Enhancing the Performance of Microperforated Panel
Absorbers by Designing Custom Backings
Authors:
X. Hua, and D. W. Herrin
Department of Mechanical Engineering
University of Kentucky
P. Jackson
American Acoustical Products Inc.
ABSTRACT
Micro-perforated (MPP) panels are acoustic absorbers that are non-combustible,
acoustically tunable, lightweight, and environmentally friendly. They are normally
spaced from a wall, and the spacing determines the frequency range where the
absorber performs well. The absorption is maximized when the particle velocity
in the perforations is high.
Accordingly, the absorber performs best when
positioned approximately a quarter acoustic wavelength from the wall.
At
multiples of a half acoustic wavelength, the absorption is minimal. Additionally,
the absorption is minimal at low frequencies due to the limited cavity depth
behind the MPP. By partitioning the backing cavity, the cavity depth can be
strategically increased and varied. This will improve the absorption at low
frequencies and can provide absorption over a wide frequency range. In this
paper, backing substrates have been developed to enhance the absorption
performance. Maa’s model is first reviewed. Then, different MPP and backing
substrate combinations are analyzed using Maa’s theory and the boundary
element method.
The broadband absorptive behavior is validated via
measurement in a square impedance tube. Measured results agree reasonably
well with analysis.
1. INTRODUCTION
Micro-perforated panel (MPP) absorbers are a promising replacement for
traditional sound absorbing materials like fibers and foams since they are light
weight, nonflammable, cleanable, durable, and fiber free. Because of these
advantages, MPP absorbers have been utilized in mufflers [Allam, 2009; Masson,
2008], HVAC ducts [Wu, 1997], building interiors [Kang, 2005], engine enclosures
[Corin, 2005], noise barriers [Pan, 2004; Asdrubali, 2007], and in space crafts
[Omrani, 2010].
MPP absorbers are normally manufactured from steel, aluminum or plastic and
have uniformly distributed sub-millimeter sized holes. Porosity typically ranges
from between 0.5% to 2%.
The first generation MPP absorbers were metal
panels with circular perforations. In order to reduce the manufacturing cost, slitshaped perforations have become prevalent recently. Slits are pressed or
sheared into the metal.
The absorbers are most often spaced from a rigid wall. In fact, it is best to think
of the absorber as a system consisting of both the panel and backing cavity. Air
oscillates back and forth in the perforations. Since the perforations are small, the
acoustic resistance is high. Accordingly, frictional losses are greatest when the
particle velocity in the pores is maximized. This roughly corresponds to spacing
the panels a quarter acoustic wavelength from the wall.
Hence, large cavity
depths are required to extend absorption to lower frequencies.
Whereas foams and fibers provide excellent broadband absorption above a
certain frequency, the frequency range that MPP absorbers perform acceptably in
is much narrower due to the sound absorbing mechanism.
Absorption is
minimal, when the particle velocity in the perforations is low. Consequently, the
absorption is low in frequency ranges corresponding roughly to multiples of a half
acoustic wavelength.
However, MPP absorbers generally do not perform as well as a fiber or foam
even at the frequencies they are designed for. Yairi et al. [10], Toyoda and
Takahashi [11], and Liu et al. (first SAE Paper) demonstrated that performance is
comparable to traditional absorbing materials only when the backing cavity is
partitioned. The most common partitioning has been a honeycomb.
Liu and Herrin investigated the mechanism for the improved performance with
partitioning using boundary element simulation. The study confirmed the earlier
findings, and provided an explanation for the improvement in performance. The
study indicated that partitioning improves the performance of the absorber by
disrupting wave propagation behind the MPP. It was shown that acoustic waves
that propagated normal to the panel were attenuated equally well with or without
a partitioned substrate.
However, the sound pressure due to grazing waves
(propagating parallel to the MSP absorber) was essentially unaffected if
partitioning was not used.
Efforts have been aimed at improving the broadband absorption of MPP
absorbers. For example, double layer [Tao, 2005; Sakagami, 2006 & 2009] or
multi-layer [Ruiz, 2011] MPP absorbers have been suggested.
Though the
absorptive performance is improved, multi-layer MPP absorbers effectively
double the materials cost and are more difficult to install. Additionally, partitioning
must also be installed between the layers to improve their effectiveness.
Others have considered varying the cavity depth. For example, Jiang [16] and
Liu et al. [first SAE Paper] investigated using a triangular prism backed cavity.
Sum [17] recommended a parallel stepped configuration.
However, these
approaches do not lend themselves to typical box-shaped enclosures. Moreover,
large cavity depths are still required to enhance low frequency absorption.
The work in this paper focuses on improving single-layer MPP absorbers by
developing a partitioned backing. The backing is designed so that the cavity
depth is increased which provides lower frequency absorption without adding
additional volume.
Moreover, the cavity depth is varied so that broadband
frequency absorption is achieved.
The newly designed backings are
demonstrated using a square impedance tube and using simulation.
2. REVIEW OF MAA’S MODEL AND EQUIVALENT PARAMETERS
2.1 Maa’s Models for Micro-perforated Panel
For a single layer MPP, the particle velocity can be assumed to be continuous
due to the small thickness. Accordingly, Maa [Maa, 1975 and 1998] modeled the
MPP as a transfer impedance and expressed the impedance as:
Ztr =
32ht æ
2 d ö wt æ
dö
2
1+
b
/
32
+
b ÷+ j ç 1+1/ 9 + b 2 / 2 + 0.85 ÷
ç
2
sr0 cd è
32 t ø s c è
tø
(
)
where η is the dynamic viscosity of air, t is the thickness of the MPP, σ is the
porosity, ρ0 is the density of the air, c is the sound speed, d is the single hole
diameter, ω is the angular frequency, and   d  0 / 4 is the perforate
constant. For MPP absorbers with rectangular perforations, Maa [Maa, 2000]
proposed a modified model which is similar to the circular perforation MPP
model,
Z tr 
12t
 0 cd 2



2 d
t
1
d
 1   2 / 18 
  j  1  1 / 25  2  2  F e  


12 t 
c 
2
t

where e  1 
d2
is the eccentricity of the ellipse and F is the incomplete elliptic
4l 2
integral of the first kind which is expressed as
F e  


2
0
d
1  e 2 sin 2 
The impedance at the surface of a conventional MPP absorber is expressed as
 D 
Z  Z tr  j cot

 c 
where D is the air cavity depth behind the MPP. It can be observed that the
porosity, thickness, hole diameter, and the cavity depth govern the performance
of the MPP absorber. The effect of varying the cavity depth is shown in Figure 1.
Notice that an extremely low absorption band cannot be avoided when the cavity
depth is approximately half of an acoustic wavelength. Moreover, increasing the
cavity depth lowers the frequency of the high absorption band.
1 inch cavity depth
1.0
Absorption Coefficient
2 inch cavity depth
0.8
0.6
0.4
0.2
0.0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 1 Typical absorption coefficients for a same MPP with different cavity depth.
2.2 Equivalent Parameters Based on Maa’s Model
At first glance, Maa’s model seems difficult to apply to second-generation MPP
absorbers where slits are pressed or cut into metal to reduce the cost. Although
the sound absorbing mechanism is the same, Maa’s equations cannot be applied
directly due to the slit not being irregular in shape (i.e., not circular or elliptical).
Moreover, the slit is often pressed through the material at an angle and the
dimensions of the slit are not consistent through the thickness.
However, Liu (2013) used a nonlinear least-square data-fitting algorithm to
estimate equivalent parameters (usually porosity and hole size) from the
measured absorption. The absorption of a sample with a given cavity depth is
first measured and then a least-square data fitting algorithm is used to select a
porosity and hole diameter that minimize the error. The algorithm has been used
to investigate the effect of manufacturing variability and dust contamination. In
the current effort, the method was used to determine the porosity and hole size of
a MPP with slit perforations so that Maa’s theory could be used for the simulation
models.
Hence, the sound absorption of an MPP sample was measured using ASTM
E1050 [Ref.] with a four-inch cavity behind it. Maa’s model was fitted to the
measured data from 200 to 1500 Hz. The fitted equivalent hole size and porosity
were 0.26 mm and 8% respectively. The fitted and measured absorption are
compared in Figure 16.
Absorption Coefficient
1.0
Data-fitted
0.8
Measured
0.6
0.4
0.2
0.0
0
500
1000
1500
2000
Frequency (Hz)
Figure 16 The Data-fitted and measured absorption of a selected MSP with 102 mm
empty backing cavity.
3. SIMULATION OF BACKING MPP PLUS BACKING SUBSTRATES
3.1 Plane Wave Model for Parallel Cavities
Using the electrical analogy, the MPP and the backing air cavity can both be
modeled as impedances in series.
The MPP is both resistive and reactive
whereas the backing air cavity is purely reactive. Figure 3a illustrates the
analogous electrical circuit. Next, consider the case where the cavity is
partitioned into a series of channels with varying depths as shown in Figure 3b.
It can be assumed that the sound pressure is constant on the front surface of the
MPP. In that case, the combined transfer impedance and reactance for each
cavity may be considered in parallel with its neighbor as shown in Figure 3b.
Ztr
Plane
Wave
Zcavity
Cavity
(a)
1
Ztr
2
Zi
…
Plane
Wave
n
(b)
Figure 3 Electrical analogy of traditional MPP absorber (a) and multi-channel MSP
absorber (b).
The total volume velocity at the surface of the MPP is equal to the sum of the
volume velocities in the individual channels. Thus,
uS 
n
u S
i
i
i
where u is the particle velocity at the surface of the MPP and ui are particle
velocities directly behind the MPP in each channel. By assuming sound pressure
is constant across the surface of the MPP, the velocities can be written in terms
of specific acoustic impedances and sound pressure. Thus,
PS cavity
Z cavity

 PSi

i 1  Z i
n




where Zi and Si are the acoustic impedance (Equation X) and cross-sectional
area of the ith air channel respectively. Therefore, the combined impedance of
the series of air cavities can be expressed as
Z cavity 
1
n
Z
si
i 1
i
where si is the cross-sectional area ratio between the ith channel and the whole
cavity.
The normal incident absorption coefficient of the multi-channel MPP
absorber can be calculated using the equation

4 R 0 c0
R   0 c0 2  X 2
Where R and X are real and imaginary part of Z respectively.
The simplest case is a two-channel cavity. In that case, the impedance can be
expressed as
Z cavity 
1
s1 s2

Z1 Z 2
In a similar manner, a three-channel cavity can be expressed as
Z cavity 
3Z1Z 2 Z 3
Z1Z 2  Z1Z 3  Z 2 Z 3
where the impedances for each channel are governed by the lengths of the
channels.
3.2 Modeling a Three-Channel Cavity
In order to better evaluate the folded three-channel MSP absorber, a modified
model, J-shape three-channel model, is investigated. In this model, the three
channel share the same cross-sectional area as in the three-channel model. The
maximal thickness D of the absorber is pre-determined. The first and second
channels behavior maintains the same as the three-channel model. The third
channel, which is folded, is not modeled simply as a straight duct but as a simple
expansion chamber as shown in Figure 10.
D1
D2
D
Figure 10 Simple expansion approximation for the third channel.
The cross-sectional area of each channel is assumed as a square with the
dimension a. And then, the length lu and area su of the upstream duct of the third
channel is (D+D2)/2 and a2 respectively. The length lc and area sc of the chamber
is 2a and a(D-D2) respectively. The length ld and area sd of the downstream duct
is (D+D2)/2-D1 and a2. According to the transfer matrix theory, the surface
impedance at the outlet of the third channel can be expressed as
Z3 
T11
su T21
where
T11  cos(k lu ) cos(k lc ) cos(k ld ) 

sc
sin( k lu ) sin( k lc ) cos(k ld )
su
sd
s
cos(k lu ) sin( k lc ) sin( k ld )  d sin( k lu ) cos(k lc ) sin( k ld )
sc
su
T21  su sin( k lu ) cos(k lc ) cos(k ld )  sc cos(k lu ) sin( k lc ) cos(k ld )

su s d
sin( k lu ) sin( k lc ) sin( k ld )  s d cos(k lu ) cos(k lc ) sin( k ld )
sc
3.3. BEM SIMULATION OF BACKING SUBSTRATES
In order to validate the plane wave models described before, the two-channel
and three-channel MSP absorbers are simulated numerically using boundary
element models, as shown in Figure 13. A 95 mm by 95 mm square duct is
created with an MSP placed 100 mm away from the top. The MSP is modeled as
a transfer relationship that relates the respective particle velocities (vn1 and vn2)
and sound pressures (P1 and P2) on either side of the MSP, which can be written
as
 1
 v n1   Z tr
  1
v n 2   
 Z tr
1 
Z tr   P1 

1   P2 
Z tr 

where Ztr is the transfer impedance of the MSP. The cavity behind the MSP is
partitioned ideally by rigid elements without any thickness. A unit velocity
boundary condition is applied on the Red surfaces. Couple field points are
randomly picked to calculate the impedance and further acquire the absorption
coefficient.
(a)
(b)
Figure 13 Boundary element models of two-channel (a) and three-channel (b) MSP
absorbers. Unit velocity boundary condition on red and MSP transfer relation on green.
Besides boundary element method, the plane wave model for MSP absorbers is
also validated experimentally in a square impedance tube. The tube is made of
aluminum with the cross-sectional area 95 mm by 95 mm; and the cutoff
frequency is approximately 1800 Hz. A two-microphone method with random
noise excitation is applied to measure the absorption coefficient based on ASTM
1050.
Two different backing cavities are built and tested, a two-channel cavity and a
three-channel cavity that are shown in Figure 15. The partition sheets is made of
steel with 0.12 mm thick that can be ignored compared with the acoustic wave
length. All potential gaps are sealed with plumber’s putty.
(a)
(b)
Figure 15 Photos of two-channel (a) and three-channel (b) cavity design inside square
impedance tube.
Figure 17 shows the absorption comparison of the two-channel MSP absorber.
The plane wave model agrees well with the boundary element model and the
measured result below 1200 Hz. The absorption behavior at low frequency is
improved to a great extent compared with the traditional MSP absorber; however
the low absorption at around 800 Hz occurs, which cannot be avoided with twochannel design as analyzed in the section 3.2. The absorption comparison of the
three-channel MSP absorber is shown in Figure 18. The plane wave model
compares well with the other two results. It is also shown that the low frequency
absorption is further improved; and the behavior at around 800 Hz is enhanced
significantly.
1.0
Absorption Coefficient
0.8
0.6
0.4
0.2
BEM
Plane Wave
Measurement
Empty Backing
0.0
0
500
1000
1500
2000
Frequency (Hz)
Figure 17 The absorption comparison of the two-channel MSP absorber among plane
wave model, boundary element model measurement and measurement with empty cavity.
1.0
Absorption Coefficient
0.8
0.6
0.4
BEM
Plane Wave
0.2
Empty Backcing
Measurement
0.0
0
500
1000
1500
2000
Frequency (Hz)
Figure 18 The absorption comparison of the three-channel MSP absorber among plane
wave model, boundary element model measurement and measurement with empty cavity.
5. CONCLUSION
By partitioning the backing cavity to create parallel multi-channels, the traditional
MSP absorber can be improved without increasing the cavity volume. Maa’s
model is utilized to characterize irregular-shaped MPP or MSP by nonlinear data
fitting. According to the fitted hole diameter and porosity, plane wave theory is
effective to help design the backing channels below cutoff frequency.
The two-channel design improves the absorption at low frequency. For a 102 mm
material space, the absorption reaches 0.7 at around 340 Hz. However, a low
absorption in higher frequency range is inevitable. A three-channel design is able
to achieve the both targets, high absorption at low frequency and a broadband
absorption thereafter. By folding the longest channel, the material space can be
utilized much more efficiently. A two-channel and a three-channel MSP absorber
are investigated numerically and experimentally. Both of the results compares
well with the plane wave model.
ACKNOWLEDGMENTS
This work was supported
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