Muffler with Perforates

The original version of this model was developed by Dr. Sabry Allam and Prof. Mats Åbom at the Marcus Wallenberg Laboratory for Sound and Vibration Research, Royal Institute of Technology, Stockholm, Sweden. Dr. Allam and Prof. Åbom also provided the experimental data used in the model.

There are two basic types of mufflers:

•

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 and thermal losses in fibrous materials or flow-related (resistive) losses in perforated pipes.

Reflective 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. 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.

In the industry, exhaust systems are typically analyzed with nonlinear 1D gas-dynamics codes. Such codes, however, do not capture 3D acoustic effects such as higher-order duct modes, and the modeling of fibrous materials is not satisfactory. In practice, there is therefore a need to use linear acoustic models of exhaust and intake systems to enable detailed modeling and optimization of the acoustic response, at the cost of neglecting nonlinear effects. The Port boundary conditions enable the analysis of plane waves and higher order modes above their cutoff frequency propagating through the inlet and outlet. In this model, we stay below the cutoff frequency of the first mode and only model the plane wave propagation.

Model Definition

The muffler you analyze here is an example of a complex hybrid muffler in which the dissipative element is created completely by flow through perforated pipes and plates. When designing a model for a muffler without fibrous materials you need to consider the following aspects:

•

Geometry — The design for this model is based on a modular muffler developed for research purposes. It closely resembles commercially available automotive mufflers, and was used as a test case for muffler modeling in an EC-project (ARTEMIS).

•

Mean flow distribution — The Mach number in an exhaust system is normally less than 0.3. This means that in mufflers with flow expansions, the average Mach number is quite small (less than 0.1). For such cases, you can neglect the convective flow effects, and the only important effect of the mean flow is its influence on the impedance of perforated pipes/plates. This model treats the case where there is no mean flow in the muffler.

•

Temperature distribution — In a running engine, the air temperature inside the muffler is typically in the range 300–400 °C. There is also a temperature gradient through the muffler. However, the acoustic effect of this gradient is small and the average temperature is normally used to calculate the speed of sound. In this case, the experiments were performed at room temperature (20 °C). The model therefore assumes the temperature in the muffler to be constant and uses the default values for air density and speed of sound at 1 atm and 20 °C.

A schematic cross section of the muffler geometry is shown in Figure 1.

Figure 1: Muffler geometry cross section.

The detailed design and dimensions of the outlet pipe and the four baffles (as seen from the right in Figure 1) are given in Figure 2 through Figure 5.

Figure 2: Outlet pipe. A stainless steel sleeve is located above the left perforated section with 288 holes. The other two pipe perforates contain 144 holes each.

Figure 3: Baffle number 1, outlet side to the left and inlet side to the right.

Figure 4: Baffles number 2 and 3.

Figure 5: Baffle number 4.

The model geometry is supplied as a CAD file. To reduce computation time only the upper half of the muffler is included; a symmetry condition is added at z = 0. (A careful inspection of the drawings in Figure 3–Figure 5 shows that the reflection symmetry is not perfect for the perforates on the outlet side. However, the asymmetry is so minor that it is safe to neglect its effects.)

In the CAD geometry, the perforated regions are outlined by edges drawn on the corresponding boundaries. This is illustrated for baffle number 1 in Figure 6 where the perforated regions have been shaded for emphasis.

Figure 6: Perforated regions of baffle number 1.

You model the acoustic effects of the perforates by applying the Acoustics Module’s Interior Perforated Plate boundary condition on these regions. A thorough description of the perforated plate impedance models is available in the section Theory for the Interior Impedance Models in the Acoustics Module User’s Guide. The expression for the transfer impedance Zi of the plate depends on the chosen model type, but is in general a function of the plate geometry, the porosity pattern, the material parameters of the fluid, and the wave number k. The last is given by 2π f/c where f denotes the frequency. You run the simulation for a range of frequencies between 20 Hz and 600 Hz.

The Interior Perforated Plate boundary condition also allows user defined contributions to the impedance. For example, you will specify the additional resistance caused by a mean flow in the muffler to include the effects of the metallic sleeve above pipe perforate number 1. For details about the effect of flow see Ref. 1, Ref. 2, and Ref. 3.

The relevant input parameters for the model are listed in Table 1. The porosity values were obtained by dividing the total area of the holes in each perforate with the area of the corresponding region in the CAD geometry. The resistance of the metallic sleeve was experimentally measured.

Table 1: Model Input parameters.
Property	Value	Description
tp	1.5 mm	Plate thickness
dh	5 mm	Hole diameter in perforates
σp	0.22	Porosity, pipe perforates
σbi	0.46	Porosity, baffle perforates on inlet side
σbo	0.30	Porosity, baffle perforates on outlet side
θsleeve	1	Specific resistance, metallic sleeve

The parameters related to the material are taken from the surrounding fluid material data.

Results and Discussion

The transmission loss in the muffler is defined as

(1)

where Pin and Pout denote the total acoustic power at the inlet and the outlet, respectively. The incident and outgoing power flow is automatically calculated by the Port boundary conditions (variables acpr.port1.P_in and acpr.port2.P_out). The Port boundary condition automatically registers the symmetry condition, and calculates the correct total power. For plane wave excitation it is relatively straightforward to calculate the power (it is proportional to p2), however this becomes much more complicated if higher order modes are included. This is taken care of by the built-in expressions available with the Port boundary conditions. Figure 7 displays the Acoustics Module modeling results for the transmission loss as a function of sound frequency together with experimentally measured values.

Figure 7: Transmission loss versus frequency: model simulation results and experimentally measured values.

As the figure shows, the agreement is excellent except in the range 170–300 Hz. The relatively limited agreement in the range of 170–300 Hz could be caused by the effects of the flow on the acoustic field, as the measurements were done with air flowing through the muffler. The current model only considers the additional resistance of the perforated plates due to the flow, while in reality the flow will have a more complex effect.

As the acoustic excitation goes higher in frequency, it will start to induce vibrations in the outer shell. The agreement at high frequencies can be improved by including the effect of the shell vibrations as illustrated in the Application Gallery model “Absorptive Muffler with Shells” (www.comsol.com/model/absorptive-muffler-with-shells-14717).

You can get a better sense of the results by studying the sound pressure level field inside the muffler for selected frequencies. The plots in Figure 8 display this field for the frequencies 530 Hz and 555 Hz, respectively. As Figure 7 shows, the former frequency corresponds to a local maximum for the transmission loss whereas the latter gives a local minimum. In Figure 8 you can see these how these properties are related to the sound pressure level distributions near the muffler inlet and outlet.

Figure 8: Sound pressure level distributions at 530 Hz (left) and 555 Hz (right).

The propagation of the acoustic energy through the muffler system is illustrated in Figure 9 as a streamline plot of the acoustic intensity vector.

Figure 9: Streamline plot of the acoustic intensity field. The color scale represents the magnitude of the acoustic intensity vector.

References

1. E.J. Rice, “A Theoretical Study of the Acoustic Impedance of Orifices in the Presence of a Steady Grazing Flow,” NASA report TM X-71903, 1976.

2. T. Elnady, Modelling and Characterization of Perforates in Lined Ducts and Mufflers, doctoral dissertation, Dept. Aeronautical and Vehicle Eng., Royal Institute of Technology, Stockholm, 2004.

3. R. Kirby, “Transmission Loss Predictions for Dissipative Silencers of Arbitrary Cross Section in the Presence of Mean Flow,” J. Aoust. Soc. Am., vol. 114, pp. 200–209, 2003.

Acoustics_Module/Automotive/perforated_muffler

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

A horizontal symmetry plane through the muffler means that it is sufficient to model only half of the geometry. Here, the entire geometry is imported as a sequence from the geometry file. The instructions to the geometry can be found in the appendix at the end of this document.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file perforated_muffler_geom_sequence.mph.

In the toolbar, click

You should now see the upper half of a muffler in the drawing area.

Root

Enter the parameters needed for the model or load them from the file perforated_muffler_parameters.txt.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
p0	1[Pa]	1 Pa	Inlet pressure
t_w	1.5[mm]	0.0015 m	Wall thickness
d_h	5[mm]	0.005 m	Hole diameter
sigma_p	0.22	0.22	Porosity, pipe perforates
sigma_bi	0.46	0.46	Porosity, baffles on inlet side
sigma_bo	0.3	0.3	Porosity, baffles on outlet side

Definitions

Define the following interpolation function (measurement data) and variable definitions so that you can compute the transmission loss and compare it with experimental data.

Interpolation 1 (int1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click

Browse to the model’s Application Libraries folder and double-click the file perforated_muffler_exp_data.txt.

Click

In the text field, type TL_exp.

Materials

The walls of the muffler are considered perfectly rigid and will not use any material properties. The only material you need to select is air.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Pressure Acoustics, Frequency Domain (acpr)

With all selections, variables, and material settings completed, it is time to set up the physics.

All boundaries that are exterior to the geometry are by default assigned the Sound Hard boundary condition. Start by applying the Interior Sound Hard condition to the interior rigid walls.

Interior Sound Hard Boundary (Wall) 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

You need four Interior Perforated Plate conditions, one for each different set of parameters. The Thin plate (default) model type is appropriate here since the value ktp is much less than 1.

Interior Perforated Plate 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the σ text field, type sigma_p.

In the tp text field, type t_w.

In the dh text field, type d_h.

Locate the section. From the list, choose .

Interior Perforated Plate 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the σ text field, type sigma_p.

In the tp text field, type t_w.

In the dh text field, type d_h.

Locate the section. From the list, choose .

Locate the section. Select the check box.

In the θ(user) text field, type 1.

The user-defined resistance represents the impedance contribution from the metallic sleeve outside the pipe.

Interior Perforated Plate 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the σ text field, type sigma_bi.

In the tp text field, type t_w.

In the dh text field, type d_h.

Locate the section. From the list, choose .

Interior Perforated Plate 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the σ text field, type sigma_bo.

In the tp text field, type t_w.

In the dh text field, type d_h.

Locate the section. From the list, choose .

Use the Port boundary conditions at the inlet and outlet. Note that the first port at the inlet has an incident field excitation. Both ports capture ingoing and outgoing plane-wave modes. If the frequency range is extended above the cutoff frequency of the first non-plane mode, then add more ports at the inlet and outlet to capture these. The cutoff frequency of a port can be evaluated in postprocessing.

Port 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the Ain text field, type p0.

Circular Port Reference Axis 1

In the toolbar, click

and choose .

Select two points at the inlet boundary to define the reference for the azimuthal angle. This becomes important if higher-order modes are included in the model. Note that because of the symmetry only certain symmetric modes should be selected.

In the window for , locate the section.

Click

Select Points 5 and 8 only.

Port 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Circular Port Reference Axis 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Points 1 and 4 only.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Mesh

Proceed and generate the mesh using the functionality. The frequency controlling the maximum element size is per default taken . Set the desired in the study step. In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. In this model, we use 8 elements per wavelength; the default is to have 5.

Study 1

Select to solve for the frequency range from 20 Hz to 600 Hz, in steps of 5 Hz.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(20,5,600).

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type 8.

Click

Study 1

In the toolbar, click

Results

Create a Mirror 3D dataset in order to visualize the solution on the full geometry.

Mirror 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Acoustic Pressure (acpr)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

The plot shows the pressure distribution on the surface of the muffler. Follow the instructions to see the sound pressure level on a slice inside it.

Surface 1

In the window, expand the node.

Right-click and choose .

Sound Pressure Level (acpr)

In the window, click .

In the window for , locate the section.

From the list, choose .

Multislice 1

In the toolbar, click

and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. In the text field, type 5.

Find the subsection. In the text field, type 0.

In the toolbar, click

At 600 Hz, most of the sound leaves the inlet pipe through the perforate. Take a look at some other frequencies to reproduce Figure 8.

Sound Pressure Level (acpr)

In the window, click .

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

Locate the section. From the list, choose .

In the toolbar, click

Locate the section. From the list, choose .

In the toolbar, click

Acoustic Pressure, Isosurfaces (acpr)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

The last default plot, depicting pressure isosurfaces should look like the figure below.

To study the transmission loss as a function of the frequency and reproduce Figure 7, add a 1D plot.

Transmission Loss

In the toolbar, click

and choose .

In the window for , type Transmission Loss in the text field.

Locate the section.

Select the check box. In the associated text field, type Frequency (Hz).

Select the check box. In the associated text field, type Transmission loss (dB).

In the window, click .

Octave Band 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type acpr.port1.P_in.

In the text field, type acpr.port2.P_out.

Locate the section. From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Computed

Transmission Loss

Right-click and choose .

Global 1

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
TL_exp(freq)		Transmission loss

Click to expand the section. Find the subsection. From the list, choose .

Click to expand the section. From the list, choose .

In the table, enter the following settings:


Legends
Measured

Transmission Loss

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Finally, create a streamline plot of the acoustic intensity vector (SI unit: W/m2) in order to visualize the direction of propagation of the acoustic energy in the muffler system.

Intensity

In the toolbar, click

and choose .

In the window for , type Intensity in the text field.

Locate the section. From the list, choose .

Streamline 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. From the list, choose .

Color Expression 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

The result should look like the plot in Figure 9.

Appendix: Geometry Sequence Instructions

Add Component

In the toolbar, click

and choose .

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Ellipse 1 (e1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.145.

In the text field, type 0.05.

In the text field, type 180.

Extrude 1 (ext1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Distances (m)
0.225
0.225+0.128
0.225+0.128+0.098
0.225+0.128+0.098+0.098
0.225+0.128+0.098+0.098+0.171

Work Plane 2 (wp2)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type -0.1.

Locate the section. Clear the check box.

Work Plane 2 (wp2)>Plane Geometry

In the window, click .

Work Plane 2 (wp2)>Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 28.5[mm].

In the text field, type 180.

Locate the section. In the text field, type -22.5[mm].

Work Plane 2 (wp2)>Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 22.5[mm].

In the text field, type 180.

Locate the section. In the text field, type 63[mm].

Extrude 2 (ext2)

In the window, under right-click and choose .

In the window for , locate the section.

In the list, select .

Click

Select the object only.

Locate the section. In the table, enter the following settings:


Distances (m)
0.1
0.128
0.272
0.325
0.453
0.47
0.542
0.551
0.564

Extrude 3 (ext3)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Distances (m)
0.1
0.325
0.347
0.437
0.453
0.551
0.649
0.73

Click the

button in the toolbar.

Work Plane 3 (wp3)

In the toolbar, click

In the window toolbar, click

next to

, then choose .

Click the

button in the toolbar.

On the object , select Boundaries 18, 13, 8, 3, 5, 10, 15, and 20 only.

Click the

button in the toolbar.

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 6 only.

Work Plane 3 (wp3)>Plane Geometry

In the window, click .

Work Plane 3 (wp3)>Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type -117[mm] -137.5[mm] -137.5[mm] -117[mm] -108[mm] -96.5[mm] -96.5[mm] -117[mm].

In the text field, type -25[mm] -25[mm] -13.5[mm] -13.5[mm] -4.5[mm] -4.5[mm] -25[mm] -25[mm].

Work Plane 3 (wp3)>Chamfer 1 (cha1)

In the toolbar, click

On the object , select Points 2 and 7 only.

In the window for , locate the section.

In the text field, type 2.5[mm].

Work Plane 3 (wp3)>Polygon 2 (pol2)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type 136.5[mm] 136.5[mm] 136.5[mm] 131[mm] 131[mm] 121[mm] 121[mm] 113.5[mm] 113.5[mm] 105.5[mm] 105.5[mm] 97.3[mm] 97.3[mm] 90[mm] 90[mm] 80.8[mm] 80.8[mm] 72.4[mm] 72.4[mm] 72.4[mm] 72.4[mm] 60[mm] 60[mm] 55[mm] 55[mm] 45.9[mm] 45.9[mm] 37[mm] 37[mm] 28[mm] 28[mm] 28[mm] 28[mm] 37[mm] 37[mm] 45.9[mm] 45.9[mm] 55[mm] 55[mm] 63.8[mm] 63.8[mm] 72.4[mm] 72.4[mm] 80.8[mm] 80.8[mm] 89[mm] 89[mm] 97.3[mm] 97.3[mm] 105.5[mm] 105.5[mm] 113.5[mm] 113.5[mm] 121[mm] 121[mm] 126[mm] 126[mm] 123.3[mm] 123.3[mm] 123.3[mm] 123.3[mm] 136.5[mm] .

In the text field, type -25[mm] -20.5[mm] -20.5[mm] -12.2[mm] -12.2[mm] -4[mm] -4[mm] -0.1[mm] -0.1[mm] 3.3[mm] 3.3[mm] 6[mm] 6[mm] 8[mm] 8[mm] 10.3[mm] 10.3[mm] 12[mm] 12[mm] 12[mm] 12[mm] 14[mm] 14[mm] 15[mm] 15[mm] 15.7[mm] 15.7[mm] 16.8[mm] 16.8[mm] 17.5[mm] 17.5[mm] 12.5[mm] 12.5[mm] 11.8[mm] 11.8[mm] 10.7[mm] 10.7[mm] 10[mm] 10[mm] 8.5[mm] 8.5[mm] 7[mm] 7[mm] 5.3[mm] 5.3[mm] 3.5[mm] 3.5[mm] 1[mm] 1[mm] -1.7[mm] -1.7[mm] -5.1[mm] -5.1[mm] -9[mm] -9[mm] -12.2[mm] -12.2[mm] -20.5[mm] -20.5[mm] -25[mm] -25[mm] -25[mm] .

Work Plane 4 (wp4)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 11 only.

In the tree, select .

Work Plane 4 (wp4)>Plane Geometry

In the window, click .

Work Plane 4 (wp4)>Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type -137.5[mm] -137.5[mm] -135[mm] -117[mm] -108[mm] -96.5[mm] -90[mm] -81[mm] -69.5[mm] -63[mm] -60.5[mm] -63[mm] -63[mm] -69.5[mm] -69.5[mm] -96.5[mm] -117[mm] -137.5[mm].

In the text field, type -25[mm] -13.5[mm] -13.5[mm] -13.5[mm] -4.5[mm] -4.5[mm] -4.5[mm] 4.5[mm] 4.5[mm] 4.5[mm] 2[mm] -0.5[mm] -3.5[mm] -9[mm] -25[mm] -25[mm] -25[mm] -25[mm].

Work Plane 4 (wp4)>Polygon 2 (pol2)

In the toolbar, click