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A Compound Micro-perforated Panel Sound Absorber with Partitioned Cavities of Different Depths
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 78 (2015) 1617 – 1622
6th International Building Physics Conference, IBPC 2015
A compound micro-perforated panel sound absorber with partitioned cavities of different depths Wencheng Guoa, Hequn Mina * a
Key Laboratory of Urban and Architectural Heritage Conservation, Ministry of Education School of Architecture, Southeast University, Nanjing 210096, China
Abstract The micro-perforated panel (MPP) sound absorber is becoming a good replacement of conventional porous absorbers in building noise control. The MPP absorber is fiber-free and avoids potential harms such as powdered fibers and damage to the person’s health. This paper presents a new compound MPP sound absorber architecture with excellent sound absorption performance both on higher sound absorption coefficient and broader absorption band compared with single MPP absorber. The absorber consists of an array of parallel-arranged MPP sub-absorbers with different depths, where different local resonances are apparent and dominate. In this paper, firstly, an analytical method is proposed to look into the mechanism of the absorber architecture design and this method is validated through simulations based on a finite element procedure. Secondly, the acoustic properties from different MPP absorber dimension sets are investigated through simulations with the proposed theoretical method. Results show that the presented MPP absorber can have normal incidence sound absorption coefficients higher than 0.5 over the frequencies from 370Hz to 2520 Hz, with maximum value of 0.9. This indicates that the proposed MPP absorber provides a good alternative for indoor sound absorption applications in commercial and industrial buildings. © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license ©2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL. Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL Keywords: Sound absorber; Micro-perforated panel; Quadratic residue diffuser; Noise control
1. Introduction Micro-perforated panel absorber (MPP) was first proposed by Maa in 1975 [1], which has many excellent performances, such as fireproofing, moisture-proofing and can adapt to high-speed air flow. This absorber also has enough acoustic resistance and sufficiently low acoustic reactance to provide good absorption properties by reducing * Corresponding author. Tel.: +86-15996367978 E-mail address: [email protected]
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL doi:10.1016/j.egypro.2015.11.238
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
the orifice diameter to sub-millimetre scale. A typical MPP absorber consists of two sections, one is the front perforated panel and the other is the backing cavity, based on Helmholtz absorption effect. Some researchers studied the possibility of extending the absorption bandwidth of the MPP absorber through modifications upon the backing cavity depths [2, 3]. However, the MPP absorber performance dependent of the backing cavity architecture and dimensions are not completely clear yet for the practical applications. This motivates the present study in this paper. The quadratic residue diffuser (QRD) is widely used to increase the diffusivity in auditoria [4]. In one period, the depths of its elements vary according to a pseudo-stochastic sequence, as shown in Eq. 1. This sequence may provide a good way to design the MPP absorber backing cavities, and more importantly, QRD is also having absorption properties first observed by Fuiwara and Miyajima [5, 6]. Their findings were then verified by Kuttruff [7]. F. P. Mechel thoroughly investigated the absorption mechanism of QRD [8], showing the potential of such diffusers to be used as sound absorbers with an extra resistive layer covered on the diffuser opening. T. Wu, T. J. Cox, and Y. W. Lam extended Mechel’s work and proposed a numerical optimization method to tune the QRD depths sequence for high absorption performance of it [9]. In this paper, a compound sound absorber is proposed through covering MPP over a QRD opening, whose onedimensional model is shown in Fig. 1. To look into the absorber’s absorption mechanism analytically, in this paper a QRD with seven wells (N0=7 for one period) is employed, in which the ratio of depths between those wells are calculated from Eq. 1, with 0:1:4:2:2:4:1 for the prime number 7. For simplicity, the well with zero depth is excluded in the QRD-backed MPP absorber. That is, only six wells are considered in the QRD design, whose depths are determined through 2
c mod ( n , N 0 )
, n 0,1, " N 0 1. (1) N 0 (2 f r ) where ln is the depth of the well, N0 is the prime number of diffuser, c is the speed of sound in air, and fr is the design frequency. ln
Fig.1. One-dimensional QRD-backed MPP absorber with two periods
2. Theoretical method 2.1. Theoretical basis The acoustic impedance by the MPP and QRD wells is first discussed to predict the normal incidence absorption coefficient of the presented sound absorber. The MPP can be regarded as an extra resistive layer applied on the opening of the QRD to enhance the absorption. For normal incidence of sound wave on the panel, the specific acoustic impedance is [1]: 'p Z1 (2) u where ǻp is the sound pressure difference between the ends of the panel, u is average particle velocity. The relative (to the characteristic impedance ȡ0c in air) acoustic impedance of the MPP, ZMPP, can be evaluated according to studies [1]. The QRD is actually a compound of several parallel arrangement narrow wells with different depths, Morse and Ingard derived the complex wave number in a cylindrical tube and it also can be used for those narrow wells. The inward impedance on the surface of the well with length ln can be derived as [9, 10]:
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
j Ue c
Z (l n )
k0 kt
(3)
cot( kt ln ),
where: 2
2
§ S q · | k k0 (1 j )[ d (J 1) d ], ¸ v h 0 2b © b ¹
k0 ¨
kt
k0
Z / c,
U e U 0 [1 (1 j ) d v / b ] in which Ȗ=7/5 for air. Parameters dv and dh are [10]: 2u
dv
1
| 0.21
U 0Z
dh
,
f
2K
U 0Z C p
| 0.25
(4) (5)
1
(cm).
(6)
f
where f is sound frequency, K, ȝ and Cp are the properties of gas. K is the thermal conductivity, and ȝ is the coefficient of viscosity, Cp is the heat capacity per unit mass at constant pressure. Thus the inward normalized specific impedance of the well with length ln is [9]: Z (l n ) (7) j{1 (1 j )[ d v (J 1) d h ] / 2b}cot( kt ln ). ] (l n ) Uo c 2.2. Absorption prediction model of the QRD-backed MPP absorber As shown in Fig. 1, the sound field in front of the absorber is decomposed into the incident plane wave pe(x,z), and scattered field ps(x,z), including arbitrary incident angles, but only șe=0 (the normal incidence) is investigated in this paper. The followed analysis method is proposed by Mechel [8]: (8) p ( x, z ) pe ( x, z ) ps ( x, z ),
pe ( x, z ) where k x
Pe e
j ( x kx z kz )
k 0 sin T e , f
ps ( x , z )
¦ Ae
kz
, k 0 cos T e ,
j ( x En z J n )
n
(9)
.
(10)
n f
Since the absorber is periodic, the scattered field is also periodic in x. Therefore the wave numbers in the x and z directions of the spatial harmonics are (the first from the periodicity, the latter from the wave equation): 2S En kx n ; T 2
2
Jn
2
k0 E n
j k0
§ sin T n O · 1 , ¨ ¸ e T¹ ©
(11)
where Ȝ=2ʌ/k0 is the wavelength. For Eq. 11, when the quantity under the square root is negative, those radiating harmonics, corresponding to the indices ns, can propagate to the far field: T T (1 sin T e ) d ns d (1 sin T e ). (12)
O
O
An admittance function G(x) is employed to describe the diffuser, which can be given through combining the normalized specific impedance of MPP and QRD: 1 (13) , G ( x) z MPP ] (ln ) Then together with the admittance function G(x)ˈthe relation of pressure and particle velocity along the
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
positive z direction on the surface is U 0 cvz ( x, 0) f
cos T e Pe
Jn
¦k
n f
An e j x n 2S /T
0
ª ¬
G ( x) « Pe
G ( x ) p ( x, 0) .This gives
f
¦ Ae
j x n 2S / T
n
n f
º »¼ .
(14)
Since G(x) is periodic with a period T, we apply a Fourier analysis: f
¦ge
G ( x)
j n (2S / T ) x
n
1 T G ( x)e j n (2S /T ) x dx. ³ 0 T
; gn
n f
(15)
This inserted into the boundary condition given by Eq.14, after multiplication by ejm (ʌ7) x and integration over T:
§
f
¦ A ¨g n
n f
©
mn
§ Jn ·· ¸ ¸ Pe (G m ,0 cos T e g m ); © k0 ¹ ¹
G m,n ¨
m
f, "" , f
(16)
where
G m,n
^
1,
m n
0,
mzn
.
The value of m, n will be terminated at ±2*N for convergence, where N is the number of wells in one period. By solving the above equations, the coefficients An can be obtained, and the absorption coefficient of QRD-backed MPP absorber is then
D (T e ) 1
A0 Pe
2
1 cos T e
¦
ns z 0
An
2 s
Pe
1 (sin T e ns O / T ) 2 .
(17)
the summation runs over radiating spatial harmonics only. Evidently the second term is the specular reflection, and the third term is due to scattering. 3. Results and discussions 3.1. Numerical Validations Prediction of the normal incidence absorption coefficient of the presented absorber is numerically validated through a finite element procedure [3]. The mesh geometry in the finite element model is shown in Fig. 2, the effect caused by vibration is neglected because the panel is assumed to be rigid.
Fig.2. Mesh geometry in the finite element model of the QRD-backed MPP absorber with six parallel arrangement cavities
There are two acoustic domains (the incident duct and backing cavity) and one acoustic structure (the MPP) as shown in the finite element model. The sound field in the model is determined by the Helmholtz equation:
§ 2 1 w2 · ¨ c w t 2 ¸ I 0, © ¹
(18)
where I is the total velocity potential of the sound field and is determined by sound pressure p and acoustic particle velocity u through [3]:
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
U0
p
wI wt
,
u
I .
(19)
Table. 1. Properties of the QRD-backed MPP absorber used in the investigations Orifice diameter Panel Thickness Perforation ratio
Cavity depths (mm)
Case 1
0.5mm
0.5mm
1%
5, 20, 10, 50, 100, 25
Case 2
0.5mm
0.5mm
1%
25, 100, 50, 50,100, 25
A commercial finite element software, COMSOL Multiphysics, was used to implement this numerical model. The model parameters of the proposed absorber are listed in Table 1. Take Case1 as example, the computational domains are discretized with triangular quadratic-Lagrange elements, and a total of 3429 elements in one period are defined. Another two and ten periods model is also simulated by this procedure. Analytical and numerical results of the normal incidence absorption coefficients of the absorber are compared and shown in Fig. 3. It is shown that, results from the analytical prediction method agree excellently with those from the finite element model at frequencies lower than about 1800 Hz. Beyond this frequency, distinct deviation between these two predictions can be observed. For the corresponding reason, extensive numerical simulations by the finite element model are conducted, in which the considered period number (T) in the model geometry increases to be 2 and 10. The corresponding numerical results are also shown in Fig. 3. It is shown that, with the period number increasing, the deviation between the analytical and numerical prediction models decreases markedly. This indicates that, such deviation comes from the numerical errors in the finite element model, in which the proposed absorber with infinite period numbers has to be simulated as one with a finite but larger enough period numbers. It also indicates that the analytical prediction method presented in Sec. 2 is valid for the normal incidence absorption coefficients of the proposed absorber. 1
Normal Incidence Absorption coefficient
Normal Incidence Absorption coefficient
1
0.8
0.6
0.4
Case 1 FEM 1,T=1 FEM 2,T=2 FEM 3,T=10
0.2
0
0.6
0.4
0.2
0 0
500
1000
1500 Frequency,Hz
2000
2500
3000
Fig.3. Comparison of the predicted normal incidence absorption coefficients by the proposed method (Case1) and the finite element model (FEM). FEM1-3 denotes the period of simulation T=1, 2, 10, respectively
Case 1 Case 2
0.8
0
500
1000
1500 2000 Frequency,Hz
2500
3000
Fig.4. Comparison of the predicted normal incidence absorption coefficients by typical QRD-backed MPP absorber (Case2) and modified QRD-backed MPP absorber (Case1).
3.2. Simulation cases Sound absorption performance of the proposed absorber is investigated in two cases based on the analytical prediction method, to preliminarily understand the property influence from the absorber parameter set. Parameters in both cases are shown in Fig. 4,with different depths of the backing well cavities. In the Case 2, well parameters follow strictly those of the pseudo-stochastic sequence, while in the Case 1, some modifications are conducted to broaden local resonances at a wider frequency band. The parameters of the two cases are listed on Table 1. In both cases, parameters of the MPP are the same. As shown in Fig. 4, the dash curve has three absorption peaks laid on 480Hz, 670Hz and 970Hz, corresponding to 100mm, 50mm and 25 mm depth respectively. Each absorption peak is above 0.9 due to a good impedance matching condition. But aiming for the enhanced absorption performance of the QRD-backed MPP absorber, the solid curve is preferred, whose absorption bandwidth is almost twice of the dash curve. The main difference
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
between case 1 and case 2 is the cavity depths, indicating the great potential of it to obtain a better absorption performance. 4. Conclusion A compound fiber-free sound absorber architecture was presented through covering MPP over the opening of a QRD. An analytical prediction method was proposed to look into the theoretical mechanism of this absorber and this method was validated through numerical simulations based on the finite element method. Simulation cases have been carried out to investigate the absorption performance dependent of the absorber architecture and dimension set. Results show that the presented MPP absorber can have normal incidence sound absorption coefficients higher than 0.5 over the frequencies from 370Hz to 2520 Hz. Acknowledgements This work is supported by the Natural Science Foundation of China (Grant No. 51408113) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140632). References [1] D. Y. Maa. Theory and design of microperforated panel sound absorbing construction. (in Chinese) 1975; Sci. Sin. 18:55–71. [2] D. Y. Maa. Micorperforated Panel wide-band absorber. Noise Control Eng. J. 1987; 29: 77–84. [3] Chunqi Wang and Lixi Huang. On the acoustic properties of parallel arrangement of multiple micro-perforated panel absorbers with different cavity depths. J. Acoust. Soc. Am 2011; 130(1): 208-218. [4] M. R. Schroeder. Binaural dissimilarity and optimum ceilings for concert halls: More lateral sound diffusion. J. Acoust. Soc. Am. 1979; 65: 958–963. [5] K. Fujiwara and T. Miyajima. Absorption characteristics of a practically constructed Schroeder diffuser of quadratic-residue type. Appl. Acoust. 1992; 35:149–152. [6] K. Fujiwara and T. Miyajima. A study of the sound absorption of a quadratic-residue type diffuser. Acustica 1995; 81: 370–378. [7] H. Kuttruff. Sound absorption by pseudostochastic diffusers (Schroeder diffusers). Appl. Acoust. 1994; 42: 215–231. [8] F. P. Mechel. The wide-angle diffuser—A wide-angle absorber?. Acustica 1995; 81:379–401 [9] T. Wu, T. J. Cox, and Y. W. Lam. From a profiled diffuser to an optimized absorber. J. Acoust. Soc. Am 2000; 108(2): 643-650. [10] P. M. Morse and K. Ingard. Theoretical Acoustics (McGraw-Hill, New York) 1968; Chap. 9:pp. 519–522. [11] P. M. Morse and K. Ingard, Theoretical Acoustics (McGraw-Hill, New York) 1968; Chap. 6: pp. 285–291.
ScienceDirect Energy Procedia 78 (2015) 1617 – 1622
6th International Building Physics Conference, IBPC 2015
A compound micro-perforated panel sound absorber with partitioned cavities of different depths Wencheng Guoa, Hequn Mina * a
Key Laboratory of Urban and Architectural Heritage Conservation, Ministry of Education School of Architecture, Southeast University, Nanjing 210096, China
Abstract The micro-perforated panel (MPP) sound absorber is becoming a good replacement of conventional porous absorbers in building noise control. The MPP absorber is fiber-free and avoids potential harms such as powdered fibers and damage to the person’s health. This paper presents a new compound MPP sound absorber architecture with excellent sound absorption performance both on higher sound absorption coefficient and broader absorption band compared with single MPP absorber. The absorber consists of an array of parallel-arranged MPP sub-absorbers with different depths, where different local resonances are apparent and dominate. In this paper, firstly, an analytical method is proposed to look into the mechanism of the absorber architecture design and this method is validated through simulations based on a finite element procedure. Secondly, the acoustic properties from different MPP absorber dimension sets are investigated through simulations with the proposed theoretical method. Results show that the presented MPP absorber can have normal incidence sound absorption coefficients higher than 0.5 over the frequencies from 370Hz to 2520 Hz, with maximum value of 0.9. This indicates that the proposed MPP absorber provides a good alternative for indoor sound absorption applications in commercial and industrial buildings. © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license ©2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL. Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL Keywords: Sound absorber; Micro-perforated panel; Quadratic residue diffuser; Noise control
1. Introduction Micro-perforated panel absorber (MPP) was first proposed by Maa in 1975 [1], which has many excellent performances, such as fireproofing, moisture-proofing and can adapt to high-speed air flow. This absorber also has enough acoustic resistance and sufficiently low acoustic reactance to provide good absorption properties by reducing * Corresponding author. Tel.: +86-15996367978 E-mail address: [email protected]
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the CENTRO CONGRESSI INTERNAZIONALE SRL doi:10.1016/j.egypro.2015.11.238
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
the orifice diameter to sub-millimetre scale. A typical MPP absorber consists of two sections, one is the front perforated panel and the other is the backing cavity, based on Helmholtz absorption effect. Some researchers studied the possibility of extending the absorption bandwidth of the MPP absorber through modifications upon the backing cavity depths [2, 3]. However, the MPP absorber performance dependent of the backing cavity architecture and dimensions are not completely clear yet for the practical applications. This motivates the present study in this paper. The quadratic residue diffuser (QRD) is widely used to increase the diffusivity in auditoria [4]. In one period, the depths of its elements vary according to a pseudo-stochastic sequence, as shown in Eq. 1. This sequence may provide a good way to design the MPP absorber backing cavities, and more importantly, QRD is also having absorption properties first observed by Fuiwara and Miyajima [5, 6]. Their findings were then verified by Kuttruff [7]. F. P. Mechel thoroughly investigated the absorption mechanism of QRD [8], showing the potential of such diffusers to be used as sound absorbers with an extra resistive layer covered on the diffuser opening. T. Wu, T. J. Cox, and Y. W. Lam extended Mechel’s work and proposed a numerical optimization method to tune the QRD depths sequence for high absorption performance of it [9]. In this paper, a compound sound absorber is proposed through covering MPP over a QRD opening, whose onedimensional model is shown in Fig. 1. To look into the absorber’s absorption mechanism analytically, in this paper a QRD with seven wells (N0=7 for one period) is employed, in which the ratio of depths between those wells are calculated from Eq. 1, with 0:1:4:2:2:4:1 for the prime number 7. For simplicity, the well with zero depth is excluded in the QRD-backed MPP absorber. That is, only six wells are considered in the QRD design, whose depths are determined through 2
c mod ( n , N 0 )
, n 0,1, " N 0 1. (1) N 0 (2 f r ) where ln is the depth of the well, N0 is the prime number of diffuser, c is the speed of sound in air, and fr is the design frequency. ln
Fig.1. One-dimensional QRD-backed MPP absorber with two periods
2. Theoretical method 2.1. Theoretical basis The acoustic impedance by the MPP and QRD wells is first discussed to predict the normal incidence absorption coefficient of the presented sound absorber. The MPP can be regarded as an extra resistive layer applied on the opening of the QRD to enhance the absorption. For normal incidence of sound wave on the panel, the specific acoustic impedance is [1]: 'p Z1 (2) u where ǻp is the sound pressure difference between the ends of the panel, u is average particle velocity. The relative (to the characteristic impedance ȡ0c in air) acoustic impedance of the MPP, ZMPP, can be evaluated according to studies [1]. The QRD is actually a compound of several parallel arrangement narrow wells with different depths, Morse and Ingard derived the complex wave number in a cylindrical tube and it also can be used for those narrow wells. The inward impedance on the surface of the well with length ln can be derived as [9, 10]:
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
j Ue c
Z (l n )
k0 kt
(3)
cot( kt ln ),
where: 2
2
§ S q · | k k0 (1 j )[ d (J 1) d ], ¸ v h 0 2b © b ¹
k0 ¨
kt
k0
Z / c,
U e U 0 [1 (1 j ) d v / b ] in which Ȗ=7/5 for air. Parameters dv and dh are [10]: 2u
dv
1
| 0.21
U 0Z
dh
,
f
2K
U 0Z C p
| 0.25
(4) (5)
1
(cm).
(6)
f
where f is sound frequency, K, ȝ and Cp are the properties of gas. K is the thermal conductivity, and ȝ is the coefficient of viscosity, Cp is the heat capacity per unit mass at constant pressure. Thus the inward normalized specific impedance of the well with length ln is [9]: Z (l n ) (7) j{1 (1 j )[ d v (J 1) d h ] / 2b}cot( kt ln ). ] (l n ) Uo c 2.2. Absorption prediction model of the QRD-backed MPP absorber As shown in Fig. 1, the sound field in front of the absorber is decomposed into the incident plane wave pe(x,z), and scattered field ps(x,z), including arbitrary incident angles, but only șe=0 (the normal incidence) is investigated in this paper. The followed analysis method is proposed by Mechel [8]: (8) p ( x, z ) pe ( x, z ) ps ( x, z ),
pe ( x, z ) where k x
Pe e
j ( x kx z kz )
k 0 sin T e , f
ps ( x , z )
¦ Ae
kz
, k 0 cos T e ,
j ( x En z J n )
n
(9)
.
(10)
n f
Since the absorber is periodic, the scattered field is also periodic in x. Therefore the wave numbers in the x and z directions of the spatial harmonics are (the first from the periodicity, the latter from the wave equation): 2S En kx n ; T 2
2
Jn
2
k0 E n
j k0
§ sin T n O · 1 , ¨ ¸ e T¹ ©
(11)
where Ȝ=2ʌ/k0 is the wavelength. For Eq. 11, when the quantity under the square root is negative, those radiating harmonics, corresponding to the indices ns, can propagate to the far field: T T (1 sin T e ) d ns d (1 sin T e ). (12)
O
O
An admittance function G(x) is employed to describe the diffuser, which can be given through combining the normalized specific impedance of MPP and QRD: 1 (13) , G ( x) z MPP ] (ln ) Then together with the admittance function G(x)ˈthe relation of pressure and particle velocity along the
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
positive z direction on the surface is U 0 cvz ( x, 0) f
cos T e Pe
Jn
¦k
n f
An e j x n 2S /T
0
ª ¬
G ( x) « Pe
G ( x ) p ( x, 0) .This gives
f
¦ Ae
j x n 2S / T
n
n f
º »¼ .
(14)
Since G(x) is periodic with a period T, we apply a Fourier analysis: f
¦ge
G ( x)
j n (2S / T ) x
n
1 T G ( x)e j n (2S /T ) x dx. ³ 0 T
; gn
n f
(15)
This inserted into the boundary condition given by Eq.14, after multiplication by ejm (ʌ7) x and integration over T:
§
f
¦ A ¨g n
n f
©
mn
§ Jn ·· ¸ ¸ Pe (G m ,0 cos T e g m ); © k0 ¹ ¹
G m,n ¨
m
f, "" , f
(16)
where
G m,n
^
1,
m n
0,
mzn
.
The value of m, n will be terminated at ±2*N for convergence, where N is the number of wells in one period. By solving the above equations, the coefficients An can be obtained, and the absorption coefficient of QRD-backed MPP absorber is then
D (T e ) 1
A0 Pe
2
1 cos T e
¦
ns z 0
An
2 s
Pe
1 (sin T e ns O / T ) 2 .
(17)
the summation runs over radiating spatial harmonics only. Evidently the second term is the specular reflection, and the third term is due to scattering. 3. Results and discussions 3.1. Numerical Validations Prediction of the normal incidence absorption coefficient of the presented absorber is numerically validated through a finite element procedure [3]. The mesh geometry in the finite element model is shown in Fig. 2, the effect caused by vibration is neglected because the panel is assumed to be rigid.
Fig.2. Mesh geometry in the finite element model of the QRD-backed MPP absorber with six parallel arrangement cavities
There are two acoustic domains (the incident duct and backing cavity) and one acoustic structure (the MPP) as shown in the finite element model. The sound field in the model is determined by the Helmholtz equation:
§ 2 1 w2 · ¨ c w t 2 ¸ I 0, © ¹
(18)
where I is the total velocity potential of the sound field and is determined by sound pressure p and acoustic particle velocity u through [3]:
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Wencheng Guo and Hequn Min / Energy Procedia 78 (2015) 1617 – 1622
U0
p
wI wt
,
u
I .
(19)
Table. 1. Properties of the QRD-backed MPP absorber used in the investigations Orifice diameter Panel Thickness Perforation ratio
Cavity depths (mm)
Case 1
0.5mm
0.5mm
1%
5, 20, 10, 50, 100, 25
Case 2
0.5mm
0.5mm
1%
25, 100, 50, 50,100, 25
A commercial finite element software, COMSOL Multiphysics, was used to implement this numerical model. The model parameters of the proposed absorber are listed in Table 1. Take Case1 as example, the computational domains are discretized with triangular quadratic-Lagrange elements, and a total of 3429 elements in one period are defined. Another two and ten periods model is also simulated by this procedure. Analytical and numerical results of the normal incidence absorption coefficients of the absorber are compared and shown in Fig. 3. It is shown that, results from the analytical prediction method agree excellently with those from the finite element model at frequencies lower than about 1800 Hz. Beyond this frequency, distinct deviation between these two predictions can be observed. For the corresponding reason, extensive numerical simulations by the finite element model are conducted, in which the considered period number (T) in the model geometry increases to be 2 and 10. The corresponding numerical results are also shown in Fig. 3. It is shown that, with the period number increasing, the deviation between the analytical and numerical prediction models decreases markedly. This indicates that, such deviation comes from the numerical errors in the finite element model, in which the proposed absorber with infinite period numbers has to be simulated as one with a finite but larger enough period numbers. It also indicates that the analytical prediction method presented in Sec. 2 is valid for the normal incidence absorption coefficients of the proposed absorber. 1
Normal Incidence Absorption coefficient
Normal Incidence Absorption coefficient
1
0.8
0.6
0.4
Case 1 FEM 1,T=1 FEM 2,T=2 FEM 3,T=10
0.2
0
0.6
0.4
0.2
0 0
500
1000
1500 Frequency,Hz
2000
2500
3000
Fig.3. Comparison of the predicted normal incidence absorption coefficients by the proposed method (Case1) and the finite element model (FEM). FEM1-3 denotes the period of simulation T=1, 2, 10, respectively
Case 1 Case 2
0.8
0
500
1000
1500 2000 Frequency,Hz
2500
3000
Fig.4. Comparison of the predicted normal incidence absorption coefficients by typical QRD-backed MPP absorber (Case2) and modified QRD-backed MPP absorber (Case1).
3.2. Simulation cases Sound absorption performance of the proposed absorber is investigated in two cases based on the analytical prediction method, to preliminarily understand the property influence from the absorber parameter set. Parameters in both cases are shown in Fig. 4,with different depths of the backing well cavities. In the Case 2, well parameters follow strictly those of the pseudo-stochastic sequence, while in the Case 1, some modifications are conducted to broaden local resonances at a wider frequency band. The parameters of the two cases are listed on Table 1. In both cases, parameters of the MPP are the same. As shown in Fig. 4, the dash curve has three absorption peaks laid on 480Hz, 670Hz and 970Hz, corresponding to 100mm, 50mm and 25 mm depth respectively. Each absorption peak is above 0.9 due to a good impedance matching condition. But aiming for the enhanced absorption performance of the QRD-backed MPP absorber, the solid curve is preferred, whose absorption bandwidth is almost twice of the dash curve. The main difference
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between case 1 and case 2 is the cavity depths, indicating the great potential of it to obtain a better absorption performance. 4. Conclusion A compound fiber-free sound absorber architecture was presented through covering MPP over the opening of a QRD. An analytical prediction method was proposed to look into the theoretical mechanism of this absorber and this method was validated through numerical simulations based on the finite element method. Simulation cases have been carried out to investigate the absorption performance dependent of the absorber architecture and dimension set. Results show that the presented MPP absorber can have normal incidence sound absorption coefficients higher than 0.5 over the frequencies from 370Hz to 2520 Hz. Acknowledgements This work is supported by the Natural Science Foundation of China (Grant No. 51408113) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140632). References [1] D. Y. Maa. Theory and design of microperforated panel sound absorbing construction. (in Chinese) 1975; Sci. Sin. 18:55–71. [2] D. Y. Maa. Micorperforated Panel wide-band absorber. Noise Control Eng. J. 1987; 29: 77–84. [3] Chunqi Wang and Lixi Huang. On the acoustic properties of parallel arrangement of multiple micro-perforated panel absorbers with different cavity depths. J. Acoust. Soc. Am 2011; 130(1): 208-218. [4] M. R. Schroeder. Binaural dissimilarity and optimum ceilings for concert halls: More lateral sound diffusion. J. Acoust. Soc. Am. 1979; 65: 958–963. [5] K. Fujiwara and T. Miyajima. Absorption characteristics of a practically constructed Schroeder diffuser of quadratic-residue type. Appl. Acoust. 1992; 35:149–152. [6] K. Fujiwara and T. Miyajima. A study of the sound absorption of a quadratic-residue type diffuser. Acustica 1995; 81: 370–378. [7] H. Kuttruff. Sound absorption by pseudostochastic diffusers (Schroeder diffusers). Appl. Acoust. 1994; 42: 215–231. [8] F. P. Mechel. The wide-angle diffuser—A wide-angle absorber?. Acustica 1995; 81:379–401 [9] T. Wu, T. J. Cox, and Y. W. Lam. From a profiled diffuser to an optimized absorber. J. Acoust. Soc. Am 2000; 108(2): 643-650. [10] P. M. Morse and K. Ingard. Theoretical Acoustics (McGraw-Hill, New York) 1968; Chap. 9:pp. 519–522. [11] P. M. Morse and K. Ingard, Theoretical Acoustics (McGraw-Hill, New York) 1968; Chap. 6: pp. 285–291.