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Jones_Kessissoglou2010-StudyofMufflers.pdf
A NUMERICAL AND EXPERIMENTAL STUDY OF
THE TRANSMISSION LOSS OF MUFFLERS USED
IN RESPIRATORY MEDICAL DEVICES
P. W. Jones and N. J. Kessissoglou
School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney NSW 2052, Australia
[email protected], [email protected]
Mufflers are incorporated into continuous positive airway pressure (CPAP) devices to reduce noise in the air paths to and
from the flow generating fan. The mufflers are very small, irregularly shaped, and are required to attenuate noise up to 10kHz.
It is important that the acoustic performance of these mufflers is reliably predicted and optimised, in order to improve the user
experience and maximise compliance with the CPAP therapy. In this study, the acoustic performance of three reactive muffler
designs similar to those used in CPAP devices is presented. Transmission loss predictions obtained using analytical and finite
element methods are compared with experimental data measured using a test rig based on the two-microphone acoustic pulse
method. The analytical methods were found to be unsuitable for predicting the transmission loss of CPAP muffler designs
due to the complexity of the muffler geometries. Good agreement between the finite element and experimental results were
obtained.
INTRODUCTION AND BACKGROUND
Obstructive sleep apnoea (OSA) is a medical condition whereby
the smooth muscles of the upper airway lose sufficient condition
during sleep that the airway becomes constricted, resulting in partial
or complete obstruction of the airway. OSA can be successfully
managed through the application of a positive pressure to the airway
during sleep. This elevated airway pressure is produced by a flow
generating fan within a continuous positive airway pressure (CPAP)
device. Noise from the flow generator is controlled using mufflers
situated in the flow path at the fan inlet and the flow generator outlet.
The mufflers are very small and are required to attenuate noise up
to 10 kHz. Often these mufflers are irregularly shaped and consist
of a number of interconnected volumes. They are predominantly
reactive, although absorptive materials have been utilised. An
assessment of the air path noise characteristics of three ResMed®
CPAP device designs has identified that the most significant noise
levels are present at frequencies below 4 kHz. This frequency
range encompasses the dominant noise sources associated with the
rotation of the blower shaft, blade pass frequency and shaft bearing
harmonics. With the exception of the narrow peak sound levels
associated with these discrete sources, the region of greatest benefit
for targeting noise attenuation lies below 1.5 kHz.
The most common type of linear acoustic model used to predict
the performance of mufflers applies classical electrical filter theory
and is most widely known as the transfer matrix method, although
it is also referred to as the two port approach or 4-pole parameter
method [1-3]. The acoustic parameters of many of the individual
design elements that are frequently used in mufflers have been well
characterised [4-6]. Kim and Soedel [7] and Wu et al. [8] present a
transfer matrix method which rearranges the variables used in the
original method such that only the velocity boundary condition is
used in the calculation of the matrix parameters. This improved
method offers several advantages over the original method when
applied with the finite element method to evaluate transmission loss
Acoustics Australia
[9]. Other methods include the 3-point method [10] and impedance
method (also known as transmission line theory) [11]. Numerical
approaches used to predict the performance of mufflers include
the finite element method (FEM) [12-14], the boundary element
method (BEM) [15, 16], and computational fluid dynamics (CFD)
[17]. Barbieri et al. [13, 14] implemented the transfer matrix method
to predict the acoustic performance of expansion chambers using
both the original parameter formulation and the improved method.
A comparison of the various numerical methods has been given by
Bilawchuk and Fyfe [18].
This study considers the application of the analytical impedance
method and transfer matrix method to three reactive muffler
designs having dimensions and geometric complexity similar to
those found in CPAP devices. These designs correspond to a single
expansion chamber design, an integrated multi-chamber design and
a multi-chamber design consisting of three interconnected volumes.
Acoustic finite element modelling of the designs was conducted by
the authors [19] and the results obtained using the ANSYS package
are reproduced in this paper. The transmission loss of each of the
mufflers was measured using a two-microphone acoustic pulse
method which was based on the procedure developed by Seybert
and Ross [20]. Experimental results for the three muffler designs are
compared with the analytical and computational results.
MUFFLER DESIGNS
The first design shown in Figure 1 consists of a single unbaffled
chamber having coaxial inlet and outlet ports. The close proximity
between the inlet and outlet ports creates a narrow, short opening
between the ports and the muffler chamber. While the level of
geometric detail in the design is high, the underlying configuration
is that of a Helmholtz resonator. The acoustic characteristics of this
design are well known and, as such, it serves as a suitable design
for initial comparison between the analytical, computational and
Vol. 38 April (2010) No. 1 - 13
experimental results. The characteristic dimension of the muffler
chamber will result in the propagation of some higher order modes
within the chamber at frequencies within the desired attenuation
range.
ANALYTICAL APPROACHES
Figure 4: Five volume expansion chamber reactive muffler
Figure 1: Cross-section and air
volume of single chamber design
Figure 2: Cross-section
and air volume of integrated
multi-chamber design
Figure 3: Air volume of three chamber muffler design
The second design shown in Figure 2 consists of two integrated
chambers and presents a complex path between the inlet and outlet
ports. If air is flowing through the device it would be deflected
around a vertical internal baffle before passing through a narrow
slot into the final chamber. The third design shown in Figure 3
consists of three interconnected expansion chambers each having
orthogonal inlet and outlet ports. The chambers are geometrically
simple and contain no internal baffles. A cylindrical pipe of 43mm
length and 18mm internal diameter connects each chamber to isolate
through-wall noise transmission between adjacent chambers. The
dimensions of each muffler design are given in Table 1. The lengths
are measured in the direction normal to the chamber inlet and the
cross-sectional areas are calculated by dividing the total chamber
volume by its length.
Chamber 2
Chamber 3
Design
2
L1 (mm)
S 1 (mm )
L2 (mm)
S 2 (mm )
L3 (mm)
S 3 (mm2)
1
35
5,728
-
-
-
-
2
50
6,489
28
2,770
-
-
3
40
5,920
47
7,077
27
7,247
14 - Vol. 38 April (2010) No. 1
Impedance Method
The impedance method is based on the premise that equality
of acoustic impedance, Z, is maintained at a change of section.
Development of the final equation for transmission loss requires
calculating the impedance and pressure at each of the locations 1 to
12 as identified in Figure 4.
Impedance calculations proceed from the outlet (point 12) to
the inlet (point 1). The acoustic impedance at point 12 is given by
Z12 = z12 / S12, where z12 is the specific acoustic impedance and S12 is
the cross-sectional area of the outlet pipe. By applying an anechoic
termination to the outlet duct at point 12, the specific acoustic
impedance at that location is equal to the characteristic impedance
(ρc), where ρ is the fluid density and c is the speed of sound. Thus
the acoustic impedance is given by
Z12 2
z12
c
S12 S12
(1)
The acoustic pressures and volume velocities at points 11 and 12
are equal, hence the acoustic impedances at these points are equal.
The specific acoustic impedance at point 11 is then given by z11 =
Z11 S11 or
S
z11 z12 11
(2)
S12
The specific acoustic impedance at point 10 can be found from that
at point 11 by considering the impedance formula for undamped
plane acoustic waves in a gas column. The complex representation
of the actual acoustic pressure is obtained by the superposition of
the acoustic pressures associated with the positive and negative
travelling one dimensional plane acoustic waves. Thus [2]
p( x, t ) P
e j (
Table 1: Dimensions of muffler designs
Chamber 1
Two analytical approaches have been considered: the impedance
method and the transfer matrix method. The development of each
method is introduced by considering a design comprising three
coaxial expansion chambers connected in series by small lengths
of pipe, as shown in Figure 4. Unique cross-sectional areas are
provided at each point (1 to 12) to allow for the representation of
the non-uniform cross-sectional areas that are present in the designs
being considered in this study.
t kx )
P e j(
t kx )
(3)
where the amplitude and phase information has been grouped as
P=PejΦ. P+ and P- are the complex amplitudes associated with the
positive and negative travelling waves, respectively, k is the wave
number and ω is the radian frequency. The corresponding particle
velocities associated with the positive and negative travelling waves are
related to the acoustic pressures by the characteristic impedance. Thus
the complex representation of the particle velocity is given by [2]
Acoustics Australia
(4)
P
j ( t kx ) P j ( t kx )
e
e
c
c
At x5 = L5 (corresponding to point 11 in Figure 4), the specific
acoustic impedance becomes
u ( x, t ) j ( t kL5 )
j ( t kL5 )
z11 p11
Pe
c j(
u11
P
e
P e
P e j(
t kL5 )
(5)
t kL5 )
(1 c / z11 ) j 2 kL5
(6)
e
(1 c / z11 )
Similarly, at x5 = 0 (corresponding to point 10 in Figure 4), the
specific acoustic impedance can be found as
P P
(7)
p
P P
z10 10 c u10
P
P
Equations (2), (6) and (7) can be used to show that
1
cS12 jkL5
e 1
z12 S11
cS12
e
z12 S11
1
cS12 jkL5
e
z12 S11
cS12
e
z12 S11
1
Z9 c
S10
c
c
e jkL5 1
e
Z12 S11
Z12 S11
1
c
e jkL5
Z12 S11
1
c
e
Z12 S11
jkL5
(8)
jkL5
jkL5
(9)
jkL5
Equation (9) may also be re-stated in non-complex form as
c Z12 S11 cos(kL5 ) j c sin( kL5 )
(10)
S10 c cos(kL5 ) jZ12 S11 sin(kL5 )
The acoustic impedance at points 1 to 8 may be obtained by following
the same methodology used in the development of equations (2) to
(10), and considering the muffler to comprise five volumes (three
expansion chambers plus two interconnecting pipes). The resulting
five sets of equations can be combined using the overlap which
occurs at the interface between each volume.
Pressure calculations proceed from inlet (point 1) to outlet (point
12). Considering the first volume shown in Figure 4, the pressure
and specific acoustic impedance at point 1 are given by
Z9 Pi Pr
(11, 12)
Pi Pr
which can be rearranged to give the acoustic pressure at point 1
P1 Pi Pr
P1 ,
z1 c
2Pi
(1 c / z1 )
(13)
Unity pressure amplitude of the incident wave (Pi = 1) has been
considered. Equality of the acoustic pressures at points 1 and 2 leads
to P2 = P1. At x1 = 0 (corresponding to point 2 in Figure 4), the
acoustic pressure given by equation (3) and the specific acoustic
impedance can be arranged to give
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P P2 ( z 2
c)
2 z2
(14, 15)
P2
(1 c / z 2 ) e
2
jkL1
(1
c / z 2 ) e jkL1
(16)
Equality of the acoustic pressures at points 3 and 4 leads to P4 = P3
and, given z2 = Z1 S2, the pressure at point 4 can be written in terms
of that at point 1 by
P4 P1
(1 c / Z1 S 2 ) e
2
jkL1
(1
c / Z1 S 2 ) e jkL1 (17)
Equation (17) may also be re-stated in non-complex form as
P
P4 1 ( Z1 S 2 cos(kL1 ) j c sin(kL1 ))
(18)
Z1 S 2
The specific acoustic impedance at point 10 is given by z10 = Z10
S10. The acoustic pressure and volume velocity at points 9 and 10
are equal, hence the acoustic impedances at these points are equal
Z9=Z10. The acoustic impedance at point 9 can then be obtained as
1
P2 ( z 2 c)
,
2 z2
The acoustic pressure at point 3 can be obtained from that at point 2
by substituting these relationships into equation (3) defined at point
3 (x1 = L1).
P3 which can be rearranged to give
z10 c
P
The acoustic pressure at points 5 to 12 in response to an incident
wave of unity pressure amplitude may be obtained by following the
same methodology used in the development of equations (16) to
(18). As the outlet duct is anechoically terminated, the magnitude of
P12 is simply the transmitted pressure, Pt.
Transmission loss is calculated using the ratio of the incident
acoustic power at the inlet of a system and the transmitted acoustic
power at the outlet. The sound power of a travelling harmonic plane
acoustic wave is defined as [21]
2
prms
Wrms dS
(19)
c
S
prms is the root mean square pressure and S is the area of the surface
through which the sound power is passing. The corresponding form
of the transmission loss equation is
W
TL 10 log10 i
(20)
Wt
Wi and Wt are the incident and transmitted sound power, respectively.
For unity pressure amplitude of the incident wave (Pi = 1) the sound
transmission loss (TL) for the reactive muffler shown in Figure 4
can now be found from
S
TL 10 log10 P122 12
(21)
S1
Note that the above derivation accommodates differing cross
sectional areas at each of the points 1 to 12. For the simplified case
of a single expansion chamber of constant cross section and having
inlet and outlet ducts of equal cross section, the above process can
be shown to produce a well known form of the transmission loss
[20]
2
1 1
TL 10 log10 cos 2 (kL) m sin 2 (kL)
(22)
4 m
where m is the expansion ratio (chamber cross section divided by
duct cross section).
Vol. 38 April (2010) No. 1 - 15
Transfer Matrix Method
The transfer matrix method (also known as the two port method)
uses 2 x 2 matrices to relate two variables at planes on either side of
an acoustic component [2]. Matrices for individual components can
be readily combined to form a single, overall matrix that describes
the behaviour for a multi-component muffler system.
Adopting acoustic pressure (p) and volume velocity (U) as the
two state variables, the following general transfer matrix may be
written to relate the state variables on either side of an expansion
chamber reactive muffler.
p1
A B p2
(23)
U1
C D U2
For the case of the simple cylindrical expansion chamber reactive
muffler, only the transfer matrices for (i) a uniform tube, (ii) a sudden
expansion and (iii) a sudden contraction need to be considered. For
a uniform tube, the transfer matrix is given by [2, 4]
cos(kLc )
A B
S
C D
j c sin(kLc )
c
j
c
Sc
j LK
1
LK ,
sin(kLc )
(24)
cos(kLc )
8
3
2
rp
H
rp
rc
(25)
where LK is the analogous acoustical inductance, rp is the radius of
the pipe, rc is the radius of expansion chamber and H(rp/rc) is given
by Figure 2b in Miwa and Igarashi [5] or may be approximated by
H ( rp / rc ) 0.6857(rp / rc ) 3
0.4312(rp / rc ) 2 1.2501(rp / rc ) 1.0006
(26)
With reference to the series connected expansion chambers shown
in Figure 4, the final transfer matrix, T, can be derived using simple
matrix multiplication of the appropriate combination of the above
matrices
AT
CT
BT
1 BK 1, 2 A2,3
C 2,3
DT
0
1
B2,3 1 BK 3, 4
A10,11
D2,3 0
C10,11
1
B10,11 1 BK 11,12
D10,11 0
1
(27)
For the case of a non-reflecting termination of the system, the
corresponding form of the transmission loss equation incorporating
the transfer matrix constants can be shown to be [2]
TL 20 log10
1 S1
2 S12
1/ 2
AT BT
S12
S
c
CT
DT 12
c
S1
S1
(28)
For the simplified case of a single expansion chamber of constant
cross section having inlet and outlet ducts of equal cross section
and only a very small expansion ratio (m ~ 1), the correction matrix
given by equation (25) simplifies to become a unit matrix and the
above process can be shown to produce the same result as given by
16 - Vol. 38 April (2010) No. 1
A D cos 2 (kLm ) sin 2 (kLm )
B/ j
C/ j
R
where Lc is the length of the expansion chamber. When the muffler
cross section, Sc, is small compared with the wavelength, and in the
absence of air flow, the sudden expansion and contraction at the
discontinuities (ends) of the expansion chamber may be represented
by simple unit matrices [2]. When this assumption cannot be made,
additional elements referred to as “Karal’s correction” [5] should be
introduced at the discontinuities. The correction matrix is given by
A B
1
C D
0
equation (22).
As the single chamber design presented in Figure 1 is expected
to perform as a Helmholtz resonator rather than a simple expansion
chamber, it is appropriate to also consider the transfer matrix
for a resonator. Miwa and Igarashi [5] define the transfer matrix
parameters for a resonator comprised of a side branch with a closed
cavity as:
c
Sm
c
Sm
sin( kLm ) cos(kLm )
R
Sm
(29)
2 sin( kLm ) cos(kLm )
sin 2 (kLm )
R
Sm
(30)
2 sin(kLm ) cos(kLm ) cos 2 (kLm )
R
Sm
(31)
S b sin(kLb ) cos(kLc ) S c cos(kLb ) sin(kLc )
cos(kLb ) cos(kLc ) ( S c / S b ) sin(kLb ) sin(kLc )
(32)
where Lm is the length of the main pipe extending equally either side
of the resonator, Sm is the cross-section of the main pipe, Lb is the
length of the side branch (measured from main pipe centreline to
start of cavity), Sb is the cross-section of the connecting branch, Lc
is the length of the resonator cavity and Sc is the cross-section of the
cavity. For the case of a non-reflecting termination of the system,
parameters A, B, C and D given by equations (29) to (31) may be
substituted into equation (28) to obtain the transmission loss for the
resonator.
FINITE ELEMENT APPROACH
Acoustic finite element models of the three muffler designs have
been developed using the finite element analysis (FEA) software
package ANSYS (Version 11). Transmission loss is calculated
by implementing the transfer matrix methodology using ANSYS
Parametric Design Language (APDL), a scripting language that
may be used to customise the FEA workflow. Kim and Soedel [7]
and Wu et al. [8] present a method for the calculation of the four
pole parameters which presents the general transfer matrix equation
in the form
p1
A* B * U 1 / S1
*
(33)
p2
C D* U 2 / S 2
S1, S2 are the cross-sectional areas of the inlet and outlet pipes,
respectively. Utilising this method, the transfer matrix parameters
in equation (33) may be calculated by applying the following two
load cases
Case 1:
A* p1 u 1, u
Case 2:
B * p1 u 0, u
1
1
2 0
2 1
,
C * p2 u 1, u
1
,
2 0
D * p 2 u 0 , u
1
2 1
(34,35)
(36,37)
where u1 and u2 are particle velocities on either side of the acoustic
component. It is then possible to calculate the original four pole
parameters by combining equations (23) and (33) to give
Acoustics Australia
A
A*
C*
B B*
A* D *
C*
1
S2
C
1
S1
C*
D
D*
C*
S1
S2
(38-41)
While each load case requires an acoustic particle velocity to be
specified, ANSYS does not accept velocity as an applied boundary
condition. Instead, the velocity must be converted to a displacement
[22] using the relationship X = -ju / ω. The inlet boundary condition
can thus be specified with a velocity magnitude equal to unity (u =
1). Acoustic load cases are applied to the finite element model at
single frequencies, and analyses are conducted across the frequency
range of interest at regular frequency intervals, with the number
of computational runs being dictated by the frequency resolution
required. The output from the finite element analysis includes the
acoustic pressure and particle velocity at each node in the finite
element model. The acoustic pressure data from nodes located at
the muffler inlet and outlet can be used to calculate the complex
transfer matrix parameters using equations (34) to (41). The
matrix parameters calculated at each frequency are substituted into
equation (28) to obtain the transmission loss spectrum over the
desired frequency range.
The finite element model for each muffler design was meshed
using tetrahedral FLUID30 elements with mesh controls applied
to adequately resolve the fine details and tight radii in the muffler
geometries. The resulting mesh size produced 15-25 elements per
acoustic wavelength at the upper bound of the frequency range
being analysed (limiting case). This is very high compared with a
widely accepted minimum acceptable mesh density of 6 elements
per wavelength. The fluid (air) was assumed to be non-flowing
and inviscid and acoustic damping was not utilised at the fluidstructure interface. That is, the walls were treated as acoustically
hard boundaries. The lack of air flow and the absence of both air
and structural damping from the models represent a simplification
of the actual conditions present in a CPAP device muffler during
operation. Further work is being conducted to incorporate damping
into the finite element models and to assess the impact of typical
CPAP device air flow rates on the acoustic performance of the
muffler designs.
shows a schematic diagram of the experimental set-up used in the
current study.
A transient acoustic pulse was generated from the Brüel & Kjær
LAN-XI Pulse front end and fed to two TU-650 horn drivers via
a PA-25E power amplifier. The pulse propagated down the 18mm
diameter PVC conduit where it was measured by the upstream
microphone, M1, before continuing to the muffler inlet. The pressure
of the corresponding pulse transmitted from the outlet of the muffler
was measured by the downstream microphone, M2. Utilising
long lengths of pipe in the system provided a time separation of
approximately 15ms between the arrival of the initial pulse and the
arrival of the subsequent reflections of the pulse at the muffler and
pipe ends. This time delay was sufficient to facilitate extraction of
the time intervals that captured only the initial positive travelling
wave from the total time histories recorded by the two microphones.
Rectangular windowing with leading and trailing cosine tapers
was applied to the time history measured by M1 and exponential
windowing with a leading cosine taper and 5ms decay constant (τ)
was applied to the time history measured by M2. These extracted
time histories were captured for 100 individual pulses, Fourier
Transformed, and the results averaged in the frequency domain.
The transmission loss for the muffler was then obtained using
TL 10 log10
FFT1
FFT2
(42)
where FFT1 and FFT2 are the Fourier Transforms of the time
histories of the incident and transmitted waves, respectively.
RESULTS AND DISCUSSION
Single chamber design
EXPERIMENTAL APPROACH
Figure 6: Single chamber muffler comparing analytical (transfer
matrix method), FE and experimental results
Figure 5: Schematic diagram of the two-microphone acoustic pulse
experimental set-up (dimensions are mm)
Experimental data was obtained using a two-microphone
technique based on a short duration acoustic pulse [19]. Figure 5
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Figure 6 contains the transmission loss obtained experimentally for
the single chamber muffler and the transmission loss predicted by
the analytical transfer matrix method (both as an expansion chamber
and side branch Helmholtz resonator) and the finite element model.
While both analytical models exhibit a poor correlation with the
experimental results, the resonator model aligns more closely with
the measured results than the expansion chamber model. This is
attributed to the close proximity between the coaxial inlet and outlet
ports. The FE results show excellent agreement with the experimental
results over the frequency range assessed, with the exception that
the magnitude at resonant frequencies is over-predicted by the FE
model. This is attributed to the FE model assuming an inviscid fluid
Vol. 38 April (2010) No. 1 - 17
and rigid walls. The inclusion of damping in the FE model would
result in a reduction in the peaks at the resonances.
Integrated chamber design
Figure 7: Integrated chamber muffler comparing analytical (transfer
matrix method), FE and experimental results
Figure 7 contains the transmission loss obtained experimentally for
the integrated chamber muffler and the transmission loss predicted
by the analytical transfer matrix method and the finite element
model. The analytical approach modelled the muffler chambers
as two expansion chambers connected in series. While similarities
may be observed, the analytical results show poor agreement
with the experimental results. This is attributed to the simplified
geometric representation used in the analytical model and the
influence of the higher order modes as the frequency increases. The
FE results show good agreement up to approximately 3 kHz. The
underlying trend followed by both sets of results is similar over the
remaining frequency range with the exception of the large double
peak predicted by the FE model.
Interconnected chamber design
Figure 8: Interconnected chamber muffler comparing analytical
(impedance and transfer matrix methods), FE and experimental
results
Figure 8 contains the transmission loss obtained experimentally
for the interconnected chamber muffler and the transmission loss
predicted by the analytical impedance method, the analytical transfer
matrix method, and the finite element model. Both analytical methods
model the muffler as series-connected expansion chambers. The
analytical results show reasonable agreement with the experimental
results up to 1.6 kHz, with the transfer matrix method exhibiting
closer agreement than the impedance method. Based on a limiting
chamber diameter of 95mm, the plane wave cut-on frequency is
approximately 2.1 kHz. Application of the analytical approach to
18 - Vol. 38 April (2010) No. 1
this muffler design is complicated by the orthogonal alignment of
the inlet and outlet connections on each chamber as the volumes are
no longer simply coaxial. While the impedance method provides
for differing chamber cross sectional areas at inlet and outlet, both
of the analytical methods presented assume one primary path for
forward and reverse travelling waves. The FE model results show
good agreement with the experimental results for the majority
of the frequency range. Departure between the FE results and
experimental data at higher frequencies is attributed to assumptions
of totally rigid walls and no acoustic damping in the FE model.
Altering wall compliance has been shown to significantly affect
resonant frequencies in the transmission loss results [23].
During experimental testing, it was noted that pressures in the
FFT spectrum for the downstream microphone (M2) above 500Hz
were less than 20 x 10-6 Pa, resulting in poor coherence. These
observations highlight the importance of producing an acoustic
pulse of short duration which still has sufficient energy at high
frequencies to provide an acceptable signal-to-noise ratio. A further
challenge presented by the two-microphone method is the need to
weight the time domain results to capture the initial incident (M1) and
transmitted (M2) pulses while excluding any subsequent reflections.
While use of long lengths of duct goes some way towards providing
adequate time spacing, it must be balanced against the higher system
losses attributable to the longer ducts. Internal reflection within each
of the muffler chambers complicates the separation of the initial
transmitted and subsequent reflected pulses due to the length of time
decay and lack of clarity in the form of the pressure signal. Care was
also required to avoid leakage errors associated with the FFT of the
microphone results.
CONCLUSIONS
In this study, the acoustic performance of three reactive muffler
designs similar to those used in CPAP devices have been
numerically and experimentally compared. Analytical expressions
for the transmission loss based on both impedance formulae and the
transfer matrix method were developed. Finite element models of
the three muffler designs were also generated based on the transfer
matrix method. Experimental validation of the computational
results was conducted using a test rig based on the two-microphone
acoustic pulse method.
At lower frequencies, the analytical results showed reasonable
agreement with the finite element and experimental results.
However, they departed significantly before reaching the first cuton frequency, beyond which the plane wave assumption is not valid.
The analytical methods are not suitable for CPAP designs due to
the complexity of the muffler geometries and the non plane wave
behaviour that must be considered when the design incorporates
orthogonal inlet and outlet ports.
In general, good agreement between the finite element and
experimental results were obtained. The FE models over-predict
the transmission loss at resonant frequencies and this is attributed
to simplifying assumptions corresponding to the use of inviscid
fluid and rigid walls. The experimental results predict that the
interconnected chamber design has the most desirable transmission
loss characteristics, which is attributed to the combined effect of three
discrete chambers, the isolation provided by the interconnecting
pipes and a 250% increase in total chamber volume compared to
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the single chamber design. The average transmission loss for the
integrated chamber design is similar to that of the single chamber
design despite a 60% increase in total chamber volume. However,
the frequency response is more uniform and the design outperforms
the single chamber muffler across much of the frequency range. The
proximity between the inlet and outlet ports of the single chamber
muffler design results in this design behaving as a Helmholtz
resonator, producing narrow transmission loss peaks centred at
specific frequencies but having significantly lower performance
at other frequencies. Variants of this design may be useful where
discrete frequencies are to be targeted but it has limited application
over a broad frequency range.
Further refinement of the work covered by this study will include
an assessment of the acoustic interaction of adjacent chambers and
integration of wall compliance into the FE models. The acoustic
performance of the predominantly reactive mufflers used in CPAP
devices can be enhanced with the inclusion of dissipative materials.
Current work is investigating appropriate experimental techniques
for the acoustic characterisation of polyurethane foams with the
aim of incorporating significant foam volumes into the muffler FE
models.
ACKNOWLEDGMENT
Financial assistance for this work was provided as part of an ARC
Linkage Project jointly funded by the Australian Research Council
and ResMed.
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