼ being the unity matrix with same order as ½T . Therefore, the same expression as Eq. (8) may be obtained, here S1 ðz~ =z0 ÞT S23 1 . ½Z ¼ ½zp Ip T33 3. Acoustic properties of sound-absorbing material and perforated facing
p
fPpS2 g ¼ z~ ½T S2 fVpS2 g,
9 8 < PpS2 = , fPpS23 g ¼ : PpS3 ;
S2 T21
Combining Eqs. (16), (17), (20) and (21) leads to
2.2. Expansion chamber with packed end-plates
31
¼ z~ 4
9 38 S2 < S2 = Vp1 T12 5 S2 : S2 ;, T22 Vp3
S2 T11
fPpS33 g ¼ z0 ½T S3 fVpS33 g:
S1 where ½Z ¼ ½zp Ip T33 ðz~ =z0 ÞT S2 1 , and [Ip] being the unity S1 . Eq. (8) defines the impedance with same order as ½T33
p
: PpS23 ;
2
fPpS1 gfPpS23 g ¼ z0 zp fVpS1 g:
ð13; 14Þ
Eqs. (9), (12)–(14) may be combined to obtain the impedance matrix between the inlet and outlet of the silencer, which is same S1 ðz~ =z0 ÞT S23 1 . as Eq. (8) with ½Z ¼ ½zp Ip T33 2.3. Expansion chamber with full-packed housing The expansion chamber with full-packed housing (i.e. packed cylindrical shell and two end-plates) as shown in Fig. 1(c) is divided into two substructures: the center air domain S1 and the outer sound-absorbing material domain S2. Then, the same treatment as the expansion chamber with packed cylindrical shell may be used to obtain the impedance matrix between the inlet and outlet of the silencer.
The complex-valued acoustic impedance and wavenumber of sound-absorbing material may be calculated usually by empirical expressions [19] or determined by experiments [20]. The present study uses texturized fiberglass roving as the sound-absorbing material. The complex acoustic impedance z~ and the complex wavenumber k~ of the sound-absorbing material with the filling density of 100 kg/m3 are expressed as [21] z~ =z0 ¼ 1:0 þ 33:20f 0:7523 j28:32f 0:6512 ,
ð23Þ
0:6841 ~ k=k j38:39f 0:6285 , 0 ¼ 1:0 þ 39:20f
ð24Þ
where f is the frequency. In order to predict the acoustic attenuation performance of silencers with perforated facing, the specific acoustic impedance of perforated facing must be obtained first. The specific acoustic impedance of perforated plate is defined [22] as
zp ¼
pi po
r0 c0 ui
or
zp ¼
pi po
r0 c0 uo
,
ð25Þ
where pi, po, ui and uo are the average sound pressures and particle velocities on each side of the perforated plate, r0 the density of air and c0 the speed of sound. Combining the mass conservation equation, Eq. (25) may be rewritten as
zp ¼
pi po uh
r0 c0 uh ui
¼
zh Rh þjXh ¼ , f f
ð26Þ
where uh is the particle velocity within the hole, zh the specific acoustic impedance of a single hole, f the porosity, j the imaginary unit and Rh, Xh the specific acoustic resistance and reactance of a single hole, respectively, which may be expressed as [22,23] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ Rh ¼ ð1 þ tw =dh Þ 8k0 m=z0 ,
2.4. Straight-through hybrid expansion chamber Xh ¼ k0 ðtw þ adh Þ, The straight-through hybrid expansion chamber as shown in Fig. 1(d) is divided into three substructures: the center air passage S1, sound-absorbing material domain S2 and outer air cavity S3. Eq. (3) combined with the rigid wall boundary condition yields 8 S 9 2 S1 S1 S1 38 S1 9 T11 T12 T13 > > V > P1> > > < i > < i > = = 6 S1 S1 S1 7 S1 7 T T T , ð15Þ PoS1 ¼ z0 6 V o 21 22 23 5 4 > S > > S > > > > S1 S1 S1 > 1 ; :P 1 ; : V T31 T32 T33 p1 p1
ð28Þ
where m is the dynamic viscosity of air, tw the thickness of perforated plate, dh the diameter of hole and a the end correction coefficient of a single hole. Based on the piston-driven model, Ingard [24] derived the formulation for the end correction coefficient of a circular hole in a rectangular tube as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 X J12 p ðmxÞ2 þðnZÞ2 X 4 1 0 a¼ 2 e , ð29Þ p ðxZÞ1=2 m ¼ 0 n ¼ 0 mn ½m2 ðh=bÞ þ n2 ðb=hÞ3=2
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zp ¼
~ ~ =z0 Þdh g Rh þ jk0 ftw þ 0:5a½1þ ðk=k 0 Þðz
f
:
50
Transmission loss (dB)
where x ¼dh/b, Z ¼dh/h, b and h are the distances between two neighboring holes in two directions, the prime on the summation sign implies that the term representing the (0, 0) fundamental mode is excluded, emn ¼1 if ma0 and na0 or emn ¼ 1/2 if otherwise, and J1 is the Bessel function of the first kind of order 1. In the presence of sound-absorbing material, Eq. (26) should be modified accordingly. In view of the fact that one side of the perforated facing is the air and another side is the soundabsorbing material, expression (26) is modified as
693
Experiment BEM using Eqs. (26)-(29) BEM using Eq. (31)
40 30 20 10
ð30Þ
0 0
500
1000
1500 2000 Frequency (Hz)
4. Results and discussions
2500
3000
Fig. 3. Transmission loss of straight-through perforated tube reactive silencer (f ¼8.4%).
L
50
Transmission loss (dB)
In order to validate the present substructure boundary element approach as well as to assess the applicability and accuracy of expressions (26)–(30) for the acoustic attenuation predictions of silencers with perforated facing, the straightthrough perforated tube silencer as shown in Fig. 2 (which may be considered as a special case of the configuration 1(a), where the diameter of perforated tube equals to diameters of inlet and outlet tubes) is considered. The silencer has a cylindrical concentric configuration, and the dimensions are D¼164.4 mm and L¼257.2 mm for the inner diameter and length of expansion chamber, respectively, d¼ 49.0 mm and tw ¼0.9 mm for the inner diameter and wall thickness of perforated tube, respectively, dh ¼4.98 mm for the diameter of hole. Transmission loss of the straight-through perforated tube silencer without and with sound-absorbing material filling (100 kg/m3) is predicted by the present approach. In view of the axisymmetrical configuration of the silencer, the axisymmetrical BEM [7] is applied and the three-node quadratic elements are used to discretize the boundary generator. The maximum size of elements is less than 12.5 mm. Figs. 3–6 compare the transmission loss predictions from the present BEM using Eqs. (26)–(30) and experimental results in the literature [25] for two different porosities (f ¼ 8.4% and 25.7%). The BEM predictions for the straight-through perforated tube reactive and dissipative silencers show reasonable agreements with experiments in the entire frequency range of interest. The discrepancy between BEM predictions and experimental results may possibly be attributed to (1) the texturization condition and uniformity of soundabsorbing material filling inside the silencer in the experiments, and (2) the approximations involved in the expressions for the perforate specific acoustic impedance and sound-absorbing material properties. The following empirical expression for the specific acoustic impedance of perforate suggested by Sullivan and Crocker [26] is widely used to calculate the acoustic attenuation performance of
Experiment BEM using Eqs. (26)-(29) BEM using Eq. (31)
40 30 20 10 0 0
500
1000
1500 2000 Frequency (Hz)
2500
3000
Fig. 4. Transmission loss of straight-through perforated tube reactive silencer (f ¼25.7%).
50
Transmission loss (dB)
4.1. Validation
Experiment BEM using Eq. (30) with Eqs. (27) & (29) BEM using Eqs. (26)-(29)
40 30 20 10 0 0
500
1000
1500 2000 Frequency (Hz)
2500
3000
Fig. 5. Transmission loss of straight-through perforated tube dissipative silencer (f ¼8.4%).
silencers with perforated elements [27,28]
d
D
zp ¼
Fig. 2. Straight-through perforated tube silencer (dissipative expansion chamber).
0:006 þ jk0 ðtw þ0:75dh Þ
f
:
ð31Þ
This expression was obtained for a small (1600 mm2) sample of perforated plate. The test sample consisted of a 0.81 mm thick plate with 2.49 mm diameter holes drilled on a 9.65 mm 9.65 mm
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60 Experiment BEM using Eq. (30) with Eqs. (27) & (29) BEM using Eqs. (26)-(29)
40
Reactive 2" packing 4" packing Dissipative
50 Transmission loss (dB)
Transmission loss (dB)
50
30 20 10
40 30 20 10 0
0 500
1000
1500 2000 Frequency (Hz)
2500
0
3000
Fig. 6. Transmission loss of straight-through perforated tube dissipative silencer (f ¼25.7%).
center-to-center pattern, resulting in a percentage open area of 4.2%. When the geometric parameters of perforates differ from their sample plate, the specific acoustic impedance of perforates may deviate from that of Sullivan and Crocker. In order to examine the accuracy of BEM predictions using the Sullivan and Crocker formula, the transmission loss predictions using Eq. (31) are included in Figs. 3 and 4. It may be seen that the transmission loss predictions using Eq. (31) deviate from the measurements in the mid to high frequency range. The comparisons demonstrated that Eqs. (26)–(29) are more suitable and accurate than Sullivan and Crocker formula for the acoustic attenuation prediction of perforated tube reactive silencers. The effect of sound-absorbing material correction on the acoustic attenuation predictions of the straight-through perforated tube dissipative silencer is illustrated in Figs. 5 and 6. It may be seen that the transmission loss predictions using Eq. (30) with Eqs. (27) and (29) show better agreements with the experiments than Eqs. (26)–(29) due to the consideration of sound-absorbing material correction in Eq. (30), especially for the lower porosity. Based on the above analyses and discussions, Eq. (30) will be used to calculate the acoustic attenuation performance of the hybrid silencers in the next section.
4.2. Acoustic attenuation analysis of hybrid silencers The hybrid expansion chambers are commonly used in the intake and exhaust systems of high power internal combustion engines as well as the intake and discharge systems of blowers in view of their acoustic and aerodynamic performance with economy consideration. This section will investigate numerically the acoustic attenuation characteristics of the typical hybrid expansion chamber silencers as shown in Fig. 1 and examine the effects of internal structure on the acoustic attenuation performance of the silencers. For all configurations, the present study in this section considers D¼457.2 mm (1800 ) and L¼609.6 mm (2400 ) for the inner diameter and length of expansion chamber, respectively, d ¼152.4 mm (600 ) for the inner diameter of inlet and outlet tubes, tw ¼1.0 mm for the wall thickness of perforated facing, dh ¼4.0 mm for the hole diameter, f ¼15% for the porosity of perforation, and the filling density of sound-absorbing material is 100 kg/m3 for the hybrid and dissipative silencers. Fig. 7 compares the transmission loss of the simple (reactive) expansion chamber, the expansion chamber with packed cylindrical shell and the dissipative expansion chamber. The
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 7. Transmission loss of expansion chamber silencer with packed cylindrical shell.
50
Transmission loss (dB)
0
Reactive Packed inlet end-plate Packed outlet end-plate Packed inlet & outlet end-plates
40 30 20 10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 8. Transmission loss of expansion chamber silencer with packed end-plate (s).
reactive expansion chamber exhibits the period attenuation behavior in the low frequency range and the very low attenuation in the high frequency range. The packed cylindrical shell changes the acoustic attenuation characteristics of the expansion chamber and improves the acoustic attenuation at most frequencies. Increasing the thickness of sound-absorbing material improves the acoustic attenuation performance of the expansion chamber and shifts the attenuation peak in transmission loss curve to lower frequency. Fig. 8 shows the effects of packed end-plate (s) on the transmission loss of expansion chamber. The packed end-plate (s) improves the acoustic attenuation of expansion chamber at most frequencies and raises the troughs in transmission loss curve. The expansion chamber with packed outlet end-plate reveals the similar behavior to the expansion chamber with packed inlet end-plate. Packed both inlet and outlet end-plates lead to higher acoustic attenuation than the packed single endplate only at most frequencies. The numerical results with varying thickness of sound-absorbing material for the expansion chamber with packed inlet end-plate are depicted in Fig. 9. Increasing the thickness of sound-absorbing material improves the acoustic attenuation at lower frequencies particularly at the first trough, and changes transmission loss in the middle frequency range, while the effect of thickness of sound-absorbing material on the high frequency acoustic attenuation is marginal. Transmission loss predictions of the simple reactive expansion chamber, the expansion chamber with the full-packed housing
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Transmission loss (dB)
50
expansion chamber provides a relative flat broad band acoustic attenuation, and deletes the pass frequencies of the reactive expansion chamber and improves the acoustic attenuation performance at most frequencies. For all BEM computations in Figs. 7–11, the axisymmetrical BEM is applied and the three-node quadratic elements are used to discretize the boundary generator. The maximum size of elements is less than 40 mm.
Reactive 2" packing 4" packing 6" packing
40 30 20
5. Conclusions
10 0 0
250
500
750
1000
1250
1500
Frequency (Hz) Fig. 9. Effect of thickness of sound-absorbing material on transmission loss of expansion chamber silencer with packed inlet end-plate.
60
Reactive 2" packing 4" packing Dissipative
Transmission loss (dB)
50 40
The substructure boundary element approach is developed to predict and analyze the acoustic attenuation characteristics of four typical hybrid expansion chamber silencers with perforated facing. The hybrid expansion chambers change the acoustic attenuation behavior of the simple reactive expansion chamber by switching to a broad-band acoustic attenuation, raise the troughs in transmission loss curves and improve the acoustic attenuation at most frequencies. Increasing the thickness of sound-absorbing material may improve the acoustic attenuation of the hybrid expansion chambers in the specific frequency range.
Acknowledgments
30
The author wishes to thank Professor A. Selamet at The Ohio State University for providing the experimental results.
20
References
10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 10. Transmission loss of expansion chamber silencer with full-packed housing.
60
Reactive 2" packing 4" packing Dissipative
50 Transmission loss (dB)
695
40 30 20 10 0 0
250
500
750 1000 Frequency (Hz)
1250
1500
Fig. 11. Transmission loss of straight-through perforated tube hybrid silencer.
and the dissipative expansion chamber are presented in Fig. 10. Similarly, the full-packed housing changes transmission loss shape and improves the acoustic attenuation performance at most frequencies. Fig. 11 compares the transmission loss of the simple reactive expansion chamber, the straight-through hybrid expansion chamber and the dissipative expansion chamber. The hybrid
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