COMSOL 6.3 - Absorptive Muffler

This example describes the pressure-wave propagation in a muffler for an internal combustion engine. The approach used here is generally applicable when analyzing the damping of propagating pressure waves as well as determining the transmission properties of a given system. The model uses the port boundary conditions to model the inlet and outlet of the muffler. The model shows how to analyze both inductive and resistive damping in pressure acoustics. The main output is the transmission loss for the frequency range 50 Hz–2800 Hz. It is represented both as a continuous curve and given in 1/3 octave bands.

See also the tutorial model Eigenmodes in a Muffler, which computes the propagating modes in the main chamber of the muffler.

Model Definition

The muffler—schematically shown in Figure 1—consists of a 24-liter resonator chamber with a section of the centered exhaust pipe included at each end. The model is first set up assuming that the chamber is empty. In a second step, it is lined with 15 mm of absorbing glass wool.

Figure 1: Geometry of the lined muffler; the liner is the outer layer in the main muffler volume. The exhaust fumes enter through the left pipe and exit through the right pipe.

Domain Equations

This model solves the problem in the frequency domain using the Pressure Acoustics, Frequency Domain interface. The model equation is a slightly modified version of the Helmholtz equation

where p is the acoustic pressure, ρ is the density, c is the speed of sound, and ω is the angular frequency. The subscript c refers to that these material properties can be complex valued.

In the absorbing glass wool, modeled as a domain, 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. This is an equivalent fluid model for the porous domain where the losses are modeled in a homogenized way.

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. This class of poroacoustic models is named in the user interface. Using the original coefficients/constants of Delany and Bazley (Ref. 1), the expressions are

where Rf is the flow resistivity, and where k0 = ω/c and Z0 = ρf c are the free-space wave number and characteristic impedance of air, respectively. Several different coefficients/constants for the empirical fitting model can be selected in the feature, each representing different porous or fibrous materials. The model is the default selected in the domain feature. Several porous models can be selected here depending on the situation at hand.

You can find flow resistivities in tables (see, for example, Ref. 3) or by measuring it. For glass-wool-like materials, Bies and Hansen (Ref. 2) give an empirical expression

where ρap is the apparent density of the material and dav is the mean fiber diameter. This model uses a rather lightweight glass wool with ρap = 12 kg/m3 and dav = 10 μm.

The Delany-Bazley constants are valid for values of

up to

. The frequency limit of 2800 Hz gives a value of X = 2.36, meaning this is at the upper limit of validity. Other variants of the Delany-Bazley-Miki model are built into the Acoustics Module; these have different validity regions or are used for other fibrous materials. For example, the Miki constants can be selected, they extend the region of applicability of the model in the low X limit, as compared to the Delany-Bazley constants.

Boundary Conditions

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At the solid boundaries, which are the outer walls of the resonator chamber and the pipes, the model uses sound hard (wall) boundary conditions. The condition imposes that the normal velocity at the boundary is zero.

•

The model uses port boundary conditions to model the inlet and outlet of the muffler. In waveguides, the port conditions are superior to radiation conditions as they can capture complex wave fields that involve several propagating modes. The port uses a plane-wave (0,0) mode to excite the system. Since the analysis is carried out above the cutoff frequency of the first nonplane mode (above 2514 Hz), several port conditions are required at the inlet and outlet to capture these modes. Note that when including an azimuthal mode in a circular port, it is necessary to add both possible azimuthal dependencies (sine and cosine) as different Port conditions. The cosine and sine variants are orthogonal modes and need to be included for a full wave field decomposition. Note also that each port condition generates a postprocessing variable that defined the cutoff frequency of its mode, for example, for the variable is acpr.port2.fc.

Results and Discussion

The pressure distribution in the absorptive muffler without the lining material is shown in Figure 2 for the frequency f = 2800 Hz. From the figure, it is seen that at this frequency not only longitudinal standing waves exist but also transverse modes are present.

Figure 2: Pressure represented as an iso-barometric surface plot in the muffler.

An important parameter for a muffler is the transmission loss or attenuation. It is defined as the ratio between the incoming and outgoing acoustic energy. The attenuation or transmission loss L (in dB) of the acoustic energy is defined by

Here, Pin and Pout denote the incoming power at the inlet and the outgoing power at the outlet, respectively. These values are readily derived from the port boundary variables as acpr.port1.P_in, and the sum of acpr.port4.P_out, acpr.port5.P_out, and acpr.port6.P_out (the three ports at the outlet). These variables can be directly used in postprocessing.

Figure 3: Comparison of the transmission loss as function of frequency for the empty muffler and the muffler with absorptive lining. (top) The transmission loss depicted as a continuous curve. The first four dips are due to longitudinal resonances. In the muffler with absorbing lining the dips are still present, but the general trend is that the higher the frequency, the better the damping. (bottom) The same data but depicted in 1/3 octave bands.

Figure 3 (top and bottom) shows the result of a parametric frequency study. The two graphs represent the case of an empty muffler without any absorbing lining material (blue lines) and the case with a layer of glass wool lining on the chamber’s walls (green lines). In the top figure, the transmission loss is depicted as a continuous curve (pure tone sweep) while it is depicted in 1/3 octave bands in the bottom figure. Both graphs are created using the Octave Band plot of the Acoustics Module.

The graph for the undamped muffler shows that damping works rather well for most low frequencies. At frequencies higher than approximately 1000 Hz, the behavior is more complicated and there is generally less damping. Above this frequency a whole range of modes that are combinations of propagating mode and longitudinal modes become available, making the damping properties increasingly unpredictable. For an analysis of these modes, see the related model Eigenmodes in a Muffler. The glass-wool lining improves attenuation at the resonance frequencies as well as at higher frequencies.

Figure 4: Intensity streamlines at 2800 Hz without the liner.

The flow of energy in the muffler without the liner is shown in Figure 4 at 2800 Hz. The plot represents the intensity field depicted as streamlines. The intensity is per definition the time average of the intensity and thus represents the average energy flow in the system. Here from the inlet to the outlet. Change between solutions and frequencies to study and visualize the sound-absorbing properties of the muffler.

Figure 5: Normalized power balance for the muffler with absorptive lining.

The normalized power balance for the muffler with absorptive lining, shown in Figure 5,shows how plane modes are the main mechanisms for the transmitted and reflected power. Note how the azimuthal modes are only present above 2514 Hz, which is the cutoff frequency for this mode.

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. 13, issue 5, pp. 357–391, 1980.

3. T.J. Cox and P.D’Antonio, Acoustic Absorbers and Diffusers, 2nd ed., Taylor and Francis, 2009.

Acoustics_Module/Automotive/absorptive_muffler

Modeling Instructions

This section contains the modeling instructions for the Absorptive Muffler model. They are followed by the Geometry Sequence Instructions section.

The instructions take you through two versions of the model, first one with a completely hollow chamber with rigid walls, then one where the chamber is lined with glass wool.

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

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

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

The parameters define the physical values of the system.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

To save some time import the geometry sequence from a file. The instructions for setting up the geometry can be found in the Geometry Sequence Instructions section at the bottom of this document.

In the toolbar, click and choose .

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

In the toolbar, click

Having imported the geometry, it can be easily modified as it is parameterized. Simply change the value of a dimension in the parameters list: this will update the geometry automatically. The imported geometry parameters are automatically added to the node.

The geometry should look like the one depicted in Figure 1.

Definitions

Create selections for the inlet and outlet of the muffler.

Inlet

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundary 1 only.

Outlet

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundary 28 only.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

By default, the first material you add applies on all domains so you do not need to alter the geometric scope settings.

In the second version of this model, you will use a lining material in Domain 2. Add such a material with an empty selection.

Materials

Absorptive Liner

In the window, under right-click and choose .

In the window for , type Absorptive Liner in the text field.

Pressure Acoustics, Frequency Domain (acpr)

Pressure Acoustics 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type T0.

In the pA text field, type p0.

Use the boundary condition to define the inlet and outlet. The port condition is superior to the classical radiation condition in waveguide configurations. This is particularly the case when nonplane modes start to propagate. This happens above the first cutoff frequency. For the present model, the cutoff frequency for the first nonplane mode (m = 1 and n = 0) is at 2514 Hz so this mode should be included. Note that the variable acpr.port2.fc gives the cutoff frequency of the mode (here for the mode defined on the condition).

Note that when including an azimuthal mode in a circular port, it is necessary to add both possible azimuthal angle dependencies (sine and cosine), each as a different condition. To get a complete basis for the modal expansion at the boundary, both the since and cosine variants of the mode need to be included, as these modes are orthogonal.

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 Apin text field, type p_in.

Port Reference Axis 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Points 1 and 4 only.

Select two points that define a reference axis for the azimuthal angle. The setting is in general necessary when higher-order azimuthal modes are used and/or necessary.

Port 2

In the window, under > right-click and choose .

In the window for , locate the section.

In the m text field, type 1.

Locate the section. From the list, choose .

Port 3

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Port 1, Port 2, Port 3

In the window, under > , Ctrl-click to select , , and .

Right-click and choose .

Inlet Ports

In the window for , type Inlet Ports in the text field.

Port 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Port Reference Axis 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select two points at the outlet in the same way.

Select Points 37 and 40 only.

Port 5

In the window, under > right-click and choose .

In the window for , locate the section.

In the m text field, type 1.

Port 6

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Port 4, Port 5, Port 6

In the window, under > , Ctrl-click to select , , and .

Right-click and choose .

Outlet Ports

In the window for , type Outlet Ports in the text field.

Now, add a poroacoustics model for the absorptive liner domain. You will deactivate this domain when configuring the first study step.

Poroacoustics 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

In the T text field, type T0.

Locate the section. From the list, choose .

The material data for the flow resistivity will now be picked up from the Absorptive Liner material. Enter the data in the material.

Materials

Absorptive Liner (mat2)

In the window, under > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Flow resistivity	Rf_iso ; Rfii = Rf_iso, Rfij = 0	R_f	Pa·s/m²	Poroacoustics model

Mesh 1

Proceed and generate the mesh using the functionality. 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, use the default option, which gives 5 elements per wavelength.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the fmax text field, type fmax.

Click

Study 1 - No Liner

In the window, click .

In the window for , type Study 1 - No Liner in the text field.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(50,25,fmax).

Locate the section. Select the checkbox.

In the tree, select > > .

Right-click and choose .

In the toolbar, click

Results

Acoustic Pressure (acpr)

The first one of the default plots shows the pressure distribution on the walls of the muffler at the highest frequency, 2800 Hz.

The two other default plot groups show the sound pressure level on the wall surface and the pressure inside the muffler as isosurfaces.

Acoustic Pressure, Isosurfaces (acpr)

In the window, click .

Proceed to plot the transmission loss of the muffler system. Use the Octave Band plot, which makes it possible to plot any transfer function both as band plots and as continuous curves (sweeps).

Transmission Loss, Continuous

In the toolbar, click

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

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

Locate the section.

Select the checkbox. In the associated text field, type Power, incoming wave (dB, rel. outgoing wave).

Locate the section. From the list, choose .

Octave Band 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Start by locating and inspecting the postprocessing variables available for the port boundary conditions. Add the variable for the power of the incident mode at Port 1. Then modify the expression manually to get the ratio to the power of the outgoing wave at Port 2. This will give the transmission loss.

Click in the upper-right corner of the section. From the menu, choose > > > > .

Locate the section. In the text field, type acpr.port1.P_in/(acpr.port4.P_out+acpr.port5.P_out+acpr.port6.P_out).

From the list, choose .

Locate the section. From the list, choose .

Click to expand the section. Select the checkbox.

From the list, choose .

In the table, enter the following settings:


Legends
No liner

In the toolbar, click

The plot should be a reproduction of the blue curve in Figure 3.

Proceed to solve the model including a layer of absorptive glass wool on the muffler line. Continue working from where you left off with the model developed thus far and add a second study to keep your existing results intact.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2 - Absorptive Liner

In the window for , type Study 2 - Absorptive Liner in the text field.

Locate the section. Clear the checkbox.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(50,25,fmax).

In the toolbar, click

Results

Acoustic Pressure, Isosurfaces (acpr)

You chose not to have new default plots generated. Once the solution process is finished you can use the existing plot groups and just switch the dataset to see how the damping material affects the solution.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

At 2800 Hz, the pressure in the chamber is much lower than before.

Proceed to study how the transmission loss has changed with the addition of the lining. First do a bit of formatting and then duplicate the first plot and select the new dataset.

Octave Band 2

In the window, under > right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Legends
Absorptive liner

In the toolbar, click

The plot should look like that in Figure 3 top.

Duplicate the Transmission Loss plot and change the format to 1/3 octave bands.

Transmission Loss, 1/3 Octave Bands

In the window, right-click and choose .

In the window for , type Transmission Loss, 1/3 Octave Bands in the text field.

Octave Band 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Octave Band 2

In the window, click .

In the window for , locate the section.

From the list, choose .

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

From the list, choose .

In the toolbar, click

The plot should look like that in Figure 3 bottom.

Now, create a plot that represents the energy flux through the muffler system. Use streamlines that follow the intensity vector. You can change between solutions and frequencies to study and visualize the sound-absorbing properties of the muffler.

Intensity

In the toolbar, click

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

Streamline 1

Right-click and choose .

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

Select Boundary 1 only.

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

In the text field, type 2.

Color Expression 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. From the list, choose .

From the list, choose .

In the toolbar, click

This should reproduce Figure 4.

Normalized Power Balance

In the toolbar, click

In the window for , type Normalized Power Balance in the text field.

Locate the section. From the list, choose .

From the list, choose .

In the text field, type 4.

Click the

button in the toolbar.

Locate the section. Select the checkbox.

In the text field, type 10.

In the text field, type 2850.

In the text field, type 9e-5.

In the text field, type 1.1.

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
acpr.port1.P_out/acpr.port1.P_in	1	Reflected Power - Plane mode
acpr.port3.P_out/acpr.port1.P_in	1	Reflected Power - Second azimuthal mode
acpr.port4.P_out/acpr.port1.P_in	1	Transmitted Power - Plane mode
acpr.port6.P_out/acpr.port1.P_in	1	Transmitted Power - Second azimuthal mode