Reactive Muffler Sbs

8/12/2019 Reactive Muffler Sbs

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any form without prior written consent from COMSOL AB. COMSOL, COMSOL Multiphysics, COMSOL Reac-

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are trademarks or registered trademarks of their respective holders.

ExampleReactive MufflerSOLVED WITH COMSOL MULTIPHYSICS 3.5a


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E X A M P L E R E A C T I V E M U F F L E R | 1

Examp l eRea c t i v e Mu f f l e r

Introduction

This model examines the sound-transmission properties of an idealized reactive

muffler with infinitely long inlet and outlet pipes (or a reflection-free source at the inlet

pipe and a reflection-free end of the outlet pipe) and one expansion chamber. One

measure of the transmission properties is the transmission-loss coefficient,Dtl, which

is defined as

where Wiis the time-averaged incident sound power and Wtis the transmitted sound

power. This problem has a theoretical 1D solution that you can compare with the FEM

solution.

Model Definition

In the following figure, a plane sound wave enters the inlet pipe (left) and is reflected

and attenuated in the expansion chamber. The attenuated sound wave exits through

the outlet pipe (right).

The diameter of both the inlet pipe and the outlet pipe is d, and the corresponding

cross-sectional area isS1. The expansion chamber has a diameterDwith acorresponding cross-sectional areaS2.

Dtl 10WiWt-------

log=

Expansion

L

d

D

Inlet pipe Outlet pipe

chamber

Symmetry line


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According to Ref. 1, the 1D theoretical solution for the transmission loss to this

problem is

where kis the wave number;S1andS2are the areas of the pipes and expansion

chamber; andLgives the length of the expansion chamber.

The model computes the pressure,p, for the fluid in the region defined by the above

geometry. This is a time-harmonic problem so you can use the Helmholtz equation

defined in the axisymmetric Acoustics application mode:

where = 2fis the angular frequency, 0is the fluid density, and csis the speed of

sound. The qterm is a dipole source with the dimension of force per volume.

Because this is an axisymmetric model, you need to include only half of the geometry

as indicated in the following figure:

You must apply axial symmetry boundary conditions on the line of symmetry.

Assume the walls are rigid, and thus use sound-hard (wall) boundary conditions,

which means that the normal derivative of the pressure is zero at the boundaries.

Radiation boundary conditions describe the inlet and outlet boundaries:

Dtl 10 1S1

2 S2--------------

S22 S1--------------

2

kL( )2

sin( )+log=

1

0------ p q( )

2p

0 cs2

--------------- 0=

ExpansionInlet pipe Outlet pipe

chamber

Line of symmetry

n

p0=

n 1

0------ p q( )

ik

0------

p+

ik i k n( )( )p0ei k r( )

0------------------------------------------------------------------=


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The radiation boundary condition is useful when the surroundings are merely a

continuation of the domain, which is the case in this model. The term on the

right-hand side represents an incoming pressure wave with an amplitudep0and adirection given by the wave vector, k. In this model, an incoming pressure wave with

the amplitudep0= 1 Pa enters at the inlet boundary.

To determine the transmission loss in the model, you must first calculate the incident

and transmitted time-averaged sound intensities and the corresponding sound power

values. The equation

gives the time-averaged sound intensities wherepis equal top0at the inlet and the

computed solution at the outlet.

Using the boundary integration tool, you can evaluate the incident and transmitted

sound powers, W, as:

Results and Discussion

Figure 1shows the theoretical transmission loss (square markers) and the COMSOL

Multiphysics solution (triangle markers) as a function of frequency. The theoretical

solution has an upper frequency limit for its validity. This limit is the cut-on frequency,which defines the frequency range where only plane waves can propagate; above this

frequency, also higher modes can propagate.

According to Ref. 1, the first cut-on frequency for a pipe is

.

Its value in this case is approximately 332Hz, but it is evident from the above figurethat a discrepancy exists between the theoretical and the FEM solution, even below the

cut-on frequency. The discrepancy also increases with frequency between the 1D

theoretical model and a 3D analysis, as you can see in Ref. 1. In the lower frequency

range, however, there is good agreement between the theoretical solution and the

FEM solution.

I p220 c-------------=

W I 2r( ) rd=

f01 1.841c

D--------=


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Figure 1: Muffler transmission loss versus frequency: theoretical solution (squares) andCOMSOL Multiphysics solution (triangles).

Reference

1. H.P. Wallin, Ljud och Vibrationer, Institutionen fr Farkostteknik, KTH,Stockholm, Sweden, 1999 (in Swedish).

Model Library path: COMSOL_Multiphysics/Acoustics/reactive_muffler

Modeling Using the Graphical User Interface

M O D E L N A V I G A T O R

1 Go to the Model Navigatorand select Axial symmetry (2D)in the Space dimensionlist.

2 In the list of application modes open the COMSOL Multiphysics>Acoustics>Acoustics

folder and then select Time-harmonic analysis.

3 Click OK.


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O P T I O N S A N D S E T T I N G S

1 Go to the Optionsmenu and choose Constantsto parameterize the model.

2 In the Constants dialog box enter the following constants, representing fluid

properties and some geometrical properties to calculate the cut-on frequency and

the theoretical transmission loss:

G E O M E T R Y M O D E L I N G

1Shift-click the

Rectangle/Squarebutton to specify a rectangle.

2 Go to the Rectangledialog box and type 0.3in the Widthedit field and 1in the

Heightedit field.

3 Click OK.

4 Click the Zoom Extentsbutton.

5 Shift-click the Rectangle/Squarebutton to specify another rectangle.

NAME EXPRESSION DESCRIPTION

rho_air 1.2[kg/m^3] Density of air

c_air 340[m/s] Speed of sound in air

p0 1[Pa] Pressure-source amplituded 0.3[m] Diameter, pipes

D 0.6[m] Diameter, expansion chamber

S1 pi*d^2/4 Cross-sectional area, pipes

S2 pi*D^2/4 Cross-sectional area, expansion

chamber

L 2[m] Length, expansion chamber

f01 1.841*c_air/(pi*D) First cut-on frequencyfreq 20[Hz] Sound frequency


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6 In the Rectangledialog box, type 0.6in the Widthedit field, 2in the Heightedit field,

and 1in the z edit field. Click OK.

7 Shift-click the Rectangle/Squarebutton to specify a third rectangle.

8 In the Rectangledialog box type 0.3in the Widthedit field, 1in the Heightedit field,

and 3in the z edit field. Click OK.

9 Click the Zoom Extentsbutton on the Main toolbar.

P H Y S I C S S E T T I N G S

Subdomain SettingsThis model uses the fluid properties of air, specified in SI units.

Enter these quantities:

1 From the Physicsmenu choose Subdomain Settings.

2 In the Subdomain Settingsdialog box select all subdomains from the Subdomain

selectionlist.

3 Type rho_airin the Fluid densityedit field.

4 Type c_airin the Speed of sound edit field.

5 Leave the default settings (all 0) for the Dipole sourceand the Monopole source.

6 Click OK.

QUANTITY ALL SUBDOMAINS

0 rho_air

cs c_air


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Boundary Conditions

1 From the Physicsmenu choose Boundary Settings.

2 In the Boundary Settings dialog box enter the following boundary condition types

and properties:

3 Select Boundaries 1, 3, and 5 in the Boundary selectionlist.

4 Select Axial symmetryin the Boundary conditionlist.

5 Select Boundaries 812 in the Boundary selectionlist.

6 Select Sound hard boundary (wall)in the Boundary conditionlist.

7 Select Boundary 2 in the Boundary selectionlist.

8 Select Radiation conditionin the Boundary conditionlist.

9 Type 1in the Pressure sourceedit field.

10 Finally select Boundary 7 in the Boundary selectionlist.

11 Select Radiation conditionin the Boundary conditionlist.

12 Click OK.

SETTINGS BOUNDARIES 1, 3, 5 BOUNDARIES 812 BOUNDARY 2 BOUNDARY 7

Boundary

condition

Axial symmetry Sound hard

boundary (wall)

Radiation

condition

Radiation

condition

Wave type Plane wave Plane wave

p0 1 0


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Expression Variables

1 On the Optionsmenu, point to Expressions, and then click Scalar Expressions.

2 In the Scalar Expressions dialog box enter the following:

The red brackets in the Unitcolumn for D_tlappear because P_inand P_out,

which you define as integration coupling variables shortly, do not have units.

Because the expression is, nevertheless, dimensionally correct, you can ignore this

warning.

3 Click OK.

4 Go to the Optionsmenu and choose Expressionsand then Boundary Expressions.

5 In the Boundary Expressions dialog box select Boundary 2 from the Boundary

selectionlist and enter the following boundary expression variable:

6 In the Boundary Expressions dialog box select Boundary 7 from the Boundary

selectionlist and enter the following boundary expression variable:

NAME EXPRESSION DESCRIPTION

k 2*pi*freq/c_air Wave number

D_tl_analytical 10*log10(1+(S1/(2*S2)-

S2/(2*S1))^2*(sin(k*L))^2)

Transmission loss,

theoretical 1D model

D_tl 10*log10(P_in/P_out) Transmission loss

NAME EXPRESSION

I_in real(conj(p0)*p0)/(2*rho_air*c_air)

NAME EXPRESSION

I_in

I_n real(conj(p)*p)/(2*rho_air*c_air)


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7 Click OK.

Integration Coupling Variables

1 Go to the Optionsmenu and choose Integration Coupling Variables and then Boundary

Variables.

2 In the Boundary Integration Variables dialog box select Boundary 2 and then enter

the following boundary integration expression:

3 In the Boundary Integration Variables dialog box select Boundary 7 and enter the

following boundary integration expression; when done, click OK.

M E S H G E N E R A T I O N

1 Go to the Meshmenu and choose Free Mesh Parameters.

2 In the Free Mesh Parametersdialog box, select Finerin the Predefined mesh sizes list.

3 Click Remesh, then click OK.

C O M P U T I N G T H E S O L U T I O N

1 From the Solvemenu, choose Solver Parameters.

2 In the Solver Parameters dialog box, select Parametricfrom the Solverlist.

3 Type freqin the Parameter nameedit field.

4 Type range(20,5,200)in the Parameter valuesedit field.

5 Click OK.

6 Go to the Physics menu and choose Scalar Variables.

7 In the Expressioncolumn, type freqin the edit field for the frequency.

8 Click OK.

9 Click the Solve button to start the simulation.

P O S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

The default visualization plots the magnitude of the pressure field at the final frequency

(200Hz).

NAME EXPRESSION INTEGRATION ORDER

P_in I_in*2*pi*r 4

NAME EXPRESSION INTEGRATION ORDER

P_in

P_out I_n*2*pi*r 4


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Next generate the transmission-loss plots in Figure 1.

1 Go to the Postprocessing menu and select Domain Plot Parameters.

2 On the Generalpage, select all frequencies from the Solutions to uselist in the Domain

Plot Parametersdialog box.

3Select the

Keep current plotcheck box.

4 On the Pointpage, select Point 1 from the Point selectionlist.

5 Type D_tlin the Expressionedit field.

6 Click the Line Settings button and select Trianglein the Line markerlist. Click OK.

7 Click Applyin theDomain Plot Parametersdialog box.

To make it easy to compare the two solutions, plot the theoretical solution in the

same figure.

8 Type D_tl_analyticalin the Expressionedit field.

9 Click the Line Settings button. Select Colorfrom the Line colorlist and Squarefrom

the Line markerlist. Click OK.

10 Click OKin theDomain Plot Parametersdialog box

11 In the figure window, click the Edit Plottoolbar button. Finish the plot by editing

the plot title and axis labels, and adding labels.


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Reactive Muffler Sbs