Absorptive Muffler

This example describes the pressure-wave propagation in a muffler for an internal combustion engine. The approach used here is generally applicable to 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–1500 Hz. It is represented both as a continuous curve and given in 1/3 octave bands.

See also the application 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. Secondly, 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.

In the absorbing glass wool, modeled as a Poroacoustics 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. Using the original coefficients of Delany and Bazley (Ref. 1), the expressions are

where Rf is the flow resistivity, and where ka = ω/ca and Za = ρa ca are the free-space wave number and characteristic impedance of air, respectively. This is the default selected porous model in the Poroacoustics 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 correlation

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 model is valid for values of

up to

, so the upper frequency limit of 1500 Hz ensures that the poroacoustics model applies. Other variants of the Delany-Bazley model are built into the Acoustics Module; these have different validity regions or are used for other fibrous materials. For example, the Miki parameters can be selected, they extend the region of applicability of 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.

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The model uses port boundary conditions to model the inlet and outlet of the muffler. In waveguides, the port conditions are superior to radiation condition as they can capture complex wave fields that involve several propagating modes. In this model, only plane-wave propagation is considered. This means that only one port condition needs to be added at each end. The port uses a plane-wave (0,0) mode. If the analysis is carried out above the cutoff frequency of the first non-plane mode (above 2540 Hz), then simply add more port conditions at the inlet and outlet to capture these modes. Note that each port condition generates a postprocessing variable that defined the cutoff frequency of its mode, for example, for the variable is acpr.port1.fc. The plane-wave cutoff frequency is of course 0 Hz.

Results and Discussion

The pressure distribution in the absorptive muffler without the lining material is shown in Figure 2 for the frequency f = 1500 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 defined by the port boundary conditions as acpr.port1.P_in and acpr.port2.P_out and 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 1250 Hz, the behavior is more complicated and there is generally less damping. This is because the tube supports not only longitudinal resonances but also cross-sectional propagating modes, for such frequencies. Not very far above this frequency a whole range of modes that are combinations of this propagating mode and the longitudinal modes participate, 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.

The flow of energy in the muffler without the liner is shown in Figure 4 at 1500 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 4: Intensity streamlines at 1500 Hz without the liner.

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, Sept./Oct., 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 .

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 .

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.

Explicit 1

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.

Explicit 2

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 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

Material 2 (mat2)

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 wave guide configurations. This is particularly the case when non-plane modes start to propagate. This happens above the first cutoff frequency. For the present model, the cutoff frequency for the first non-plane mode (m = 1 and n = 0) is at 2514 Hz which is much higher than the frequencies studied here. If the frequency analysis is extended above this frequency, then add more port conditions to capture these modes. Note that the variable acpr.port1.fc gives the cutoff frequency of the modes (here for the mode defined on the condition).

Port 1

In the toolbar, click and choose .