COMSOL 6.4 - Acoustics of a Particulate-Filter-Like System

Acoustics of a Particulate-Filter-Like System

This is a model of the acoustics in a particulate-filter-like system. Real systems, like diesel particulate filters (DPFs), are designed to remove/filter soot (diesel particles) from the exhaust of diesel engine vehicles. The porous medium in such systems are typically structured with long air-filled ducts. To simplify this model, the filter geometry is assumed to be axisymmetric and the ducts are represented by long cylindrical groves inside a porous material plug. Although the main function of a particulate filter is filtering of the exhaust flow, the filter also has acoustic damping properties that relate to the muffler system.

The model analyzes the acoustic properties of the simplified 2D axisymmetric particulate-filter like geometry using the Acoustic–Poroelastic Waves Interaction multiphysics interface. This interface describes the small-deformation elastic waves propagating in a porous material coupled to waves in a fluid by an Acoustic–Porous Boundary multiphysics coupling. The model accounts for the coupled displacement and is thus a fluid-structure interaction problem.

Model Definition

Three aligned cylinders make up the particulate filter system under study: an inlet, an outlet, and a main filter cylinder. The particulate filter is located inside the filter cylinder. Figure 1 shows a sketch of a cross section in the rz-plane of the 2D axisymmetric geometry. The filter in the central region is of length Lfilter = 200 mm with a filter radius of Rfilter = 150 mm. The inlet and outlet pipe radii are Rtube = 50 mm. The filter consists of a structured air-filled porous material (the brown region), which could be a silicon carbide matrix. The air-filled groves (light blue) have a width of dh = 5 mm and the porous walls are of thickness ht = 3.2 mm. At the end of each grove there is an impermeable steel plug (black). The rest of the system is filled with air.

The wide groves are used to simplify the model. In real DPF systems the groves are replaced by long slender ducts with a typical width of 1–2 mm and the porous walls have a typical width of 0.3–0.5 mm.

Figure 1: Geometry of the particulate filter with dimensions indicated.

The porous material is assumed to be isotropic with the material parameters as listed in Table 1.

Table 1: Material parameters of the porous matrix.
Parameter	Value	Description
E	20 GPa	Young’s modulus
ν	0.4	Poisson’s ratio
ρd	1000 kg/m3	Drained matrix density
αB	0.3	Biot–Willis coefficient
εp	0.3	Porosity
κp	10-11 m2	Permeability of porous matrix
τ	1	Tortuosity factor

Note that the Biot–Willis coefficient is equal to the porosity for rigid porous materials and is equal to 1 for a soft porous material (or a suspension of solid in liquid). The fluid parameters are those of air including the compressibility, χ, which for an ideal gas is equal to (p0)−1, where p0 is the absolute pressure (here 1 atm).

The filter is characterized acoustically by the transmission loss, TL (given in dB), as a function of the frequency, f. It is defined as

where pincident is the incident inlet pressure and pout is the outlet pressure. You solve the model for the frequency interval 20 Hz–2000 Hz.

When setting up the porous material model, you also need to specify whether to use the low-frequency (default) or high-frequency range approximation for the fluid viscosity. The transition between the two ranges is defined by the reference frequency fc given by the expression

where ρf is the fluid density (for air 1.2 kg/m3) and μ is the dynamic viscosity of the fluid (for air 1.8·10−5 Pa·s). Using the above material parameters gives a reference frequency of the order 100 kHz. Thus, the low-frequency range applies to the current problem.

Results and Discussion

The acoustic transmission loss TL through the axisymmetric simplified particulate filter is determined for the frequency range 20 Hz to 2000 Hz and depicted in Figure 2.

Figure 2: Transmission through the simple particulate filter as a function of frequency.

The loss is seen to be of the same order of magnitude as in real particulate filters (like diesel particulate filters, DPFs) the porous medium is often, as mentioned, structured with long ducts that decrease the acoustic damping while retaining good filtering properties. In this axisymmetric model the ducts take the form of cylindrical slits in 3D, which may introduce some nonstandard resonances in the filter. Moreover, in a real exhaust system there is an interaction between the exhaust flow and the acoustics (here the no-flow situation is studied), and the temperature is higher than 20°C (as used here). Other physical effects include acoustic–structure and poroelastic–structure interactions with the exhaust pipe system. The present simplified model enables isolating the acoustics problem from other physical phenomena.

Figure 3 depicts the pressure distribution inside the particulate filter model for 20 Hz and for 2 kHz.

Figure 3: Pressure distribution inside the particulate filter for f = 20 Hz (top) and f = 2 kHz (bottom).

Acoustics_Module/Automotive/acoustics_particulate_filter

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

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 acoustics_particulate_filter_parameters.txt.

The parameters loaded here define the geometric dimensions and the tortuosity parameter used in the poroelastic model. Because the geometry is now parameterized, changing the dimensions in the parameters list will update the geometry automatically.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rtube.

In the text field, type Ltube.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rfilter.

In the text field, type Lair.

Locate the section. In the text field, type Ltube.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rfilter.

In the text field, type Lfilter.

Locate the section. In the text field, type Ltube+Lair.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	ht

Select the checkbox.

Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rfilter.

In the text field, type Lair.

Locate the section. In the text field, type Ltube+Lair+Lfilter.

Rectangle 5 (r5)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rtube.

In the text field, type Ltube.

Locate the section. In the text field, type Ltube+2*Lair+Lfilter.

Rectangle 6 (r6)

In the toolbar, click

In the window for , locate the section.

In the text field, type dh.

In the text field, type Lfilter.

Locate the section. In the text field, type ht.

In the text field, type Ltube+Lair.

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 18.

Locate the section. In the text field, type (dh+ht).

Form Union (fin)

In the window, click .

In the window for , click

Click the

button in the toolbar.

The geometry of the diesel particulate filter should look like the figure below.

Definitions

All domains

In the toolbar, click

In the window for , type All domains in the text field.

Poroelastic Waves Domains

In the toolbar, click

In the window for , type Poroelastic Waves Domains in the text field.

Select Domains 3–5, 11–13, 17–19, 23–25, 29–31, 35–37, 41–43, 47–49, 53–55, 59–61, 65–67, 71–73, 77–79, 83–85, 89–91, 95–97, 101–103, 107–109, and 113–115 only.

Steel Domains

In the toolbar, click

In the window for , type Steel Domains in the text field.

Select Domains 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, and 110 only.

Air Domains

In the toolbar, click

In the window for , type Air Domains in the text field.

Locate the section. Under , click

In the dialog, select in the list.

Click .

In the window for , locate the section.

Under , click

In the dialog, in the list, choose and .

Click .

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.

By default, the first material applies to all domains except where overridden by subsequent materials in the node.

Next, create a poroelastic material with user-defined material parameters for the solid matrix.

Materials

SiC matrix

In the window, under right-click and choose .

In the window for , type SiC matrix in the text field.

The material properties for the SiC matrix will be defined after the physics interface settings.

Add Material

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Steel AISI 4340 (mat3)

In the window for , locate the section.

From the list, choose .

The steel plugs are the domains selected in the figure below.

Poroelastic Waves (pelw)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Having defined the materials, you can specify the domain settings.

Poroelastic Material 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

Use a linear elastic material model for the steel-plug domains.

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Poroelastic Waves (pelw)

In the window, under click .

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundaries 273–275 only.

The porous matrix is assumed to be glued to an outer casing at the boundary highlighted in the figure below; hence the fixed constraint boundary condition. This boundary is also sound hard.

Pressure Acoustics, Frequency Domain (acpr)

In the window, under click .

In the window for , locate the section.

From the list, choose .

The selected air domain where Pressure Acoustics apply is depicted in the figure below. It consists of the inlet and outlet as well as the thin air groves inside the particulate filter.

Port 1

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

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

Port 2

In the toolbar, click

and choose .

Select Boundary 15 only.

In the window for , locate the section.

From the list, choose .

You have now specified the domain settings. The red cross decoration for the Air and the SiC matrix nodes under indicates that there are still undefined material parameters. To discover what is missing, return to the node before proceeding with the boundary conditions.

Materials

Air (mat1)

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
Compressibility of fluid	chif	1/1[atm]	1/Pa	Basic

Recall that the compressibility of an ideal gas at the pressure P0 equals 1/P0.

SiC matrix (mat2)

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Permeability	kappa_iso ; kappaii = kappa_iso, kappaij = 0	1e-11	m²	Basic
Tortuosity factor	tau_iso ; tauii = tau_iso, tauij = 0	tauP	1	Poroacoustics model
Young’s modulus	E	20[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.4	1	Young’s modulus and Poisson’s ratio
Density	rho	1000	kg/m³	Basic
Porosity	epsilon	0.3	1	Basic
Biot-Willis coefficient	alphaB	0.3	1	Poroelastic material