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Kumar2007-Thesis.pdf
Linear Acoustic Modelling
And Testing of Exhaust Mufflers
Sathish Kumar
Master of Science Thesis
Stockholm, Sweden 2007
Linear Acoustic Modelling And Testing of
Exhaust Mufflers
By
Sathish Kumar
MWL
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
2
Abstract
Intake and Exhaust system noise makes a huge contribution to the interior
and exterior noise of automobiles. There are a number of linear acoustic tools
developed by institutions and industries to predict the acoustic properties of
intake and exhaust systems. The present project discusses and validates,
through measurements, the proper modelling of these systems using BOOSTSID and discusses the ideas to properly convert a geometrical model of an
exhaust muffler to an acoustic model. The various elements and their properties
are also discussed.
When it comes to Acoustic properties there are several parameters that
describe the performance of a muffler, the Transmission Loss (TL) can be useful
to check the validity of a mathematical model but when we want to predict the
actual acoustic behavior of a component after it is installed in a system and
subjected to operating conditions then we have to determine other properties like
Attenuation, Insertion loss etc,.
Zero flow and Mean flow (M=0.12) measurements of these properties were
carried out for mufflers ranging from simple expansion chambers to complex
geometry using two approaches 1) Two Load technique 2) Two Source location
technique. For both these cases, the measured transmission losses were
compared to those obtained from BOOST-SID models.
The measured acoustic properties compared well with the simulated model for
almost all the cases.
Key Words: Acoustic Modelling, Exhaust System, Muffler, End Corrections,
Transfer Matrix, Acoustical two port, Acoustical one port, Two Microphone
Method (TMM), Transmission Loss, Attenuation, Reflection Co-efficient, AVL
Boost
3
Acknowledgements
This work was mostly carried out at Acoustic Competence Centre (ACC),
Austria and partly at the Marcus Wallenberg Laboratory for Sound and Vibration
Research at the Royal Institute of Technology in Stockholm, Sweden during the
period July to December 2006.
First I would like to thank the entire staff at ACC, Graz for making my short
stay of six months knowledgeable and enjoyable. Thanks to Hans-H.Priebsch for
giving me the opportunity to work at ACC. Special Thanks to Andreas Dolinar,
my supervisor at ACC for explaining me the thesis in the earlier stages and for
providing support throughout my thesis both analytically and financially. Danke
für die gute Zusammenarbeit.
Thanks to Hans Bodén my guide at KTH and Mats Åbom for being really
helpful in appreciating the intricacies in the later stages of the thesis. Thank you
both of you for allowing me to use the flow test rig at MWL for the mean flow
measurements. I am also thankful to the lab technicians who helped me during
the course of work at ACC & MWL.
Special thanks to Eric for your help and support in and out of this project.
Finally, I would like to thank my family and my friends for giving me the moral
support that helped me to carry out the thesis with the same zeal and enthusiasm
with which I had started it.
Vielen Dank!
4
Table of Contents
1.
INTRODUCTION ......................................................................................................................... 7
2.
MUFFLER AND PROPERTIES.................................................................................................. 9
2.1
2.1.1
2.1.2
2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.4
MUFFLER: ....................................................................................................................................... 9
REACTIVE:........................................................................................................................................ 9
DISSIPATIVE: .................................................................................................................................. 10
ACOUSTIC PROPERTIES OF MUFFLERS:...................................................................................... 10
WAVE REFLECTION IN FLOW DUCTS.......................................................................................... 12
END CORRECTION: ......................................................................................................................... 13
DETERMINATION OF END CORRECTION: ........................................................................................ 13
OPEN OUTFLOW END CORRECTIONS: ............................................................................................ 14
EXPANSION AND CONTRACTION END CORRECTION: ..................................................................... 16
RESONATOR NECK END CORRECTION: .......................................................................................... 17
MODES OF WAVE PROPAGATION IN LINEAR ACOUSTIC THEORY: ............................................ 19
3.
MEASUREMENT OF ACOUSTIC PROPERTIES: ............................................................... 21
3.1
3.2
3.3
3.3.1
3.3.2
TWO-MICROPHONE WAVE DECOMPOSITION: ............................................................................ 21
ACOUSTICAL ONE-PORTS:............................................................................................................ 24
ACOUSTICAL TWO-PORTS: .......................................................................................................... 25
TWO-SOURCE LOCATION TECHNIQUE: .......................................................................................... 27
TWO-LOAD TECHNIQUE: ................................................................................................................ 28
4.
TEST SET-UP .............................................................................................................................. 30
4.1
4.2
4.3
4.4
4.5
ONE PORT MEASUREMENT:......................................................................................................... 30
TWO PORT MEASUREMENTS WITH ZERO MEAN FLOW:............................................................. 31
TWO PORT MEASUREMENTS WITH MEAN FLOW: ...................................................................... 32
MICROPHONE CALIBRATION: ...................................................................................................... 34
FLOW NOISE SUPPRESSION: ......................................................................................................... 36
5.
RESULTS AND DISCUSSION .................................................................................................. 37
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
OPEN END REFLECTION: ............................................................................................................. 37
SIMPLE EXPANSION CHAMBER: .................................................................................................. 41
EXPANSION CHAMBERS WITH EXTENSIONS:.............................................................................. 52
EXPANSION CHAMBERS WITH WALLS AND EXTENSIONS:......................................................... 61
MUFFLERS WITH FLUSH ECCENTRIC INLET AND OUTLET PIPES:............................................ 67
MUFFLERS WITH FLOW REVERSAL: ........................................................................................... 73
MUFFLERS WITH HELMHOLTZ RESONATORS:........................................................................... 79
MUFFLERS WITH BENDED EXTENSIONS: .................................................................................... 83
5
5.9
EXPANSION CHAMBERS WITH HORN: ......................................................................................... 88
6.
CONCLUSIONS:......................................................................................................................... 93
7.
REFERENCES: ........................................................................................................................... 95
8.
APPENDIX- ................................................................................................................................. 97
8.1
8.1.1
8.1.2
8.1.3
BOOST SID .................................................................................................................................. 97
THEORY:......................................................................................................................................... 97
TRANSFER MATRIX METHOD: ....................................................................................................... 98
SUMMARY OF ELEMENTS:............................................................................................................ 100
6
1. Introduction
Road traffic noise is caused by the combination of rolling noise (arising from
tyre road interaction) and propulsion noise (comprising engine noise, exhaust
system and intake noise). Controlling these noise sources, which contributes to
the globally emitted engine noise is the subject of stringent road noise
regulations which is being updated year by year.
The propulsion noise comprises of combustion, mechanical noise and the noise
radiated from the open terminations of the intake and exhaust systems which is
caused by
I.
The pressure pulses generated by the periodic charging and
discharging process, which propagates to the open ends of the duct
system(Pulse noise), and
II.
The mean flow in the duct system, which generates significant
turbulence and vortex shedding at geometrical discontinuities (Flow
generated noise).
Figure 1. Schematic of Engine Noise Sources.
7
Mufflers play an important role in reducing the exhaust and intake system
noise and as a result, a lot of research is done to designing these systems
effectively. The traditional “build & test” procedure which is time consuming and
expensive, can nowadays be assisted by numerical simulation models which are
able to predict the performance of several different muffling systems in a short
time. A number of numerical codes have been developed in the past few
decades, based on distinct assumptions.
Considering only one-dimensional models, the two types of simulation models
may be distinguished as
I.
Linear Acoustic models: This is based on the hypothesis of small
pressure perturbations within the ducts, and
II.
Non-linear gas dynamics models: This describes the propagation of
finite amplitude wave motion in the ducts.
Linear acoustic models are frequency domain techniques which for instance
use the four-pole transfer matrix method to calculate the transmission loss of
mufflers. This approach is very fast but the predicted results may be unreliable
because the propagating pressure perturbations generally have finite amplitude
in an exhaust system.
On the other hand, non-linear gas dynamic models are able to simulate the
full wave motion in the whole engine intake and exhaust system and are based
on time domain techniques. This simulation follow the gas flow from valves to
open terminations and so is suited to deal with finite amplitude wave propagation
in high velocity unsteady flows. The excitation source can be modeled by means
of appropriate boundary conditions for the flow in these simulations.
AVL BOOST is a 1D- gas dynamic tool which predicts engine cycle and gas
exchange simulation of the entire engine. It also incorporates the linear acoustic
prediction tool SID (Sound In Ducts) [Appendix 1] so it is possible to simulate
both the non-linear and linear acoustic behavior of the system.
8
2. Muffler and Properties
2.1
Muffler:
A muffler is a device used to reduce the sound from systems containing a
noise source connecting to a pipe or duct system such as combustion engines,
compressors, air-conditioning systems etc,. In internal combustion engines,
mufflers are connected along the exhaust pipe as a part of the exhaust system.
There are two main types of mufflers, reactive and dissipative.
Figure 2. Types of Muffler.
2.1.1
Reactive:
Reactive mufflers are usually composed of several chambers of different
volumes and shapes connected together with pipes, and tend to reflect the sound
energy back to the source, they are essentially sound filters and are mostly
useful when the noise source to be reduce contains pure tones at fixed
frequencies or when there is a hot, dirty, high-speed gas flow. Reactive muffler
for such purpose can be made quite inexpensively and require little maintenance.
9
2.1.2
Dissipative:
Dissipative mufflers are usually composed of ducts or chambers which are
lined with acoustic absorbing materials that absorb the acoustic energy and turn
it into heat. These types of mufflers are useful when the source produces noise in
a broad frequency band and are particularly effective at high frequencies, but
special precautions must be taken if the gas stream has a high speed and
temperature and if it contains particles or is corrosive.
Some mufflers are a combination of reactive and dissipative types. Selection
of these mufflers will depend upon the noise source and several environmental
factors
2.2
Acoustic properties of Mufflers:
There are several parameters which describe the acoustical performance of a
muffler. These include noise Reduction (NR), Insertion Loss (IL), Attenuation
(ATT), and the Transmission Loss (TL). Noise Reduction is the sound pressure
level difference across the muffler. It is an easily measurable parameter but
difficult to calculate and a property which is not reliable for muffler design since it
depends on the termination and the muffler. The Insertion loss is the sound
pressure level difference at a point usually outside the system, without and with
the muffler present. Insertion loss is not only dependent on the muffler but also
on the source impedance and the radiation impedance. Because of this insertion
loss is easy to measure and difficult to calculate, however insertion loss is the
most relevant measure to describe the muffler performance. Transmission loss is
the difference in sound power between the incident wave entering and the
transmitted wave exiting the muffler when the muffler termination is anechoic (no
reflecting waves present in the muffler).TL is a property fully dependent on the
muffler only. Since it is difficult to realize a fully anechoic termination (at low
frequencies) TL is difficult to measure but easy to calculate. Attenuation is the
difference in the sound power incident and the transmitted through the muffler
but the termination need not be anechoic.
The acoustic properties measured to validate the models in this thesis are
Transmission Loss (TL), Attenuation (ATT) and Reflection Coefficient (REF).
10
Why Transmission Loss (TL)?
1. It is a property of the muffler alone and It is independent of the source (its
position and strength)
2. It is easy to predict but difficult to measure since it is very difficult to achieve
an anechoic termination
P1−
P1+
P2+
⎛ P+
TL = 20 log10 ⎜ 1
⎜ P+
⎝ 2
⎞
⎟⎟
⎠ P2− =0
Figure 3. Why Transmission Loss?
It is a property independent of the inlet and outlet pipe length and solely
dependent on the geometry of the muffler itself.
Why Attenuation (ATT)?
1. It is a property dependent on the muffler and also the termination, therefore
attenuation predicts the actual behavior of the muffler after it is installed in a
system
2. Helps to verify the acoustic length of the muffler
P1−
P2−
P1+
P2+
⎛ P1+
ATT = 20 log10 ⎜
⎜ P+
⎝ 2
⎞
⎟⎟
⎠
Figure 4. Why Attenuation?
Attenuation is a property dependent on the outlet pipe length so it is helpful to
validate the models and their lengths.
11
2.3
Wave Reflection in Flow Ducts
All duct systems consist of sections of uniform duct separated by area and
other discontinuities where some of the incident wave energy is reflected, some
energy is dissipated, while the remainder is transmitted to the adjacent sections.
These area discontinuities can be classified according to their observed behavior
and they normally include terminations of length of uniform duct and sudden
expansion or contractions in duct cross-section such as those found at the ends
of the expansion chambers and side branches. It should be realized that the onedimensional models for simulation of plane wave motion in ducts does not take
account of the three-dimensional waves arising at these discontinuities.
Evaluation of the pressure reflection and transmission coefficients for each
discontinuity involves satisfying the boundary conditions associated with it,
conservation of mass, conservation of energy and momentum flux across the
discontinuity. The reflective property is expressed by a pressure reflection
coefficient R , defined in terms of the component pressure wave amplitude as
R=
p−
p+
(2.1)
Since the wave amplitudes p − & p + are complex valued, the reflection coefficient
R is also normally, a complex quantity.
A close approximation to the calculated reflection coefficient for an unflanged
pipe of radius a is expressed by [Ref-4 Davies (1988)]
R0 = 1 + 0.0133ka − 0.59079(ka)2 + 0.33576(ka)3 − 0.6432(ka) 4
(2.2)
in theuseful range 0 < ka < 1.5
Where R0 corresponds to the zero flow Mach number M
A close approximation for R on R0 for an outflow Mach number into a duct is
expressed in [Ref-10 Munt (1990).]
12
2.3.1
End Correction:
To take the three dimensional effects into account at these junctions an end
correction, l is added to the length of the duct adjoining the discontinuity. The
introduction of the end correction locates the plane of wave reflection, shifted by
a distance l away from the geometrical discontinuity plane.
l, End Correction
Incident Wave
P+
Reflected Wave
PR, Strength of
wave reflection
Figure 5. Wave reflection coefficient R and end correction l, at the open outflow end.
In other words, the end correction corresponds to the extended length
required to obtain a phase shift corresponding to the three dimensional effects
between the incident ( p + ) and reflected wave ( p − ).
2.3.2
Determination of End correction:
The above defined end correction can also be defined as the extra length
added to the pipe to produce a reflection coefficient of
R = R eiθ = R e
2k l ⎞
⎛
i⎜ π +
⎟
⎝ 1− M 2 ⎠
,
(2.3)
Where, R is the reflection coefficient at the opening.
The end-correction
l is normalized with respect of the pipe radius is given by,
2
l (θ − π ) (1 − M )
=
a
2ka
(2.4)
13
θ
Where
is the phase of the reflection coefficient R obtained theoretically
Munt’s model (Ref-10 Munt (1990)) or experimentally from the cluster technique
and a is the pipe radius.
2.3.3
Open Outflow End Corrections:
For an open end termination, the correction length l is dependent on the
value of ka, on the geometry of the open end with its surroundings and on the
Mach number M of the mean flow. There exits many theoretical models to
calculate this open outflow end correction and a few end corrections at zero flow
are given in Table 1.
Figure 6. Open Outflow End Correction
Proposed
by
P.O.A.L
Davies
P.O.A.L
Davies
Norris and
Sheng
Dalmont et.
al
End Correction Value
Valid for
l0
= 0.6133 − 0.1168( ka ) 2
a
ka < 0.5
l0
= 0.6393 − 0.1104ka
a
0.5 < ka < 2
l0 [0.6133 − 0.027(ka ) 2 ]
=
a
[1 + 0.19(ka) 2 ]
0 < ka < 3.8
l0 [0.6133 − 0.027( ka) 2 ]
=
− 0.012sin 2 (2ka)
a
[1 + 0.19(ka) 2 ]
ka < 1.5
14
Munjal &
l0
= 0.6
a
Peter et.al.
Levine and
Low
l0
= 0.6133
a
Schwinger
⎛ K ⎞
l0
= 0.6 − 0.6 ⎜
⎟
a
⎝ K 2000 ⎠
Onorati
Frequency
2
0 < f < 2000
Table 1. Open Outflow End Corrections-Zero Flow.
A close approximation for these end corrections with mean outflow with Mach
M, is given by [Ref-23 Davies (1980)]
lM = l0 [1 − M 2 ]
&
lM = l0
End corrections calculated from both these equations are plausible. Therefore
an open outflow end correction value between these two equations is likely.
Some Open outflow End corrections at zero flow are plotted in Figure 7 as a
function of Helmholtz number (ka)
Open Outflow End Corrections
1
Davies EndCorrection (ka < 0.5)
Davies EndCorrection (0.5 < ka < 2)
Norris & Sheng EndCorrection
Dalmont EndCorrection
EndCorectionLength (l) / Duct Radius (a)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
Helmholtz Number (ka)
1.4
1.6
1.8
2
Figure 7 Outflow End Corrections-Zero Flow.
15
2.3.4
Expansion and Contraction End correction:
The correction length for abrupt area changes as in the case of an expansion
or contraction in ducts are not frequency dependent but a function of geometry
only and is valid for both zero flow and non-zero mean flow. Table 2 below gives
a few end corrections which are widely used,
Figure 8. Expansion and Contraction End Correction.
Proposed by
End Correction Value
Karal
le
8 ⎡
r1 ⎤
=
⎢1 − 1.238 ⎥
r1 3π ⎣
r2 ⎦
Karal
⎛ r1 ⎞⎛
le
r1 ⎞ ⎤
8 ⎡
=
⎢0.875 ⎜ 1 − ⎟⎜ 1.371 − ⎟ ⎥
r1 3π ⎣
r2 ⎠ ⎦
⎝ r2 ⎠⎝
P.O.A.L Davies
⎛ r / r −1 ⎞
⎡
−⎜ 2 1 ⎟ ⎤
le = 0.63r1 ⎢1 − e ⎝ 1.5 ⎠ ⎥
⎣⎢
⎦⎥
Torregrosa
(
le
−1.31906 r2 / r1 )
= 2 0.26148 − e(
r1
Valid for
0<
)
r1
< 0.5
r2
0 .5 <
r1
<1
r2
0<
r1
<1
r2
0<
r1
<1
r2
Table 2. Expansion and Contraction End Corrections.
r1 and r2 are the radii of the small and large ducts respectively.
16
Some Expansion and Contraction End corrections are plotted in Figure 8 as a
function of a ratio of the duct radii
End Correction Length (l) / Radius of Smaller Duct (r1)
Expansion & Contraction End Corrections
1
Karal EndCorrection
Davies EndCorrection
Torregrosa End correction
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
r1 / r2
0.6
0.7
0.8
0.9
1
Figure 9 Expansion and Contraction End Corrections
2.3.5
Resonator Neck End Correction:
The three dimensional effects occurring at the area discontinuity ( neck-tube
and neck-cavity) of a Helmholtz resonator has to been taken into account so
there are two different end corrections applied in the case of the Helmholtz
resonator. They are given in the table below,
Figure 10. Resonator Neck End Correction.
17
Proposed by
Ingard
(neck-cavity)
Onorati
(neck-tube)
Rayleigh
(neck-tube)
End Correction
Value
Valid for
le ⎛ 8 ⎞ ⎛
r⎞
=⎜
⎟ ⎜ 1 − 1.24 ⎟
r ⎝ 3π ⎠ ⎝
R⎠
le = 0.3r
f < 500
le = 0.85r
500 < f < 1500
le = r
Onorati
(neck-tube)
Rayleigh-Empirical
(neck-tube)
1500 < f
or
more
le
= 0.5ln f − 2.68
r
400 < f < 1200
Table 3. Various Resonator Neck End Corrections.
The end correction lengths given in the table are suitable also for the calculations
with strong mean flow, since the side resonators do not experience a mean flow.
18
2.4
Modes of wave propagation in Linear Acoustic theory:
The propagation of three dimensional modes in a cylindrical duct systems,
are related to a specific pressure distribution in the duct as shown in figure below
[Ref-21 Eriksson (1980)]. Even though only plane waves are permitted in the inlet
and outlet pipes, one or more higher order modes gets cut on in the chamber in
the higher frequencies depending on the diameter of the chamber.
m , CIRCUMFERENTIAL MODES
n,
R
A
D
I
A
L
M
O
D
E
S
0
1
2
3
0
1.84
3.05
4.20
3.83
5.33
6.71
8.02
7.02
8.54
9.97
11.35
10.17
11.71
13.17
14.59
0
1
2
3
Figure 11. Higher Order Modes.
When m or n take a nonzero value then there exist higher order modes in the
ducts. Each higher order mode has an associated cutoff frequency below which it
is not possible for that mode to propagate. These frequencies may be
determined by noting that the radial component of the particle velocity must go to
zero at the wall duct. The cutoff frequencies within a given duct may be
calculated using:
19
f
xmn c
πd
c=
(2.5)
Where d is the duct diameter and c is the sound velocity, with the values of
xmn as given in the above figure. At very low frequencies only plane wave mode
can propagate, and the pressure distribution across the duct is uniform.
The first circumferential higher order mode in a circular duct is the (1, 0) mode,
which has a cutoff frequency of
f
c =
(2.6)
1 .8 4 c
πd
And the cutoff frequency of the first radial mode (0, 1) is
f
c
=
3 .8 3 c
πd
The cutoff frequencies are reduced by a factor (1 − M 2 )
(2.7)
1
2
in the presence of mean
flow, where M is the flow Mach number.
20
3. Measurement of Acoustic properties:
The standard technique today for measuring the acoustic plane wave
properties in ducts, such as absorption coefficient, reflection coefficient and
impedance is the Two Microphone Method (TMM) (Ref-7 Bodén & Åbom
(1984)).The sound pressure is decomposed into its incident and reflected waves
so that the input sound power can be calculated. Transmission Loss can in
principle be determined from measurement of the incident and transmitted power
using two-microphone method on the upstream and downstream side of the test
object provided that a fully anechoic termination can be implemented on the
outlet side, which is practically very difficult in low frequency region and
especially with flow. Instead the two load technique has been used where the
sufficient information for determining the two-port matrix is obtained from two
sets of measurements with different loads on the outlet side and for the mean
flow measurements carried out at MWL,KTH the two-source location technique
was employed by placing the source on the upstream and downstream side of
the test object.
3.1
Two-Microphone wave decomposition:
The sound field in a straight hard walled duct below the first cut-on frequency will
consist only of plane propagating waves .The sound field can be written as [Ref-2
Munjal 1987]
p( x, t ) = p+ (t − x / c) + p− (t + x / c)
(3.1)
Where p=acoustic pressure
c=Speed of sound
x=spatial coordinate along the duct axis.
The idea behind the two-microphone wave decomposition is that in the low
frequency region the sound field can be completely determined by simultaneous
21
pressure measurements at two axial positions along the duct. In the frequency
domain, the sound field can be written as
m
p( x,
l
u( x,
f ) =m
p + ( f )exp(−ik+ x) + m
p − ( f )exp(ik− x)
f)=
1 ⎡m
p ( f )exp(−ik+ x) − m
p − ( f )exp(ik− x)⎤⎥
⎦
ρ c ⎢⎣ +
(3.2)
(3.3)
p = Fourier transform of the acoustic pressure,
Where, l
ul = Fourier transform of particle velocity averaged over the duct cross-section,
x = Length coordinate along the duct axis,
f = Frequency,
k± = Complex wave number for waves propagating in the positive or negative
x-direction,
ρ = Density and
c = Speed of sound.
The complex wave numbers can be calculated using the results from Howe [Ref18 Howe (1995)] or measured [Ref-1 Allam (2004)] if two microphones on either
side are available. When the complex wave numbers are known the incident
(
p+ )
and reflected ( p− ) wave amplitude can be calculated using pressure
measurements at two microphone positions.
Figure 12. Measurement Configuration of Two-Microphone Method.
22
pˆ1 ( x, f ) = pˆ + ( f ) + pˆ − ( f )
pˆ 2 ( x, f ) = pˆ + ( f ) exp(−ik+ s) + pˆ − ( f ) exp(ik− s)
(3.4)
(3.5)
Where, s represents the microphone separation as shown in figure, using the
above equations p̂+ and p̂− can be expressed by
pˆ1 ( f ) exp(ik − s ) − pˆ 2 ( f )
exp(ik− s ) − exp(−ik+ s )
(3.6)
− pˆ1 ( f ) exp(−ik+ s ) + pˆ 2 ( f )
exp(ik− s ) − exp( −ik+ s )
(3.7)
pˆ + ( f ) =
And
pˆ − ( f ) =
According to [Ref-1 Allam (2004)] the following conditions should be fulfilled for
successful use of the method
•
The measurement must take place in the plane wave region.
•
The duct wall must be rigid in order to avoid higher order mode excitation.
•
The test object should not be placed closer than 1-2 duct diameters the
nearest microphone. This is due to the fact that spatially non uniform test
objects could excite higher order modes and therefore create near field
effects at the microphones
•
The propagation of the plane wave must be unattenuated. However in
practice it is not true even for no flow case because of various mechanisms,
mainly associated with viscosity, heat conduction, will cause deviations from
the ideal behavior.
Bodén and Åbom [Ref 3 Bodén & Ref 8 Bodén] showed that the two
microphone method has the lowest sensitivity to errors in the input data in a
region around ks = π (1 − M
2
) 2.
Åbom and Bodén [Ref 8 Åbom] stated that to avoid large sensitivity to errors
in the input data, the two microphone method should be restricted to the
frequency range.
0.1π (1 − M 2 ) < ks < 0.8π (1 − M 2 )
(3.8)
23
3.2
Acoustical one-ports:
Reflection co-efficient at the open end (x=0) of an unflanged pipe is given by
Holland and Davies [Ref-19 Holland (2000)] as
R0 ( f ) =
H12 − exp(−ik+ s)
exp(ik− s) − H12
(3.9)
Where, H12 = Transfer function between microphone 1 and microphone 2,
k+ = (2π f c − iδ ) (1 + M )
k− = (2π f c − iδ ) (1 − M )
δ = Term representing attenuation.
s = Microphone spacing.
Therefore the Reflection coefficient at an arbitrary cross section along the duct is
given by
RL ( f ) =
H12 − exp(−ik+ s)
exp ⎡⎣ 2ikL (1 − M 2 ) ⎤⎦
exp(ik− s) − H12
(3.10)
Ignoring propagation losses and flow, the two pressure signals are identical
when ks = nπ , where n is any integer; this method will yield poor results when the
distance between the microphone is close to multiples of half an acoustic
wavelength. Therefore the spacing between the microphones must be kept to
within a half wavelength of the highest frequency of interest.
24
3.3
Acoustical Two-ports:
A two-port is a linear system with an input and output. Assuming plane wave
propagation at the inlet and outlet port, the properties of these acoustical twoports can be determined from theory or by measurements by assuming two state
variables at each port. A number of different choices of state variables are
possible. However, some state variables are more convenient to use that other,
for example, to measure fluctuating density may not be easy. Two state variables
which are frequently used is pressure (p) and volume flow (q). This type of
formulation is common when having systems with one preferred of energy
transport and it is called the transfer matrix formulation and the relation between
the input and output of a time-invariant and passive two-port can be written as
y = H x
(3.11)
Where, x and y = state vectors at the input and output as shown in figure
H = is a [2x2]-matrix
Figure 13. Black Box relating two pairs of state Variable x, y.
To determine the two-port matrix H from measurements, four unknowns
must be determined. To get the four equations needed for complete experimental
determination of properties of an acoustical two-port two independent test states
(‘and “) must therefore be created. The matrix equation obtained is
⎡l
y′ l
y′′⎤ = H ⎡ xl′ xl′′⎤
⎣
⎦
⎣
⎦
(3.12)
25
The unknown two-port matrix H can be determined from this equation if and only
if
det( X ) ≠ 0
(3.13)
Where, X is the matrix containing the two-port state vectors.
Depending on the coupling of the duct system either the Transfer-matrix (If
Cascade) or the mobility-matrix (If Parallel) of the two-port is used
The transfer-matrix form uses the acoustic pressure (p) and the particle velocity
(q) i.e.
x = ⎡⎣ m
pa qla ⎤
⎦
And
y = ⎡⎣ m
pb qlb ⎤ .If there is not internal sources inside the two⎦
port element the transfer-matrix could be written in the following form
⎡ lp ⎤ ⎡Taa T ⎤ ⎡ lp ⎤
b
ab
⎢ a⎥ = ⎢
⎥⎢ ⎥
⎢⎣ q a ⎥⎦ ⎢⎣Tba Tbb ⎥⎦ ⎢⎣ q b ⎥⎦
(3.14)
The transfer matrix can be solved if the below equation is satisfied
⎛l
p′
det ⎜ b
⎜ ql′
⎝ b
l
p′′b ⎞
⎟≠0
l
q′′b ⎟⎠
(3.15)
Here “a” and “b” represents two different duct cross-sections. Three basic
assumptions concerning the sound field inside the transmission line are made
•
Only plane waves are allowed to propagate at the inlet and outlet section of
the system
•
The field is assumed to be linear, i.e. the acoustic pressure is typically less
than one percent of the static pressure [Ref-20 Åbom (1991)] so that the
analysis can be done in the frequency domain.
•
The two-port system is passive, i.e. no internal sources are allowed. This is a
problem concerning flow generated noise.
26
3.3.1
Two-Source Location Technique:
The two source location method is based on the above mentioned transfer
matrix approach. The two-port data is determined by having two independent test
states by two source location method as shown in figure below.
Figure 14. Measurement Configuration for the Two-Source Location Technique.
The first state is obtained by turning loudspeaker A on and B off and the
second independent test state is obtained by turning loudspeaker B on and A off.
From 3.12 and 3.14 we have
⎡l ′
⎢ pa
⎢ ′
⎢⎣ q a
lp ′′ ⎤ ⎡T
a ⎥
= ⎢ aa
′′ ⎥ T
q a ⎦⎥ ⎣ ba
⎡ ′
Tab ⎤ ⎢ lp b
Tbb ⎦⎥ ⎢ ′
⎣⎢ q b
lp ′′ ⎤
b ⎥
′′ ⎥
q b ⎥⎦
(3.16)
If the input and output vectors of the transfer matrix are measured, we obtain
the following matrix equation from the definition of the transfer matrix using the
two-port conditions. It is also possible to use this method using a single loud
speaker and by reversing the muffler during the second test state.
The transmission loss of the test object can be obtained from
TL( n) = 20 log10
⎛Z ⎞
⎛ Z ⎞⎤
Tab ( n)
1⎡
⎢Taa ( n) ⎜ a ⎟ +
+ Tba (n) * ( Z a * Z b ) + Tbb ( n) ⎜ b ⎟ ⎥
2⎢
(Z a * Zb )
⎝ Zb ⎠
⎝ Z a ⎠ ⎥⎦
⎣
(3.17)
27
Where
Z a = ρ c0 Aa
and
Z b = ρ c0 Ab
are the characteristic impedances at
the duct cross section at “a” and “b”.
3.3.2
Two-Load technique:
This transfer matrix based approach realizes the two independent test states
needed to determine the two-port data by changing the loads at the termination
instead of moving the sound source to the other end thus generating four
equations with four unknowns. The setup is as shown in the figure below,
Figure 15. Measurement Configuration for the Two-Load Technique.
In this two-load method, if the loads are very similar, the results will be
unstable, Generally, two loads can be two different length tubes, a single tube
with and without absorbing material, or even two different mufflers.
28
In this thesis, two loads are achieved by a having an open termination and a
closed termination. The different impedance ratios of the two loads used in this
thesis is as shown in the figure below.
Impedance Ratio for the Two Loads
250
Load-1 (Open Termination)
Load-2 (Closed Termination)
Impedance Ratio (modulus)
200
150
100
50
0
0
200
400
600
800 1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 16. Impedance Ratio of the Two Loads used in the
Two-Load Technique
The transmission loss of the test object can be obtained from
TL( n) = 20 log10
⎛Z ⎞
⎛ Zb ⎞ ⎤
Tab ( n)
1⎡
⎢Taa ( n) ⎜ a ⎟ +
+ Tba (n) * ( Z a * Z b ) + Tbb ( n) ⎜ ⎟ ⎥
Z
2⎢
Z
Z
(
*
)
⎝ b⎠
⎝ Z a ⎠ ⎥⎦
a
b
⎣
3.18
Where
Z a = ρ c0 Aa
and
Z b = ρ c0 Ab
are the characteristic impedances at
the duct cross section at “a” and “b”
29
4. Test Set-up
4.1
One Port Measurement:
All one-port experiments were carried out at room temperature in the test
setup at the Acoustic Competence Centre (ACC).The setup consists of one
loudspeaker as an acoustic source as shown in figure. Fluctuating pressures we
measured using three ½ inch condenser microphones thread mounted on the
duct wall. The measurements were carried out using burst random signal (with
60% burst time) excitation and with different number of frequency domain
averages (20, 50,100 averages).The transfer function between the three
microphone positions are measured and used to estimate the transfer matrix
components.
Figure 17. Layout of One-Port Test Object
Three microphones were used to in order to cover the frequency range 1002000 Hz while fulfilling the equation [ 0.1π (1 − M
2
) < ks < 0.8π (1 − M 2 ) ] .The
distance between the microphone 1 and 3 was 15cm giving approximately the
frequency range 100-900Hz.The distance between microphone 2 and 3 was 5cm
giving approximately the frequency range 350-2600Hz.
30
4.2
Two port Measurements with zero mean flow:
All two port measurements with zero mean flow were carried out in room
temperature in the test setup at the Acoustic Competence Center (ACC).The test
setup is as shown in Figure 18. The inlet and outlet pipes used during the
measurements were made of standard steel with wall thickness 1.5 mm. The
inlet and outlet pipe diameter was 51 mm and one loud speaker was used as an
excitation source. Fluctuating pressures were measured using six ½ inch
condenser microphones thread mounted on the duct walls. The microphone
placement and the test setup are as shown in figure below. The measurements
were carried out using Burst random (60% burst time) as the excitation signal
and with different frequency domain averages. Since the measurements were
done with zero flow, the number of averages did not really influence the quality of
the result therefore 100 FDA were used during all these measurements. The twoport data was obtained using the two load technique by altering the loads at the
termination as described in Section 3.3.2.
Figure 18. Layout of Two-Port Test Object with Zero mean flow
One of the four microphone signal is taken as the reference and the transfer
function between the reference and the other three microphones are measure
and used to estimate the transfer matrix components.
31
The same microphone separations described in the previous section for the oneport measurements were used, giving again the frequency range 100-900Hz for
15cm distance and 350-2600Hz for 5cm distance.
4.3
Two port Measurements with Mean flow:
The two-port measurements with mean flow were carried out at room
temperature using the flow acoustic test facility at The Marcus Wallenberg
laboratory for Sound and Vibration research at KTH. The muffler configurations
were the same as used in ACC but only one inlet and outlet pipe diameter
(67mm) was chosen to fit the test rig duct diameter. The loud speakers were
divided equally between the upstream and downstream side. The microphone
placement and the test setup are as shown in Figure 19. Fluctuating pressures
were measured by six ¼ inch condenser microphones flush mounted on the duct
wall. The measurements were carried out using stepped sine excitation in the
frequency range 100 -1200 Hz using different frequency steps and different
Frequency Domain averages. The two-port data was obtained using the source
switching technique as described in section 3.3.1, where the measurements were
made using the upstream loud speakers on and downstream loud speakers off
and vice- versa.
Figure 19. Layout of Two-Port Test Object at MWL, KTH
32
Figure 20 Read Clockwise-Muffler mounted on the flow test rig, SIGLAB-Data
Acquisition System and Flow Controller
The mean flow velocity was measured with a Pitot tube placed at the centre
of the duct. It was assumed that flow the fully developed a boundary layer and
the mean velocity = 0.8*Maximum Velocity (measured in the centre of the duct)
[Ref-17 Schlichting (1968)]. Once the peak velocity was measured the Pitot tube
was removed from the duct before taking the acoustic measurements as it might
disturb the flow.
33
The flow velocity on the upstream side of the test object was measured
separately before and after the acoustic measurements and average value was
used.
As an additional data, the pressure drop across the test object was also
measured for different flow speeds using the Pitot tube. The transfer function
between the reference signal and the microphone signal was measured and
used to estimate the transfer matrix components.
Two different microphone spacing were used to cover a wide frequency range
100-1200 Hz while fulfilling the equation [ 0.1π (1 − M
2
) < ks < 0.8π (1 − M 2 ) ]
.The distance between microphone 1 and 2 was 10cm giving approximately the
frequency range 170-1300 Hz and the distance between microphone 1 and 3
was 50cm giving approximately the frequency range 40-280 Hz.
4.4
Microphone calibration:
For ordinary sound pressure measurements, only amplitude calibration is
enough but for two-microphone method we need both amplitude and phase
calibration. The fluctuating pressures measured at each position have been
corrected using the relative calibration between the microphone channels.
Assuming that we have plane waves in a duct the sound pressure amplitude will
be constant over the duct cross section and the sound pressure is measured by
all microphones would give the same pressure amplitude with zero phase shifts
.However there will in practice be a deviation from this ideal case due to the
measuring chain, amplifiers, and cables etc., which introduce amplitude and
phase shifts. Relative calibration of the microphone of the microphone
measurement chain is therefore needed. In order to calculate the transfer-matrix
equation, the transfer function between the microphones and the electrical loudspeaker signal, i.e. Hr1, Hr2, Hr3, Hr4, Hr5, and Hr6 are needed. It is sufficient to
measure the transfer function between these microphones and a reference
microphone say microphone 1, H12, H13, H14, H15, and H16.
34
The calibration transfer functions, which will be used in the calculation of the
transfer matrix, can then be obtained from
H rcal
1 = H r1
H
cal
r4
H
= r4
H14
H rcal2 =
H
cal
r5
Hr2
H12
H
= r5
H15
H rcal3 =
H
cal
r6
Hr3
H13
H
= r6
H16
(4.1)
Microphones
Plastic pipe
Loud Speaker
Wooden Box
Figure 21. Calibration Tube
A special calibration tube a shown in figure above was used to measure the
transfer functions between the reference microphone and the other microphones.
The calibration tube consists of a loudspeaker, a steel pipe, which has the same
diameter as the test object and a microphone holder for six ¼ inch microphones.
The holder is made-up of a plastic material to avoid possible grounding errors
between the microphones. The length of the steel pipe is preferably short to
minimize the number of resonance in the pipe.
35
4.5
Flow noise suppression:
An efficient way of suppressing turbulent pressure fluctuations is to use a
reference signal, which is uncorrelated with the disturbing noise in the system
and linearly related to the acoustic signal in the duct [Ref-9 Åbom (1989)]. A
good choice for the reference signal is to use the electric signal driving the
external sources as a reference. Deviation from a linear relation between the
reference signal and the acoustic signal in the duct can for instance be caused
by non-linearity of the loud speakers at high input amplitudes, temperature drift
and non-linearity of the loudspeaker connections to the duct at high acoustic
amplitudes. One possibility is to put an extra reference microphone close to a
loud-speaker or even in the loud speaker box behind the membrane i.e., without
contacting the flow. Otherwise one of the measurement microphones can be
used as a reference.
The disadvantage of this technique is that one will get a minima’s at the
reference microphone at certain frequencies or poor signal to noise ratio. To
solve this problem one can use the microphone with the highest signal-to-noise
ratio as the reference. In this work the electronic signals driving the loudspeakers
were used as the reference.
36
5. Results and Discussion
5.1
Open End Reflection:
Figure 22. Test Set-up for reflection coefficient measurement at an open end
The reflection coefficient at the outlet cross section of an open pipe end
without mean flow was measured as described in Section 4.1.
Figure 23 shows the absolute value of the measured reflection coefficient as
compared with the BOOST SID theoretical model. There is very little information
available about the acoustical lengths from the absolute value of the reflection
coefficient therefore the real, imaginary and phase of the reflection coefficient
which is very sensitive for these changes are shown compared with the BOOST
SID theoretical results in Figure 24, Figure 25 and Figure 26.
We can see from these figures that the real and imaginary parts of the
measured reflection coefficient have a very good agreement (there is no
frequency shift therefore correct acoustic length) and the phase of the reflection
coefficient agrees well with the theoretical models which show that the open out
flow end corrections used in BOOST SID are correct.
37
Modulus of the Reflection Coefficient at an Pipe Open End, M=0 & T=293k
Measured
BOOST SID - With End Correction
Reflection Coefficient - Modulus
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 23. Reflection Coefficient results for open end pipe
Real part of the Reflection Coefficient at an Pipe Open End M=0 & T=293k
Measured
BOOST SID - With End Correction
1
Reflection coefficient - Real Part
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 24. Reflection Coefficient results for open end pipe
38
Imaginary Part of the Reflection Coefficient at a Pipe Open End at M=0 & T=293k
Measured
BOOST SID - With End Correction
Reflection Coefficient - Imaginary Part
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 25. Reflection Coefficient results for open end pipe
Figure 26 shows the predicted and measured phase in radians of the
reflection coefficient R at the open end of the pipe
Phase of the Reflection Coefficient at an Pipe Open End M=0 & T=293k
Reflection coefficient - Phase (radians)
15
Measured
BOOST SID
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 26 Reflection Coefficient results for open end pipe
39
By determining the complex value of the reflection coefficient R at the opening,
the phase of R is converted to an end correction as described in Section 2.3.2.
Davies gives the below expression for the reflection coefficient at a pipe opening
at zero flow,
R0 = 1 + 0.0133ka − 0.59079(ka)2 + 0.33576(ka)3 − 0.6432(ka) 4
in theuseful range 0 < ka < 1.5
Where R0 corresponds to the zero flow Mach number M
The and Davies suggests the following Open End corrections which is also used
in BOOST SID,
l0
= 0.6133 − 0.1168(ka ) 2
a
l0
= 0.6393 − 0.1104ka
a
ka < 0.5
0.5 < ka < 2
The figure below shows the measured end correction as described in, compared
with the theoretical value given in the above equation at zero flow. The predicted end
corrections agree well with the experimental value.
End Correction (Measured & Theoretical), M=0 & T=293k
20
Measured
DAVIES--Theoretical
18
End Correction (mm)
16
14
12
10
8
6
4
2
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 27 End Correction Measurement
40
5.2
Simple Expansion Chamber:
An Expansion chamber is the simplest form of reactive muffler using the
property of sound reflection caused by the change in cross sectional area to
cancel sound. A simple expansion chamber is made up of an inlet pipe, a
Chamber and an Outlet pipe as shown in figure below.
An expansion chamber has a predictable transmission loss curve having
maxima at
f =
nc
, where n = 1,3,5,..
4L
c = Speed of Sound
L = Length of Chamber
(5.1)
The position of the maxima also depends on
c = c0 T
T0
, whereT = temperaturein K
(5.2)
Where c0 is the speed of sound at T0 degree K.
For example an expansion chamber of Length, L=500mm and Diameter,
D=200mm will have a transmission loss curve as shown in the figure below at
M=0 and T=293k where the speed of sound, c=343m/s.The first four maxima
calculated from Equation 5.1 for n=1, 3, 5 & 7 matches the predicted curve at
172Hz, 515Hz, 856Hz and 1200Hz respectively.
41
Transmission Loss of an Expansion Chamber
L=500mm & D=200mm at M=0 & T=293k
25
Transmission Loss (dB)
20
f =
c
4L
f =
3c
4L
f =
5c
4L
f =
7c
4L
15
10
5
0
0
200
400
600
800
Frequency (Hz)
1000
1200
1400
Figure 28. TL of a Simple Expansion Chamber
Transmission Loss measurements were performed using the Two-load
technique as discussed in the Section 4.2. The test objects used in this
measurement are as in the table below,
Muffler ID
Inlet Pipe
Dimension in
mm
Dia
Len
Chamber
Dimension in
mm
Dia
Len
outlet Pipe
Dimension in
mm
Dia
Len
Expansion
KARAL End
correction in
mm
Contraction
KARAL End
correction in
mm
Dia 100
51
96
100
500
51
96
8
8
Dia 200
51
96
200
500
51
96
14.7
14.7
Dia 300
51
96
300
500
51
96
17
17
Table 4. Dimensions of the simple expansion chambers
Geometric Model to Acoustic model:
Converting the geometric model of a muffler to an acoustic model is an
important step in linear acoustic modelling. For example conversion of a diameter
200mm simple expansion chamber as in Table 4 to a BOOST SID model was
made as shown below,
42
Geometric model of Diameter 200mm Expansion Chamber:
Figure 29 Geometric Model
Suitable end corrections are applied to this geometric model. In our case,
KARAL end corrections as shown in Table 2 & the values as in Table 4 are
applied at the junctions where the inlet and outlet pipes are connected to the
chamber and at the open end boundary.
Geometric model with end corrections applied:
Figure 30 Geometric Model with End Corrections
As a result of these end corrections, the length of the inlet and outlet pipes
are extended by a length l, equal to the end correction and consequently, the
length of the chamber is reduced as shown in figure below. The area between
the extended pipe and the chamber forms quarter wave resonators on both ends
as shown in figure.
43
Acoustic Model:
Figure 31 Acoustic Model
Acoustic Inlet Pipe length=Geometric Inlet length + End correction.
Acoustic Chamber length =Geometric chamber length – (2 x End correction).
Acoustic Outlet Pipe length=Geometric Outlet length + End correction.
Quarter wave Resonators:
Length=14.7mm & Diameter= (2002-512)1/2mm
BOOST SID Model:
Acoustic modelling of this expansion chamber in BOOST SID can be done in
three different ways,
1. Modelling with Quarter Wave Elements:
The end correction lengths are modeled as separate quarter wave resonators
as shown in figure below as quarter wave resonators and Junctions as shown in
the figure below,
D=193
SB4
L=14.7
SB3
4
L=96+14.7
Source
L=96+14.7
2
J2
J1
Expansion QW
D=193
5
L=470.6
1
SB1
L=14.7
Chamber
Contraction QW
3
SB2
Termination
44
These models are not useful when there is mean flow inside the muffler
because the linear acoustic tool assumes that there is no flow inside the
resonator which is not true if the resonator is inside the muffler.
2. Modelling with Restrictions:
The end correction lengths are modeled as extensions in the restriction
elements as shown in figure below. Because of the flow limitation of the Quarter
Wave model, this model with restrictions and extensions are used in this thesis.
EXT=14.7
EXT=14.7
L=96+14.7
L=470.6
R1
1
2
L=96+14.7
R2
3
SB1
Source
SB2
Expansion
Chamber
Contraction
Termination
3. Modelling with Plenum Chambers:
By modelling the expansion chamber with the plenum element, No end
corrections have to be applied to these models since the higher order mode
(Both Circumferential & Radial) effects which are excited above the first cut-on
frequency are already accounted for.
L=500
L=96
D=200
PL1
1
L=96
2
SB1
Source
SB2
Chamber
Termination
There are two types of plenum chambers available in BOOST SID
•
Extended Concentric-Accounts the Radial Mode effects.
•
Flush Eccentric-Accounts the Circumferential Mode effects.
45
Figure 32 and Figure 33 compares the measured Transmission Loss results
for a Diameter 100 and Diameter 200mm muffler with the predictions of the first
two linear acoustic models from BOOST SID without considering the higher order
modes.
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 100 at , M=0 & T=293k
11
BOOST SID WITH KARAL EC & QUARTER WAVES
10
BOOST SID WITH KARAL EC AS EXTENSIONS
MEASURED
9
8
7
6
5
4
3
2
1
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 32. Transmission Loss of Simple expansion Chamber
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
30
BOOST SID WITH KARAL EC & QUARTER WAVES
BOOST SID WITH KARAL EC AS EXTENSIONS
MEASURED
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 33. Transmission Loss of Simple expansion Chamber
46
As described in Section 2.4, the cutoff frequency for circumferential and radial
higher order modes in circular ducts are given by equation 2.6 and 2.7
respectively. The table below gives the cutoff frequency above which the first
circumferential and first radial modes propagates in these three expansion
chambers at 293k & M=0.
Cutoff Frequency (Hz)
Cutoff Frequency (Hz)
1st Circumferential Mode
1st Radial Mode
100
2008
4179
200
1004
2089
300
669
1392
Muffler Diameter in mm
Table 5. Cut off frequency of Expansion chambers of different diameter.
Our frequency range of interest is 0 to 2000 Hz so we are not concerned
about the higher order modes propagating above 2000Hz. It can be seen from
Table 5 that the first circumferential mode is cut on at 1004Hz and 669 Hz for a
Diameter 200mm and 300mm expansion chambers respectively and from Figure
32 and Figure 33 we can see that there is no influence of these circumferential
higher order modes. Whereas in Figure 34 we can see the effects of the first
radial higher order mode in the diameter 300mm muffler starting from 1400Hz.
Transmission Loss of an Expansion Chamber of Diameter 300mm at , M=0 & T=293k
40
BOOST SID WITH KARAL EC & QUARTER WAVES
BOOST SID WITH KARAL EC AS EXTENSIONS
35
MEASURED
Transmission Loss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 34 Transmission Loss for Simple Expansion Chamber
47
Therefore the diameter-300mm expansion chamber is modeled using a
plenum considering the radial higher order modes and plotted in Figure 35.The
reason for the amplitude difference in higher frequency region is not known.
Transmission Loss of an Expansion Chamber of Diameter 300mm at , M=0 & T=293k
40
BOOST SID WITHOUT HIGHER ORDER MODES
BOOST SID WITH HIGHER ORDER MODES
35
MEASURED
TransmissionLoss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 35. Transmission Loss for Simple Expansion Chamber
Figure 36 shows the effect of varying chamber diameters on the
Transmission Loss of simple expansion Chambers.
Transmission Loss (dB)
Effect of Diameters on Transmission Loss of Expansion Chambers at, M=0 & T=293k
40
Dia 100-BOOST SID
Dia 100-Measured
35
Dia 200-BOOST SID
Dia 200-Measured
30
Dia 300-BOOST SID
Dia 300-Measured
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 36. Effects of Diameter on Transmission Loss for Expansion Chambers
48
Expansion Chamber with Mean flow:
Figure 37 and Figure 38 shows Transmission loss results of a Diameter
200mm expansion chamber compared with BOOST SID Theoretical models with
mean flow (M=0.06 & M=0.1) as measured in the inlet pipe of the muffler.
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.06 & T=293k
30
MEASURED
MEASURED
BOOST SID
25
20
15
10
5
0
100
200
300
400
500 600 700
Frequency (Hz)
800
900 1000 1100 1200
Figure 37. Transmission Loss for Simple Expansion Chamber
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.1 & T=293k
30
MEASURED
MEASURED
BOOST SID
25
20
15
10
5
0
100
200
300
400
500 600 700
Frequency (Hz)
800
900 1000 1100 1200
Figure 38. Transmission Loss for Simple Expansion Chamber
49
Two microphone spacings were used to cover a wide frequency and this is
the reason for two legends for the measured curves. As can be seen the
agreement between the simulation and experiment results is good. When we
take a close look at the zero and mean flow results of the diameter 200mm
expansion chamber we can see that there is an amplitude difference between the
two results, this is because of the difference in the inlet/outlet pipe diameters. For
zero flow measurements, the inlet/outlet pipe diameter was 51mm and for all the
mean flow measurements, the inlet/outlet pipe diameter was 67mm.
Attenuation Measurements:
As explained in chapter 2.2, the transmission Loss of a muffler is not
influenced by the length of the inlet and outlet pipes therefore to study and
validate the effects of the end corrections which is added as extra lengths to the
inlet and outlet pipe, we measured the attenuation of the muffler which is
dependent on the outlet pipe length.
The measured attenuation for Diameter 200mm and Diameter 300mm muffler
is as shown in Figure 39 and Figure 40.
Attenuation of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
30
MEASURED
BOOST SID WITH END CORRECTIONS
25
Attenuation(dB)
20
15
10
5
0
-5
-10
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 39 Attenuation for a simple expansion chamber
50
Just like Transmission loss measurements, the Radial higher order modes
effects are clearly visible above 1400Hz in the Attenuation curve for the 300mm
diameter muffler in Figure 40.
Attenuation of an Expansion Chamber of Diameter 300mm at , M=0 & T=293k
40
MEASURED
BOOST SID WITH END CORRECTIONS
35
30
Attenuation(dB)
25
20
15
10
5
0
-5
-10
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 40 Attenuation for a simple expansion chamber
The KARAL’s end correction model as in Table 2 for an expansion and
contraction was used in all these BOOST SID models The good agreement
between the measured and simulated attenuation results shows that the end
corrections applied are correct which in turn shows that the acoustical length of
the muffler is correct i.e. the end correction lengths added to the muffler outlet
pipes are correct.
From these attenuation results we were able to create a correct acoustical
model for a simple expansion chamber and the same technique was used to
create acoustical models other complex mufflers. Even though attenuation
measurements were performed for other mufflers during the later stages of the
thesis, they became less significant when the muffler were complex.
51
5.3
Expansion Chambers with Extensions:
The effect of inlet and outlet pipe extensions in a muffler is of great interest
therefore the inlet and outlet pipes of the simple expansion chamber of Diameter
200mm discussed in the previous chapter are extended as below.
l
l
Inlet Extension
l
l
Outlet Extension
Inlet & Outlet Extension
Figure 41 Possible Expansion Chambers with Extensions
The acoustic model of the extended muffler is derived from the geometric
model as shown in the set off figures in Section 5.2 the only change being the
increase of the length of the QW Resonator corresponding to the extended
length. The inlet and outlet extensions and the corresponding end corrections
added to the inlet and outlet pipe act together as quarter wave resonators as
shown in Section 5.2.The resonance frequency of quarter wave resonators is
given by
fQW =
nc
, n = 1,3,5,...
4l
(5.3)
c = Speed of Sound
l = Effective length of extension
Here the effective length, l is the extended length + the End Correction
The Transmission loss of the expansion chamber with various inlet and outlet
extension combination is measured with and without mean flow and the results
are compared with the BOOST SID simulation.
52
Figure 42 to Figure 46 shows the quarter wave resonator effects created by the
extended inlet and outlet pipe on the TL of an expansion chamber.
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH INLET EXT=100mm
BOOST SID WITH INLET EXT=100mm
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 42. Transmission Loss of Expansion Chamber with Extensions
TransmissionLoss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH OUTLET EXT=200mm
BOOST SID WITH OUTLET EXT=200mm
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 43. Transmission Loss of Expansion Chamber with Extensions
53
TransmissionLoss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH INLET=50mm & OUTLET=150mm EXT
BOOST SID WITH INLET=50mm & OUTLET=150mm EXT
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
50mm
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 44. Transmission Loss of Expansion Chamber with Extensions
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH INLET=150mm & OUTLET=200mm EXT
BOOST SID WITH INLET=150mm & OUTLET=200mm EXT
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
150mm
50
150mm
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 45. Transmission Loss of Expansion Chamber with Extensions
54
TransmissionLoss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH INLET=250mm & OUTLET=100mm EXT
BOOST SID WITH INLET=250mm & OUTLET=100mm EXT
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
50
250mm
250mm
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 46. Transmission Loss of Expansion Chamber with Extensions
The resonance peaks of the QW resonators can be clearly seen in these
figures and they perfectly match the theory according to Equation 5.4.
The results in Figure 47 shows the Transmission loss for an expansion
chamber of length 500mm with inlet extension 250mm and outlet extension
200mm, it is clear that the two open ends inside the chamber are too close to
each other and the acoustical length is even closer .
Transm
issionLoss(dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
80
MEASURED WITH INLET=250mm & OUTLET=200mm EXT
BOOST SID WITH INLET=250mm & OUTLET=200mm EXT
70
MEASURED WITHOUT EXTENSION
BOOST SID WITHOUT EXTENSION
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 47. Transmission Loss of Expansion Chamber with Extensions
55
This causes a shift in the measured Transmission Loss peaks towards the
lower frequency which in turn means that there is an increase in the acoustical
length of the quarter wave resonators.
More investigation about these near field effects are carried out in Section 5.8.
Expansion chamber with extensions with mean flow:
Figure 48 and Figure 49 shows the Transmission Loss of expansion
chamber with inlet extension 150mm at M=0.06 and M=0.1 respectively. It can
be seen that the amplitude of the resonance frequency of the quarter wave
resonator formed by the inlet extension is damped with mean flow and has a
good agreement with the BOOST SID predictions.
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 , M=0.06 & T=293k
INLET EXTENSION 150mm
60
MEASURED M=0.06
MEASURED M=0.06
BOOST SID M=0.06
50
MEASURED M=0
BOOST SID M=0
40
30
20
10
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 48. Transmission Loss of Expansion Chamber with Extensions
56
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 , M=0.1 & T=293k
INLET EXTENSION 150mm
60
MEASURED M=0.1
MEASURED M=0.1
BOOST SID M=0.1
50
MEASURED M=0
BOOST SID M=0
40
30
20
10
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 49. Transmission Loss of Expansion Chamber with Extensions
Figure 50 shows the effect of an outlet extension of 150mm under mean flow
(M=0.06) and it can be seen that the outlet extension with flow has the same
damping effect in the resonance peak of the quarter wave resonator but the
theoretical predictions fails to have the same damping effect.
To support this point a muffler with both inlet and outlet extensions was built
and the transmission Loss was measured with (M=0.06).and without mean flow.
The results compared with the BOOST SID predictions are shown in Figure 51
The test chamber had an inlet extension of 150mm and outlet extension of
80mm. We can see from Figure 51 that the resonance peak of the quarter wave
resonator formed by the outlet pipe extension has the same damping effect as
the inlet pipe extension.
57
Transmision Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 , M=0.06 & T=293k
OUTLET EXTENSION 150mm
60
MEASURED M=0.06
MEASURED M=0.06
BOOST SID M=0.06
50
MEASURED M=0
BOOST SID M=0
40
30
20
10
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 50. Transmission Loss of Expansion Chamber with Extensions
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 , M=0.06 & T=293k
INLET EXTENSION=150mm & OUTLET EXTENSION 80mm
80
MEASURED M=0.06
BOOST SID M=0.06
70
MEASURED M=0
BOOST SID M=0
150mm
60
80mm
50
40
30
20
10
0
300
400
500
600
700
800
Frequency (Hz)
900
1000
1100
1200
Figure 51. Transmission Loss of Expansion Chamber with Extensions
A much clearer picture of the difference in the expansion and contraction models
in BOOST SID can be obtained from Figure 52.
58
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.06 & T=293k
INLET EXTENSION=150mm & OUTLET EXTENSION 80mm
80
MEASURED M=0.06
BOOST SID M=0.06
70
Transmission Loss (dB)
60
50
40
30
20
10
0
300
400
500
600
700
800
Frequency (Hz)
900
1000
1100
1200
Figure 52 Transmission Loss of Expansion Chamber with Extensions
Figure 52 shows a clear picture of an expansion and contraction model with
extensions used in BOOST SID which predicts that both these models behave in a
different way with mean flow, whereas the measurement result contradicts this point.
Therefore it will be of great interest to compare the measurement results with other
expansion and contraction models available, one such model is the expansion and
contraction model given by Davies [Ref-4 Davies (1988)].
The above result along with the Davies model of an expansion and contraction
gives a transmission loss with mean flow (M=0.06 and M=0.1) as shown in Figure 53
and Figure 54 respectively.
We can see that the expansion models from BOOST SID & Davies have a good
agreement but the contraction models are much different. But the contraction model of
Davies has a very good agreement with the measurement results in both these flow
conditions.
59
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.06 & T=293k
INLET EXTENSION=150mm & OUTLET EXTENSION 80mm
80
MEASURED M=0.06
BOOST SID M=0.06
70
DAVIES M=0.06
Transmission Loss (dB)
60
BOOST
SID
50
40
DAVIES
30
20
10
0
300
400
500
600
700
800
Frequency (Hz)
900
1000
1100
1200
Figure 53 Transmission Loss of Expansion Chamber with Extensions
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.1 & T=293k
INLET EXTENSION=150mm & OUTLET EXTENSION 80mm
80
BOOST SID M=0.1
MEASURED M=0.1
70
DAVIES M=0.1
TransmissionLoss (dB)
60
BOOST
SID
50
40
DAVIES
30
20
10
0
300
400
500
600
700
800
Frequency (Hz)
900
1000
1100
1200
Figure 54 Transmission Loss of Expansion Chamber with Extensions
60
5.4
Expansion Chambers with Walls and Extensions:
The next configurations tested were mufflers with walls in the middle as
shown in the figures below and these mufflers can be made more complex by
having extensions in the inlet and outlet side as discussed in the previous
chapter.
Figure 55 Possible Expansion chamber with walls and Extensions
Figure 56 shows the Transmission Loss of an expansion chamber of length
500mm and diameter 200mm with a wall thickness of 2mm at 257mm from the
inlet side and a hole of diameter 67mm as shown in the figure below
Transmision Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALL AND A HOLE AT 257mm FROM THE INLETSIDE
70
MEASURED M=0
BOOST SID
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 56. Transmission Loss of Expansion Chamber with Walls and Extensions
61
Figure 57 shows the TL for the expansion chamber as shown below. The
muffler had a wall with a hole of diameter 67mm at 257mm from inlet side. The
inlet extension was 100mm and outlet extension was 50mm.
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALL AT 257mm AND WITH INLET=100mm &OUTLET 50mm EXTENSIONS
70
MEASURED M=0
BOOST SID M=0
60
TransmisionLoss (dB)
100mm
50mm
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 57. Transmission Loss of Expansion Chamber with Walls and Extensions
Figure 58 shows the Transmission Loss of a muffler as shown in the figure
below. The inlet and outlet extensions were 200mm each which means that the
distance between the pipe openings and the wall was approximately 50mm on
both sides. We can see that the open ends of the pipes are too close to the wall
for the one-dimensional theory to be applicable.
62
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALL AT 257mm AND WITH INLET=200mm &OUTLET 200mm EXTENSIONS
70
MEASURED M=0
BOOST SID M=0
60
50
dB
40
30
20
10
0
0
200
400
600
800
1000
Hz
1200
1400
1600
1800
2000
Figure 58. Transmission Loss of Expansion Chamber with Walls and Extensions
The peaks in the measured TL curve has a frequency shift towards the lower
frequency because of these near field effects which is also seen in Figure
47.and further investigations are performed in Section 5.8.
Figure 59 Shows the Transmission Loss of an expansion chamber with a wall
and a pipe as shown in figure at 257mm from inlet side
The pipe in the centre wall had a diameter of 65mm and a length of 200mm out
of which 50mm was extending in the first chamber and the remaining 150mm
was extending into the second chamber.
50
150
63
Transmision Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALL AND A PIPE AT 257mm FROM THE INLETSIDE
70
MEASURED M=0
BOOST SID M=0
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 59. Transmission Loss of Expansion Chamber with Walls and Extensions
Expansion chamber with walls and extension under mean flow:
Figure 60 and Figure 61 shows the measured Transmission Loss and the
BOOST SID prediction of an expansion chamber with a wall with a hole of
diameter 67mm at 375mm from the inlet side as shown in figure below with mean
flow (M=0 and M=0.06) measured in the inlet pipe of diameter 67mm.
It can be seen that the agreement between theory and experimental results is
good.
375
125
64
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALL AT 375mm FROM THE INLETSIDE
40
MEASURED M=0
MEASURED M=0
35
BOOST SID M=0
Transmision Loss (dB)
30
25
20
15
10
5
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 60. Transmission Loss of Expansion Chamber with Walls and Extensions
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.06 & T=293k
WITH WALL AT 375mm FROM THE INLETSIDE
40
MEASURED M=0.06
MEASURED M=0.06
35
BOOST SID M=0.06
TransmisionLoss (dB)
30
25
20
15
10
5
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 61. Transmission Loss of Expansion Chamber with Walls and Extensions
65
Figure 62 and Figure 63 shows the Transmission Loss of an expansion
chamber of length 500mm and diameter 200mm with two walls at 125mm and
375mm from the inlet with mean flow (M=0 and M=0.06).
The muffler had an outlet extension of 50mm and the first wall contained a
pipe of diameter 65mm and length 100mm. out of which 18 mm was inside the
first chamber and the remaining 82mm was inside the second chamber as shown
in figure below. The transmission loss of this muffler configuration is high at
higher frequencies and the agreement between the theory and measured result
is not good but interestingly the mean flow results in Figure 63 are better than
the no flow results in Figure 62.
18
82
125
50
250
125
Transmision Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
WITH WALLS AT 125mm & 375mm FROM THE INLETSIDE
120
MEASURED M=0
MEASURED M=0
BOOST SID M=0
100
80
60
40
20
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 62. Transmission Loss of Expansion Chamber with Walls and Extensions
66
Transmision Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0.06 & T=293k
WITH WALLS AT 125mm & 375mm FROM THE INLETSIDE
120
MEASURED M=0.06
MEASURED M=0.06
BOOST SID M=0.06
100
80
60
40
20
0
0
200
400
600
Frequency (Hz)
800
1000
1200
Figure 63. Transmission Loss of Expansion Chamber with Walls and Extensions
5.5
Mufflers with Flush Eccentric Inlet and Outlet Pipes:
Muffler with flush eccentric inlet pipe or outlet pipe or both is a common case
in modern mufflers. In this section mufflers with the Inlet and outlet pipes of
diameter 200mm expansion chamber are made flush eccentric to each other as
shown in Figure 64 and the Transmission loss was measured.
Figure 64 Possible Eccentric Inlet and Outlet pipe Muffler Configuration
The first muffler measured was the simple expansion chamber with flush
eccentric inlet and outlet pipes as shown in figure below,
67
The BOOST SID model of this flush eccentric chamber was built with the plenum
element with and without considering higher order modes as shown below,
Flush Eccentric
PL1
1
2
SB2
SB1
Termination
Source
Figure 65 shows the TL of the muffler with and without considering higher
order modes in the plenum chamber. As discussed in the previous chapters, the
first circumferential mode of a diameter 200mm expansion chamber is excited
above 1004Hz .which is visible in the measured Transmission Loss .Below this
frequency, we have a very good agreement between the measured and BOOST
SID predictions
TransmissionLoss (dB)
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
50
MEASURED
BOOST SID
45
BOOST SID with Higher Order Modes
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 65. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
68
Figure 66 shows the Transmission Loss of the flush eccentric expansion
chamber with an inlet extension of 100mm.as shown in figure below,
Since the BOOST SID code does not have a model for flush eccentric
mufflers with extended inlet/outlet pipes including higher order modes, only
concentric models with extensions and considering higher order modes are
compared with the measured Transmission loss results.
Extended Concentric
PL1
1
2
SB2
SB1
Termination
Source
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
Inlet Extension 100mm
50
MEASURED
BOOST SID (Concentric)
45
Transmission Loss (dB)
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 66. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
69
Figure 67 shows the measured Transmission loss and the BOOST SID
predictions of a flush eccentric Muffler with inlet and outlet extensions as shown
below. The inlet extension was 250mm and outlet extension was 50mm, the
same modelling technique used for the previous muffler was used here,
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
Inlet Extension 250mm & Outlet Extension 50mm
60
MEASURED
BOOST SID (Concentric)
Transmission Loss (dB)
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 67. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
From Figure 66 & Figure 67, we can see that the first circumferential mode
starts to propagate above the cut on frequency as given in Table 5.
The next configuration investigated was a muffler as shown in the figure
below. Here both the eccentric inlet and outlet pipes have their openings in the
same plane. And when the extension and contraction end corrections are
applied, the acoustical openings cross each other. Therefore it is not possible to
model this muffler using the concentric element in BOOST SID. Converting this
geometric model to a valid BOOST SID model is a bit complicated and is done as
shown in the following steps.
70
Geometric model:
In this BOOST SID model, the higher order mode effects caused by the eccentric
pipes are not considered,
Acoustic model:
PIPE 1 --- QW 1 --- PIPE 2 --- QW2 --- PIPE 3
BOOST SID Model:
SB3
QWResonator 1
Source
SB1
Pipe 1
4
1
J1
Pipe 2
J2
5
QW Resonator 2
SB4
3
2
Pipe 3
SB2
Termination
71
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
Inlet Extension 250mm & Outlet Extension 250mm
60
MEASURED
BOOST SID (Concentric)
Transmission Loss (dB)
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 68. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
The BOOST SID prediction compared well with the measured transmission
loss as shown in Figure 68. Another muffler which is of interest is when those
inlet and outlet pipe extensions are too long so that the openings cross each
other as shown in Figure 69 The inlet extension was 350mm and outlet
extension was 250mm and the chamber length being 500mm, the two pipe
openings crossed each other. The same technique described for the previous
muffler was used to build a BOOST SID model
Figure 69 Flush Eccentric Mufflers
72
Figure 70 shows the Measured and Predicted Transmission loss of the
muffler without considering higher order modes.
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
Inlet Extension 350mm & Outlet Extension 250mm
60
MEASURED
BOOST SID (Concentric )
TransmissionLoss (dB)
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 70. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
5.6
Mufflers with Flow Reversal:
Flow reversal is a common case encountered in modern mufflers and
modelling these flow reversal chambers is of great interest. Three flow reversal
chambers of length 500mm, 375mm and 250 were built as shown in figure and
Transmission Loss was measured. Inlet/outlet extensions were made to further
validate our reverse flow BOOST SID models.
Figure 71 Possible Reverse Flow Chambers
73
The BOOST SID models for the long and medium mufflers were created
using two quarter wave resonators [Ref-13 Ih(1992)] (1st for the inlet extension /
outlet extension / End Correction and 2nd for the remaining length of the
chamber) as shown in the figure below,
Source
SB1
1
J1
Chamber QW
2
J2
5
SB3
3
SB2
4
J3
Termination
6
SB4
Inlet /Outlet /EC QW
Figure 72 and Figure 73 shows the transmission loss of the long and
medium mufflers and the comparison looks good till 1000Hz after which the first
circumferential mode is excited whereas the higher order modes are not included
in the BOOST SID models.
TransmissionLoss (dB)
Transmission Loss of a Flow Reversal Chamber of Diameter 200mm at , M=0 & T=293k
Length of chamber= 500mm (LONG)
70
MEASURED
BOOST SID
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 72. Transmission Loss of Mufflers with Flow Reversal
74
TransmissionLoss (dB)
Transmission Loss of a Flow Reversal Chamber of Diameter 200mm at , M=0 & T=293k
Length of chamber= 375mm (MEDIUM)
70
MEASURED
BOOST SID
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 73. Transmission Loss of Mufflers with Flow Reversal
Figure 74 and Figure 75 shows the long and medium chambers with inlet
extension of 257mm and 167mm respectively and the same modelling procedure
is followed,
TransmissionLoss (dB)
Transmission Loss of a Flow Reversal Chamber of Diameter 200mm at , M=0 & T=293k
Length of chamber= 500mm (LONG)
70
MEASURED WITH INLET EXTENSION 257mm
BOOST SID WITH INLET EXTENSION 257mm
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 74. Transmission Loss of Mufflers with Flow Reversal
75
TransmissionLoss (dB)
Transmission Loss of a Flow Reversal Chamber of Diameter 200mm at , M=0 & T=293k
Length of chamber= 375mm (MEDIUM)
70
MEASURED WITH INLET EXTENSION 167mm
BOOST SID WITH INLET EXTESION 167mm
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 75. Transmission Loss of Mufflers with Flow Reversal
Double Flow Reversal:
The next configuration investigated was a muffler with double flow reversal as
shown in figure below.
Figure 76 Muffler with Double Flow Reversal
76
As in reverse flow chambers the whole chamber where flow reversal takes
place is modeled as a quarter wave resonator. Therefore the BOOST SID model
of a double reversal chamber just looks like two independent flow reversal
chambers in cascade. The acoustic model is as shown in figure
Ch 1
Ch 2
The BOOST SID model contains 2 QW resonators for the two chambers and 2
QW resonators for the expansion and contraction end corrections.
SB3
EC
Source
3
SB1
1
J1
Chamber 1
J3
5
8
SB5
6
9
EC
J4
7
J2
2
SB2
SB6
4
Termination
Chamber 2
SB4
Figure 77 Shows the Measured and BOOST SID predicted Transmission
Loss for the muffler. As usual the higher order modes are not considered in this
BOOST SID model therefore the comparison of results below 800Hz looks
agreeable.
77
Transmission Loss of a Flush Eccentric Chamber of Diameter 200mm at , M=0 & T=293k
With a Centre wall and pipe in the middle
90
MEASURED
BOOST SID
80
TransmissionLoss (dB)
70
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 77. Transmission Loss of Mufflers with Flush Eccentric Inlet and Outlet
The short chamber with length 250mm gives the result as in Figure 78.And
various possibilities of modelling a short reverse flow chamber is shown in the
figure and it is better to consider it as a straight pipe instead of a quarter wave
resonator if the frequency of interest is below the first cutoff frequency (800Hz).
TransmissionLoss (dB)
Transmission Loss of a Flow Reversal Chamber of Diameter 200mm at , M=0 & T=293k
Length of chamber= 125mm (SHORT)
70
MEASURED
BOOST SID-QW
60
BOOST SID-Expansion chamber
BOOST SID-expansion chamber with Higher order modes
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 78. Transmission Loss of Mufflers with Flow Reversal
78
5.7
Mufflers with Helmholtz Resonators:
Helmholtz resonator is a very important component in silencing systems. These
resonators are very effective in the low frequency region
V
S
l
The resonance frequency of a Helmholtz resonator is given by the equation
f Helm =
c
2π
S
V leff
c = Speed of Sound
S = Area of the Helmholtz neck
V = Volume of the Helmholtz resonator
leff = Effective length of the neck
5.4
The end corrections for the neck tube and neck cavity of the Helmholtz
resonator are given as in Table 3. Four Helmholtz resonators of different neck
lengths (2mm, 37mm, 100mm and 200mm) were created as shown in Figure 79
and the transmission Loss was measured.
Figure 79 Helmholtz Resonator
79
The above geometric model is converted into an acoustic model with the
Helmholtz resonator as shown below
Helmholtz
The corresponding BOOST SID model is as shown below
Termination
R2
4
SB2
6
QW Resonator
J2
SB1
1
R1
7
SB3
3
2
J1
Source
5
PL1
Helmholtz Resonator
The Helmholtz resonator had an effective volume of 7 liters. Figures 80,
Figure 81, Figure 82 and Figure 83 shows TL results fro the respective neck
lengths.
Apart from the neck lengths, the position of the neck in a Helmholtz resonator
was also investigated. For example, for a muffler in Figure 79 we would expect a
QW resonator to be formed because of the extended neck inside the Helmholtz
resonator but no such effects is seen.
Where as in Figure 82 the muffler investigated had the neck extension
outside the Helmholtz resonator (as shown in the geometric model above) and
the effect of the Quarter wave resonator can be seen around 1050Hz.What is
more important is the end corrections added to the neck on both ends.
80
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
With a Helmholtz Resonator of Volume 7 Litres and Neck Length 2mm
40
MEASURED
BOOST SID
35
Transmission Loss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 80. Transmission Loss of Muffler with Helmholtz Resonator
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
With a Helmholtz Resonator of Volume 7 Litres and Neck Length 37mm
40
MEASURED
BOOST SID
35
Transmission Loss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 81. Transmission Loss of Muffler with Helmholtz Resonator
81
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
With a Helmholtz Resonator of Volume 7 Litres and Neck Length 100mm
40
MEASURED
BOOST SID
35
TransmissionLoss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 82. Transmission Loss of Muffler with Helmholtz Resonator
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
With a Helmholtz Resonator of Volume 7 Litres and Neck Length 200mm
40
MEASURED
BOOST SID
35
TransmissionLoss (dB)
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 83. Transmission Loss of Muffler with Helmholtz Resonator
82
5.8
Mufflers with Bended Extensions:
Another frequent feature in modern day mufflers are the pipes with bends
inside a muffler as shown in Figure 84,
Figure 84 Bended Pipe in a Muffler
The acoustic model is derived as shown in the figure below. We can see that
the actual length of the bended pipe is greater than the length of the quarter
wave it is creating.
Therefore two BOOST SID models with different quarter wave resonator
lengths were created and the measurement results are compared as below,
Quarter Wave length
Pipe Length
Pipe length inside the muffler+ EC=200+14.7mm
Quarter wave length+ EC=200+14.7mm (Resonance freq=400Hz)
Figure 85 shows the BOOST SID model with a QW length equal to the
bended pipe length and we can see that the prediction curve is shifted towards
left indicating that the length of the QW resonator is too long.
83
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Bended Outlet with Extension 200mm
50
MEASURED
BOOST SID(EXT=PIPE LENGTH)
45
TransmissionLoss (dB)
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 85 Transmission Loss of Mufflers with Bended Extensions
Figure 86 shows the TL of the BOOST SID model with a shorter QW length
as shown in the acoustic model and now the measured and the BOOST SID
predictions have a good frequency match.
Quarter wave length+ EC=169+14.7mm (Resonance freq=467Hz)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Bended Outlet with Extension 200mm
50
MEASURED
BOOST SID(EXT=QW LENGTH)
45
TransmissionLoss (dB)
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 86. Transmission Loss of Mufflers with Bended Extensions
84
Then the effects of having this bended pipe close to the wall as shown in
figure are studied.
Figure 87 Bended Pipe in a Muffler (Close to the wall)
We can see in Figure 88 that there is a small shift in the resonance frequency
of the quarter wave resonator formed due to the extension, which shows that the
acoustical length of the bended pipe changes when the open end is close to a
wall whose effect is not accounted when modelling in BOOST SID.
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Bended Outlet with Extension 200mm(Close to the side wall))
50
MEASURED
BOOST SID
45
TransmissionLoss (dB)
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 88. Transmission Loss of Mufflers with Bended Extensions
85
Figure 89 Shows the measured results compared with BOOST SID simulation.
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Bended Outlet with Extension 200mm
50
BOOST SID
45
MEASURED
MEASURED (CLOSE TO SIDE WALL)
40
35
30
25
20
15
10
5
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 89. Transmission Loss of Mufflers with Bended Extensions
We can see that the measured TL peak is shifted towards the lower
frequency (measured=455Hz, simulation=467Hz) indicating that the acoustic
length of the quarter wave is long when the opening is close to a wall. The
pattern in which the shift occurs is not known which lead us to perform
measurements to calculate the correct acoustic length (End corrections) when
the pipe openings are close to a plate as shown in Figure 90
Figure 90 End Correction Measurements with a Plate
86
End correction measurements were performed as described in Section 2.3.2
by placing a plate at various distances from the open end and the measured end
corrections as a function of frequency are as shown in Figure 91
End Correction with a Plate in front of the Open End
100
Open end
Model open end
Plate 30mm away
Plate 20mm away
Plate 15mm away
Plate 10mm away
Plate 5mm away
Plate 3mm away
Plate 2mm away
Plate 1mm away
OpenEnd- EndCorrection(mm)
90
80
70
60
50
40
30
20
10
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 91 End Correction Measurement
We can see that when the plate is moved closer to the open end, the end
correction which has to be applied to the open end pipe increases which in turn
increases the acoustical length, But BOOST SID uses an end correction which is
not sufficient when the pipe is close to a wall thereby creating a frequency shift
towards the higher frequency in the predicted models.
When the distance between the plate and the wall is less than 10mm, then
the new end correction is a frequency dependent (decreasing with the increase in
frequency) but when the distance is more than 10mm (which is more commonly
found in mufflers because of the flow restrictions) the new end correction is not a
frequency component anymore, therefore one possibility to account this near
field effects is to add the extra length to the end corrections.
87
5.9
Expansion Chambers with Horn:
A muffler with a horn at one end was created and the Transmission Loss was
measured. A horn is a pipe with a different start and end diameter as shown in
the Figure 92.
Figure 92 Muffler with a Horn
The horn is modeled by splitting the length of the pipe into several segments
and then considering each segment as a straight pipe. The number of segments
is given as an input during acoustic modelling in BOOST SID.
A Horn with
3 segments
88
The BOOST SID model for the horn is created using a pipe of a length equal
to the horn length and its start and end diameters and the number of segments
and the BOOST SID models is as shown in figure
1
R1
R2
2
R3
3
4
SB1
SB2
Pipe of Varying
Diameter
(HORN)
Chamber
Source
Termination
The predicted Transmission Loss from the BOOST SID model compared to
the measured is as shown in Figure 93.
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Horn on the Outlet Side
20
MEASURED
BOOST SID
18
Transmission Loss (dB)
16
14
12
10
8
6
4
2
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 93 Transmission Loss of Expansion Chambers with Horn
89
Horn with Extensions:
The Final muffler configuration investigated in this thesis was a muffler with
horn but with an extension as shown in figure below,
For a normal horn with an extension, a conical resonator is formed between
the extended pipe and the horn wall. Whereas in our case the resonator formed
is not a perfect cone or a perfect tube but a combination of both as shown in
figure below,
l
Therefore the resonator formed in this case has a geometry as shown below
A normal tube resonator has a resonance frequency as
90
f qwCylindrical =
c
4l
A conical resonator has a resonance frequency as
f conical =
c
2l
Two BOOST SID model were created using these two resonators and the
measured Transmission Loss is compared in Figure 94
Transmission Loss (dB)
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Horn on the Outlet Side and outlet extension 100mm
60
MEASURED
BOOST SID-QUARTERWAVE RESONATOR
BOOST SID-CONICAL RESONATOR
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 94 Transmission Loss of Expansion Chambers with Horn and Extension
We can see that measured resonance frequency is in between the two
resonance frequency of the resonators, a small adjustment is done in the
geometry as shown in the figure.
91
fConical =
l
c
2l
A conical resonator of the length as given in the figure below gives the
Transmission Loss prediction as in Figure 95 compared with the measured
result.
Transmission Loss of an Expansion Chamber of Diameter 200mm at , M=0 & T=293k
Horn on the Outlet Side and outlet extension 100mm
60
MEASURED
BOOST SID
Transmission Loss (dB)
50
40
30
20
10
0
0
200
400
600
800
1000 1200
Frequency (Hz)
1400
1600
1800
2000
Figure 95 Transmission Loss of Expansion Chambers with Horn and Extension
92
6. Conclusions:
A number of muffler configurations starting from simple expansion chambers
to complex geometry have been investigated with and without the mean flow
effects.
1. When validating the acoustic models created using a linear acoustic tool
transmission loss measurement is not sufficient as it is independent of the inlet
and outlet pipe lengths which make it difficult to study the effect of the end
corrections. Sometimes other properties like attenuation, reflection coefficient
has to be measured and compared with the predictions.
2. 1-D wave theory applies for Mufflers having concentric inlet and outlet pipes
till the first radial higher order mode is excited as in equation 2.7.
For mufflers having eccentric Inlet and Outlet pipes, the circumferential higher
order modes are excited above the first cut on frequency as in equation 2.6
3. Reverse Flow Mufflers: The above mentioned condition applies for flow
reverse mufflers since the inlet or outlet or both the pipes are flush eccentric to
the chamber and the predicted transmission loss result depends on the length of
the reverse chamber.
4. Higher the Transmission Loss, the more difficult it becomes to measure.
During this thesis some of the mufflers had a very high Transmission Loss
(nearly 100dB) and the measurement results was not good.
5. The expansion model used in BOOST SID includes the flow related losses
whereas the contraction model does not include these losses which can be seen
from the mean flow measurements in Section 5.3.
6. BOOST SID does not include flow inside the Quarter Wave resonators. Since
the mufflers investigated in the later part of the thesis is entirely constructed
using QW resonators the BOOST SID predictions with and without flow will yield
the same TL result which might not agree with the measurements. For example
the influence of flow on the flow reversal and helmholtz muffler are inconclusive
from our investigations.
93
7. When the acoustical openings inside a muffler are too close to each other or
close to walls then acoustical length of the elements change therefore
measurements were performed to find the end corrections when the open end is
close to a wall.
8. Secondary or Internal sources cannot be used in BOOST SID. In a complex
muffler when the flow velocities are high then the flow inside the muffler
generates a considerable amount of noise which alters the quality of the
measurement results.
94
7. References:
1. SABRY ALLAM, Acoustic Modelling and Testing of Advanced Exhaust
System Components for Automotive Engines, 2004 Doctoral thesis, KTH,
Sweden.
2. M.L.MUNJAL, 1987 Acoustics of Ducts and Mufflers, New York: WileyInterscience.
3. H.BODEN and M.ÅBOM, Influence of Errors on the Two-Microphone method
for measuring Acoustic Properties in Ducts, 1986 Journal of the Acoustical
Society of America 79(2), 541-549.
4. P.O.A.L DAVIES, Practical Flow Duct Acoustics, 1988 Journal of Sound and
Vibration 124(1), 91-115
5. Z.TAO AND A.F.SEYBERT, A Review of Current techniques for Measuring
Muffler Transmission Loss, University of Kentucky, USA.
6. RAGNAR GLAV, On Acoustic Modelling of Silencers, 1994 Doctoral thesis,
KTH, Sweden.
7. H.BODEN and M.ÅBOM, Two-Microphone Measurements In The Plane
Wave Region of Ducts, 1984 TRITA-TAK 8401.
8. M.ÅBOM and H.BODEN, Error Analysis of Two-Microphone Measurements in
Ducts with Flow, 1988 Journal of the Acoustical Society of America 83, 24292438.
9. M.ÅBOM, Studies of Sound Propagation in Ducts and Mufflers, 1989 Doctoral
thesis, KTH, Sweden.
10. R.M.MUNT, Acoustic Transmission Properties of a Jet Pipe with Subsonic Jet
Flow, 1990 Journal of Sound and Vibration 142(3), 413-436.
11. D.E.WINTERBONE and R.J.PEARSON, Design techniques for Engine
Manifolds, Professional Engineering Publishing Limited, UK.
12. A.J.TORREGROSSA et al, Numerical Estimation of End Corrections in
Extended-Duct and Perforated-Duct Mufflers, 1999 Journal of Vibration and
Acoustics 121, 302-308.
95
13. J.G.IH, The Reactive Attenuation of Rectangular Plenum Chambers, 1992
Journal of Sound and Vibration 157(1), 93-122.
14. Y.P.SOH, E.W.T.YAP and B.H.L.GAN, Industrial Resonator Muffler Design,
The University of Adelaide, Australia.
15. M.C.A.M.PETERS et al, Damping and Reflection Coefficient Measurements
at Low mach and Low Helmholtz Numbers, 1993 Journal of Fluid Mechanics 265,
499-534.
16. S.BOIJ, Acoustic Scattering in Ducts and influence of flow coupling, 2003
Doctoral Thesis, KTH, Sweden.
17. H.SCHLICHTING, 1968, Boundary Layer Theory, McGraw-Hill Inc., New
York,
18. M.S.HOWE, The Damping of Sound by wall Turbulent Sheer Layers, 1995
Journal of the Acoustical Society of America 98(3), 1723-1730.
19. P.O.A.L.DAVIES and HOLLAND, The Measurement of Sound Power Flux In
Flow Ducts 2000, 230(4), 915-932.
20. M.ÅBOM, Measurement of the Scattering-Matrix of Acoustical Two-Ports,
1991 Journal of Mech. System and Signal Processing 5(2), 89-104.
21. L.J.ERIKSSON, Higher order mode effects in circular ducts and expansion
chambers, 1980 Journal of the Acoustical Society of America 68, 545-550
22. AVL BOOST-SID, Linear Acoustics, user Manual.
23. P.O.A.L.DAVIES et al, Reflection Coefficients for an unflanged pipe with flow,
1980 Journal of Sound and Vibration 72, 543-546.
24. J.G.IH & B.H.LEE, Analysis of Higher order modes effects in the circular
expansion chamber with mean flow.1985 Journal of the Acoustical Society of
America 77(4) 1377-1388.
25. M.ÅBOM, Derivation of Four-pole Parameters including higher order mode
effects for expansion chamber mufflers with extended inlet and outlet, 1990
Journal of Sound and Vibration 137(3), 403-418.
96
8. Appendix8.1 BOOST SID
8.1.1 Theory:
BOOST-SID linear acoustics is a frequency domain solver incorporated with AVL
BOOST to determine the acoustic properties of a system. A BOOST SID model is made
up of a source, a system and a termination. The term ‘system’ includes everything in
between the source and the termination no matter how complex or how many elements
are used in the modelling.
For application to engine exhaust or inlet systems the source is the engine, the
system is the exhaust and the termination is the tailpipe ambient.
After the calculation of the transfer matrices, pressures and volume flows for each
element in the model in the system the following main result are available:
97
•
Transmission Loss
•
Noise Reduction
•
Insertion Loss (If a reference system is defined)
•
Reflection Coefficient (at each element in the model)
8.1.2 Transfer Matrix Method:
The BOOST-SID code just like The KTH –SID code uses the transfer matrix method.
This uses the relationship between the two pairs of state variables (vectors), x and y,
coupled by a two-dimensional matrix, T
x=Ty
This is the 4-pole method
X inlet side
y outlet side
In linear acoustics, the state variables are the temporal Fourier transforms of sound
pressure p1 and sound volume velocity q1 at the inlet (=x) and the acoustic pressure p2
and volume flow velocity q2 at the outlet (=y) side and T is called the Transfer Matrix.
Provided certain assumptions, there exists a complex 2*2 matrix T, which completely
describes the sound transmission within a system at certain frequency f:
⎡ p ⎤ ⎡T
⎢ 1 ⎥ = ⎢ 11
q
⎢T
⎣⎢ 1 ⎦⎥ ⎣ 21
T12 ⎤ ⎡ p2 ⎤
⎥⎢ ⎥
T22 ⎥⎦ ⎢⎣ q2 ⎥⎦
The sound field here is assumed to be linear, so the following variables can be
separated into two parts
P = p + p,
0
ρ = ρ 0 + ρ fl ,
v =V
0
+ u,
T = T0 + τ ,
s = S 0 + sE ,
98
p
v
ρ
T
Pressure
Parameters
Units
Comments
Velocity
T11
none
---
T12
Ns/m5
Impedance
T21
m5/Ns
Admittance
T22
none
---
p1,p2
Pa
Pressure
q1,q2
m3/s
Density
Temperature
s Specific entropy
Volume
Flow
Where the first term on the right hand side is the stationary mean value and the
second term is the small disturbance. Only plane waves can propagate implies that the
sound pressure is constant over the cross section.
Using the Transfer Matrix method an actual system can be modeled as shown below
Zs and ps are the internal source impedance and source strength sound pressure ZT
is the termination impedance. The system itself can be split up into basic linear acoustic
elements and then solved.
99
8.1.3 Summary of Elements:
According to the linear acoustic theory [Ref6-Glav (1994)], the transfer matrix of the
basic elements is as given below,
8.1.3.1 Straight Ducts:
A straight duct is the simplest and the most common element encountered
automobile exhaust systems, this element is use to connect all other elements in the
system. They are usually straight or slightly curved with constant cross section and wall
thickness of 1 to 2mm. For curved pipes, the influence of bend is neglected in the
frequency range 30 to 2000 Hz.
For a hard walled pipe of length l the transfer matrix can be defined as
t11 =
t12 =
t21 =
t22 =
1
{exp(ik+l ) + exp(−ik−l )} ,
2
ρ 0 c′
2S
{exp(ik+l ) − exp(−ik−l )} ,
S
2 ρ 0 c′
{exp(ik+l ) − exp(−ik−l )} ,
1
{exp(ik+l ) + exp(−ik−l )} ,
2
8.1.3.2 Area discontinuities:
The expansion and contraction models proposed by Glav [Ref-6 Glav (1994)], treats
the extended lengths caused by the end correction at these area discontinuities as
lumped parameters .The transfer matrix of the expansion and contraction elements are
obtained by T=MxE and T=CxM respectively and the end corrections transfer matrix M
in terms of lumped parameters is a given below,
100
⎡ iωρ 0 Δ e ⎤
1
M=⎢
π ⋅ a2 ⎥
⎢
⎥
1 ⎦
⎣0
8.1.3.3 Contraction:
For an area change as shown in the figure below, the transfer matrix of the
contraction model taken from [Ref-6 Glav (1994)] is a below
c11 = 1 −
M 1 ⋅ ρ0 ⋅ c
(
z ⋅ S1 ⋅ 1 − M 12
)
⎡⎛ S1 S 2 ⎞
ρ 0 ⋅ c ⋅ M 12 ⎤
1
c12 = ρ 0 ⋅ c ⋅ ⎢⎜ − ⎟ ⋅ M 1 −
⋅
⎥
z ⋅ S 2 ⎦ S 2 ⋅ (1 − M 12 )
⎣⎝ S 2 S1 ⎠
c21 =
(
1
z ⋅ 1 − M 12
)
⎡ ⎛ 2 ρ ⋅ c ⋅ M ⎞ ⎛ S ⎞2 ⎤
1
1
1
c22 = ⎢1 − ⎜ M 1 + 0
⋅
⋅
⎥
⎟ ⎜ ⎟
z ⋅ S1 ⎠ ⎝ S 2 ⎠ ⎥⎦ (1 − M 12 )
⎢⎣ ⎝
And when the end correction is added in the form of lumped impedance, the
transfer matrix (T=CxM) of this area contraction element is given by
101
t11 = 1 −
M1 ⋅ ρ0 ⋅ c
(
z ⋅ S1 ⋅ 1 − M12
)
⎡
⎡⎛ S1 S 2 ⎞
M1 ⋅ ρ0 ⋅ c ⎤
ρ 0 ⋅ c ⋅ M 12 ⎤
t12 =
⋅ ⎢⎜ − ⎟ ⋅ M 1 −
⎥ + m12 ⋅ ⎢1 −
2 ⎥
z ⋅ S2 ⎦
S 2 ⋅ (1 − M 12 ) ⎣⎝ S 2 S1 ⎠
z
S
M
⋅
⋅
−
1
⎢⎣
1 (
1 )⎥
⎦
ρ0 ⋅ c
t21 =
(
1
z ⋅ 1 − M 12
)
⎡ ⎛ 2 ρ ⋅ c ⋅ M ⎞ ⎛ S ⎞2 ⎤
⎡
⎤
1
1
0
1
1
+ m12 ⋅ ⎢
t22 = ⎢1 − ⎜ M 1 +
⎟⋅⎜ ⎟ ⎥ ⋅
2
2 ⎥
⋅
z
S
S
−
⋅
−
M
z
M
1
1
(
)
(
⎠ ⎝ 2 ⎠ ⎥⎦
⎥
⎢⎣ ⎝
1
1
1 )⎦
⎣⎢
8.1.3.4 Expansion:
For an area expansion as shown in figure below, the turbulent losses for
expanding mean flow are considered and the expansion models is as give below
2
⎤ 2 ⎪⎫ 1
⎪⎧ ⎡ ⎛ S1 ⎞
e11 = ⎨1 + ⎢ λ ⋅ ⎜ 1 − ⎟ − 1⎥ ⋅ M 1 ⎬ ⋅
⎥⎦
⎩⎪ ⎢⎣ ⎝ S 2 ⎠
⎭⎪ det
102
e12 =
⎞ 1
2 ⋅ ρ 0 cM 1 ⎛ S1
⋅ ⎜ − 1⎟ ⋅
S2
⎝ S 2 ⎠ det
2
⎡
⎛ M 1 S1 ⎞ ⎤ 1
e21 = ⎢1 + γ ⎜
⎟ ⎥⋅
S
⎝ 2 ⎠ ⎥⎦ z ⋅ det
⎢⎣
⎧ ⎛ S1 ⎞ ⎡ 2 ⋅ S1 ⎤ 2 2 ⋅ ρ 0 cM 1 ⎛ S1 ⎞ ⎫ 1
e22 = ⎨1 + ⎜ ⎟ ⋅ ⎢1 −
⎥ ⋅ M 1 + z ⋅ S ⋅ ⎜ S ⎟ ⎬ ⋅ det
S
S
⎝ 2 ⎠⎭
⎩ ⎝ 2⎠ ⎣
2 ⎦
1
And when the end correction is added in the form of lumped impedance, the
transfer matrix (T=MxE) of this area expansion element is given by
2
⎡ ⎛ S ⎞2 ⎤ 2 m ⎡
⎤
⎛
⎞
M
S
1
12
1 1
1 + ⎢ λ ⋅ ⎜ 1 − ⎟ − 1⎥ ⋅ M 1 +
⋅ ⎢1 + γ ⎜
⎟ ⎥
S
z
S
⎝ 2 ⎠ ⎥⎦
2 ⎠
⎢ ⎝
⎥
⎢⎣
⎣
⎦
t11 =
det
t12 =
⎡ ⎛S
⎞
2 ⋅ ρ 0 cM 1 ⎛ S1
⋅ ⎜ − 1 ⎟ + m12 ⋅ ⎢1 + ⎜ 1
S2
⎝ S2 ⎠
⎣ ⎝ S2
⎞ ⎛ 2 ⋅ S1 ⎞ 2 2 ⋅ ρ 0 cM 1 ⋅ S1 ⎤
⎟ ⋅ ⎜1 − S ⎟ ⋅ M1 +
⎥
2
S
⋅
z
⎠ ⎝
⎦
2 ⎠
2
det
⎛M S ⎞
1+ γ ⎜ 1 1 ⎟
⎝ S2 ⎠
t21 =
2
z ⋅ det
⎛ S1 ⎞ ⎡ 2 ⋅ S1 ⎤ 2 2 ⋅ ρ 0 cM 1 ⎛ S1 ⎞
⎟ ⋅ ⎢1 − S ⎥ ⋅ M 1 + z ⋅ S ⋅ ⎜ S ⎟
S
⎝ 2⎠ ⎣
⎝ 2⎠
2 ⎦
1
1+ ⎜
t22 =
det
Where determinant det is given by
103
2
⎡
⎛
⎞
S1
S1 ⎤ 2 ⋅ ρ 0 cM 1
2
det = 1 + M 1 ⋅ ⎢γ − 1 + ( γ − 1) ⋅ ⎜ ⎟ + (1 − 2γ ) ⋅ ⎥ +
S2 ⋅ z
S 2 ⎥⎦
⎝ S2 ⎠
⎢⎣
8.1.3.5 Resonators:
The transfer matrix of the resonators (Helmholtz, Quarter Wave, Conical) is given
by [Ref-6 Glav (1994)] as
t11 = 1
t12 = 0
t21 =
Sr
Zr
t22 = 1
Where Z r the impedance of the resonator as seen from the flow is duct and S r is the
area of the resonator mouth. The impedances Z r for the three different resonators used
in this thesis given by [Ref-6 Glav (1994)] are as follows
i ρ0c 2 Sn
Z hr = Rtu + Rv + iωρ0 (ln + Δ i + Δ o ) −
,
ωV
⎛ exp(ik ′l ) + ℜ exp(−ik ′l ) ⎞
,
⎟
⎝ exp(ik ′l ) − ℜ exp(−ik ′l ) ⎠
Z qr = ρ 0 c ⎜
Z cr =
i ρ0c
,
cot(kl ) − (1 kl )
104
8.1.3.6 Flush Eccentric Chambers:
The transfer matrix of an expansion chamber with an offset inlet and outlet pipe
is as follows. The higher order acoustic modes arising at the area discontinuities are
included in the models,
⎡ − jkMl ⎤
cos
2 ⎥
(1
)
−
M
⎣
⎦
t11 = exp ⎢
⎡ kl ⎤
⎢ (1 − M 2 ) ⎥
⎣
⎦
⎤
⎡ − jkMl ⎤ ⎡
⎡ kl ⎤
⎡ kl ⎤
2
sin
M
sin
...
+
+
⎥
⎢ (1 − M 2 ) ⎥
⎢ (1 − M 2 ) ⎥
2 ⎥⎢
⎣ (1 − M ) ⎦ ⎣
⎣
⎦
⎣
⎦
⎦
⎡ − jkMl ⎤
⎡ kl ⎤
t12 = jZ exp ⎢
t21 =
j
exp ⎢
sin ⎢
2 ⎥
2 ⎥
Z
M
M
(1
)
(1
)⎦
−
−
⎣
⎦
⎣
⎡ − jkMl ⎤
⎡ kl ⎤
t22 = exp ⎢
cos
⎢ (1 − M 2 ) ⎥
2 ⎥
⎣ (1 − M ) ⎦
⎣
⎦
8.1.3.7 Concentric Chambers:
The transfer matrix of an expansion chamber with inlet and outlet extension
including the higher order mode propagation is much more complex than the flush
eccentric chambers .Therefore a detailed derivation and formulation of the transfer
matrix for this element used in BOOST SID is given in Reference 25.
105
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