Finite Element Analysis of Acoustic Pressure Levels and Transmission

Loss of a Muffler

OVIDIU VASILE

Department of Mechanics

Politehnica University of Bucharest

Splaiul Independentei 313, 060042 Bucharest

ROMANIA

ovidiu_vasile2002@yahoo.co.uk

GILBERT-RAINER GILLICH

Department of Mechanical Engineering

“Eftimie Murgu” University of Resita

P-ta Traian Vuia 1-4, 320085 Resita

ROMANIA

gr.gillich@uem.ro

Abstract: - The paper describes the pressure-wave propagation in a muffler for an internal combustion engine in

case of a lined muffler and for a non-lined muffler. The approach is generally applicable to analyzing the

damping of propagation of harmonic pressure waves. This paper is a complement to a study done previously,

that besides transmission losses calculated according to the 50-3000 Hz frequency, isosurface representations

were added. Also, sound pressure levels were represented for two different geometrical situations and show

finite elements analysis of both inductive and resistive damping in pressure acoustics.

Key-Words: - FEM, reactive muffler, absorptive muffler, transmission loss, glass wool

1 Introduction In this paper, we will examine two cases of reactive

and dissipative muffler, using FEM – finite element

method. The detailed design procedures for mufflers

and absorbent materials are available in the

literature [2], [3], [7] and [8].

Internal combustion engines are typically

equipped with an exhaust muffler to suppress the

acoustic pulse generated by the combustion process.

A high intensity pressure wave generated by

combustion in the engine cylinder propagates along

the exhaust pipe and radiates from the exhaust pipe

termination. The pulse repeats at the firing

frequency of the engine which is defined by

f=(engine rpm x number of cylinders)/120 for a four

stroke engine. The frequency content of exhaust

noise is dominated by a pulse at the firing

frequency, but it also has a broadband component to

its spectrum which extends to higher frequencies.

In general, sound waves propagating along a pipe

can be attenuated using either a dissipative or a

reactive muffler.

•Reflective (or reactive) mufflers - those that

reflect acoustic waves by abrupt area expansions or

changes of impedance.

•Dissipative mufflers - mufflers based on

dissipation of acoustic energy into heat through

viscous losses in fibrous materials or flow-related

(resistive) losses in perforated pipes.

Reactive mufflers are best suited for the low

frequency range where only plane waves can

propagate in the system, while dissipative mufflers

with fibers are efficient in the mid-to-high

frequency range. Dissipative mufflers based on flow

losses, on the other hand, work also at low

frequencies. A typical automotive exhaust system is

a hybrid construction consisting of a combination of

reflective and dissipative muffler elements.

Reactive silencer design is based either on the

principle of a Helmholtz resonator or an expansion

chamber, and requires the use of acoustic

transmission line theory.

In a Helmholtz resonator design a cavity is

attached to the exhaust pipe. At a specific frequency

the cavity will resonate and the waves in the exhaust

pipe are reflected back towards the source. However

there are also pass band frequencies where the

resonator has no effect and so resonator muffler

design is targeted to specific frequencies where the

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 43

majority of the attenuation is required. In some

designs, the muffler has several resonators of

different sizes to target a range of frequencies.

The reflective parts are normally tuned to

remove dominating low-frequency engine

harmonics while the dissipative parts are designed

to take care of higher-frequency noise.

2 Model Definition The muffler—schematically depicted in Figure 1—

consists of a resonator chamber with a section of the

centered exhaust pipe included at each end (Fig.1 a.)

and with non-centred exhaust pipe included at each

end (Fig. 1 b.). The exhaust fumes enter through the

left pipe and exit through the right pipe. In the first

version of the model the chamber is empty. In the

second version it is lined with 15 mm of absorbing

glass wool.

a.

b.

Fig. 1. Geometry of the lined muffler:

a) geometry 1 - lined muffler; b) geometry 2 - non-

lined muffler

3 Domain Equations This model solves the problem in the frequency

domain using the time-harmonic Pressure Acoustics

application mode. The model equation is a slightly

modified version of the Helmholtz equation for the

acoustic pressure, p:

02

2

=−

∇−⋅∇

ρω

ρsc

pp (1)

where ρ is the density, cs equals the speed of sound,

and ω gives the angular frequency.

In the absorbing glass wool, the damping enters

the equation as a complex speed of sound, cc = ω/kc,

and a complex density, ρc = kc Zc/ω, where kc is the

complex wave number and Zc equals the complex

impedance. For a highly porous material with a rigid

skeleton, the well-known model of Delany and

Bazley estimates these parameters as functions of

frequency and flow resistivity.

Using the original coefficients of Delany and

Bazley [1], the expressions are

⋅⋅−

⋅+⋅=

−− 595.07.0

189.0098.01f

a

f

a

acR

fi

R

fkk

ρρ (2)

⋅⋅−

⋅+⋅=

−− 732.0734.0

087.0057.01f

a

f

aac

R

fi

R

fZZ

ρρ (3)

where Rf is the flow resistivity, and where ka = ω/ca

and Za = ρa ca are the free-space wave number and

impedance of air, respectively. You can find flow

resistivities in tables. For glass-wool-like materials,

Bies and Hansen [2] give an empirical correlation

2

53.191018.3

av

ap

fd

Rρ⋅⋅

=−

(4)

where ρap is the material’s apparent density and dav

is the mean fiber diameter. This model uses a rather

lightweight glass wool with ρap = 12 kg/m3 and dav =

10 µm.

4 Boundary Conditions The boundary conditions are of three types:

• At the solid boundaries, which are the outer walls of the resonator chamber and the

pipes, the model uses sound hard (wall)

boundary conditions:

0=⋅

∇− nρp

(5)

• The boundary condition at the inlet involves a combination of incoming and outgoing

plane waves:

( )[ ] ( )rknk

n

⋅−

⋅−+∆=

=∆++∇⋅

ik

T

T

ep

ikpk

i

pk

i

p

pikp

0

0

0

00

12

2

1

ρ

ρ (6)

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 44

In this equation, p0 represents the applied outer

pressure, ∆T is the boundary tangential Laplace

operator, and i equals the imaginary unit [3]. This

boundary condition is valid as long as the frequency

is kept below the cutoff frequency for the second

propagating mode in the tube.

• At the outlet boundary, the model specifies an outgoing plane wave:

02

1

00

=∆++∇⋅ pk

ip

kipn Tρρ

(7)

5 Results And Conclusions We apply the required boundary conditions and then

perform the meshing for free tetrahedral option with

0,25 x – direction scale.

a.

b.

Fig. 2. Meshing of: a) geometry 1 - lined muffler;

b) geometry 2 - non-lined muffler

Figure 2 show the 3D meshing of geometry 1 for

lined muffler and geometry 2 for non-lined muffler.

The following equation defines the attenuation

(in dB) of the acoustic energy (or transmission loss),

dw [4,5]:

==

i

o

ww

wdTL log10 (8)

Here wo and wi denote the outgoing power at the

outlet and the incoming power at the inlet,

respectively. You can calculate each of these

quantities as an integral over the corresponding

surface [7,8]:

dAc

pw

s

o ∫Ω∂

=ρ2

2

(9)

dAc

pw

s

i ∫Ω∂

=ρ2

2

0 (10)

In the Figure 3 is plot the result of acoustic

pressure levels for geometry 1, absolute pressure

(Fig. 3a) for the case of an empty muffler without

any absorbing material (reactive muffler) and total

accoustic pressure field (Fig. 3b) with a layer of

lining on the chamber’s upper and lower walls

(dissipative muffler).

a.

b.

Fig.3. Acoustic pressure levels for geometry 1 at

1250 Hz: a). Absolute pressure; b). Total acoustic

pressure field

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 45

Figure 4 shows the result of geometry 1, a

parametric frequency study for the case of an empty

muffler without any absorbing material (Solution 1

– reactive muffler) and attenuation with a layer of

lining on the chamber’s upper and lower walls

(Solution 2 – dissipative muffler). The plot shows

that the damping works rather well for most low

frequencies with the exception of a few distinct dips

where the muffler chamber displays resonances.

At frequencies higher than approximately 1250

Hz, the plot’s behaviour is more complicated and

there is generally less damping, especially for

reactive muffler. This is because, for such

frequencies, the tube supports not only longitudinal

resonances but also cross-sectional propagation

modes. Not very far above this frequency a whole

range of modes that are combinations of this

propagation mode and the longitudinal modes

participate, making the damping properties

increasingly unpredictable.

In the Figure 5, is plot the result of acoustic

pressure levels for geometry 2, absolute pressure

(Fig. 4a) for the case of an empty muffler without

any absorbing material and total accoustic pressure

field (Fig. 4b) with a layer of lining on the

chamber’s upper and lower walls.

Figure 6 shows the result of geometry 2, a

parametric frequency study for the case of an empty

muffler without any absorbing material (Solution 3

– reactive muffler) and attenuation with a layer of

lining on the chamber’s upper and lower walls

(Solution 4 – dissipative muffler).

a.

b.

Fig. 5. Acoustic pressure levels for geometry 2 at

1250 Hz: a). Absolute pressure; b). Total acoustic

pressure field

Fig. 4. Transmission loss (dB) of muffler - geometry 1 - as a function of frequency

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 46

The glass-wool lining improves attenuation at

higher frequencies (Solution 2 and 4).

This model uses the Pressure Acoustics physics

interface of the Acoustics Module. This interface

has the Delany-Bazley [5] coefficients built in.

Therefore, the only damping parameter you need to

supply is the flow resistivity. The parametric solver

provides results for a range of frequencies. The

software computes integrals in the power

expressions using boundary integration coupling

variables, and it plots the resulting attenuation

versus frequency.

In the Figure 7 and 8 is plot the result of

isosurface of total acoustic pressure field for

selected geometry of the muffler for the case of an

empty muffler (see Fig. 7a and Fig. 8a) without any

absorbing material and with a layer of lining on the

chamber’s upper and lower walls (see Fig. 7b and

Fig. 8b).

For both cases, the graphical representation

(Fig. 7 and 8) corresponds to frequency of 1250 Hz.

An isosurface is a three-dimensional analog of an

isoline. It is a surface that represents points of a

constant value (in this case it is pressure but it may

be and depending on temperature, velocity or

density) within a volume of space of muffler; in

other words, it is a level set of a continuous function

whose domain is 3D-space. For the representation in

Figure 7 and 8 was set a total of 15 levels (color

table).

a.

b.

Fig. 7. Isosurface – Total acoustic pressure field in

case of geometry 1

Fig. 6. Transmission loss (dB) of muffler - geometry 2 - as a function of frequency

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 47

a.

b.

Fig. 8. Isosurface – Total acoustic pressure field in

case of geometry 2

It should be noted, however, not always possible

to use a sound-absorbing material, due to various

factors such as high temperature or high speed due

to internal gas.

We are still concerned, in the near future, the

verification method with finite element modeling

results determined in real conditions.

Usually, we have to take account the influence of

temperature, gas flow rate through the damper

characteristics as well as materials used in

construction of noise attenuation.

ACKNOWLEDGEMENT The authors acknowledge the support of the

Managing Authority for Sectorial Operational

Programme for Human Resources Development

(AMPOSDRU) for creating the possibility to

perform these researches by Grant

POSDRU/89/1.5/S/62557.

References:

[1] M. A. Delany and E. N. Bazley, “Acoustic

properties of fibrous absorbent materials”,

Appl. Acoust., vol. 3, pp. 105–116, 1970.

[2] D. A. Bies and C. H. Hansen, “Flow resistance

information for acoustical design”, Appl.

Acoust., vol. 14, pp. 357–391, 1980.

[3] D. Givoli and B. Neta, “High-order non-

reflecting boundary scheme for time-dependent

waves,” J. Comp. Phys., vol. 186, pp. 24–46,

2003.

[4] M.V. Predoi, “Finite Elements Simulations of

Noise Damping in a Muffler”, Romanian

Journal of Acoustics and Vibration, vol VI,

issue2/2009, pp. 71-74, ISSN 1584-7284.

[5] COMSOL Multiphysics, User’s Manual,

COMSOL A.B. 2008.

[6] O. Vasile, Transmission loss assessment for a

muffler by boundary element method approach,

Analele UniversităŃii “Eftimie Murgu” din

ReşiŃa – Fascicula de inginerie, Anul XVII,

No. 1, 2010, pp. 233-242.

[7] O. Vasile, N. Gillich, N. Laurentiu, "Finite

element analysis for reactive and dissipative

rectangular muffler", Proceedings of the 11th

WSEAS international conference on Signal

processing, computational geometry and

artificial vision, and Proceedings of the 11th

WSEAS international conference on Systems

theory and scientific computation, Florence,

Italy, August 23-25, 2011, ISBN 978-1-61804-

027-5, pp. 251-255.

[8] O. Vasile, “The influence of soundproofing

materials on mufflers”, Sesiunea de comunicări

ştiinŃifice a Catedrei de Mecanică Tehnică şi

Mecanisme – SIMEC 2011, ISSN 1842-8045,

pp. 281-284.

[9] F.C. Berinde, G.R. Gillich, “Consideration

Regarding the Use of the Time-Frequency

Representations in Analysis of Vibrations”,

Analele Universității Eftimie Murgu Rețița.

Fascicula de Inginerie, 13, 2006.

[10] G.R. Gillich, N. Gillich, C.P. Chioncel, F.

Cziple, “Legal aspects concerning the

evaluation of pollution effects due to vibration

in urban areas”, Journal of Enviromental

Protection and Ecology, 9 (2), 2008.

Advances in Remote Sensing, Finite Differences and Information Security

ISBN: 978-1-61804-127-2 48