COMSOL 6.3 - A 3D Model of a Diesel Particulate Filter

A 3D Model of a Diesel Particulate Filter

This example studies the averaged flow field, concentration, and temperature distribution in a homogenized model of a diesel particulate filter (DPF). The filter consists of a series of parallel channels that are closed off in an alternating fashion. The walls shared by neighboring channels are porous, allowing gas to pass through them while at the same time separating particles from the flow (Figure 1).

Figure 1: Drawing of a diesel particulate filter. At the reactor’s inlet the outlet channel is closed, while at the reactor’s outlet the inlet channel is closed.

Several factors influence a DPF’s efficiency and durability. Important issues include the removal of soot particles from the filter membranes and the influence of thermal stress on the ceramic structure, stress that arises during repeated operating cycles. Mathematical modeling provides a powerful tool for investigating such critical parameters under a wide range of operating conditions.

This example sets up and solves the coupled flow plus mass and thermal transport equations describing the transport/reaction phenomena in a diesel particulate filter. An important aspect it illustrates is how you can tailor the predefined application modes of COMSOL Multiphysics to represent a specialized mathematical model.

Model Definition

In a detailed DPF model, you can treat the channels and porous membrane with their true geometry with the only simplification being the membrane description. The membrane could, for example, be described with a porous-media approach using effective transport properties. Flow in the channels is expected to be close to fully developed laminar flow although there could be a slight deviation due to the relatively small flow of fluid over the membrane from the inlet to the outlet channels.

Figure 2: Cross section of a unit cell consisting of an inlet and an outlet channel separated by a porous membrane.

In general, a detailed model of the flow in the hundreds or even thousands of channels that make up a full filter geometry is not feasible due to the demands on computer capability. However, a homogenized model is realizable. Using such an approach, assume that the velocity profile is that of a fully developed laminar flow. The average flow rate in a cross section of one channel is proportional to the pressure gradient in the channel:

(1)

Here u denotes the flow velocity (m/s), kl represents a lumped friction factor (m2/(Pa·s)), and p is the pressure (Pa).

Velocity and Pressure

You can obtain the average velocity in the inlet channel from a mass balance. The velocity in this channel decreases as gas passes through the porous membrane to the outlet channel. The crossover of gas from an inlet to an outlet channel is assumed to be proportional to the pressure difference across the membrane.

Figure 3 shows the geometric unit cell used to set up the balance equations.

Figure 3: Geometric unit cell for the homogenized DPF model. Each inlet channel has porous walls in common with four outlet channels and vice versa.

For an inlet channel, the balance equation with respect to mass flow becomes

(2)

where κm denotes the porous membrane’s permeability (m2), H is the channel height (m), η denotes the fluid’s viscosity (Pa·s), δm gives the membrane’s thickness (m), p1 represents the pressure in the inlet channel (Pa), and p2 equals the pressure in the outlet channel (Pa).

The density is a function of pressure according to the ideal gas law:

(3)

Here M1 denotes the fluid’s average molecular weight (kg/mol), Rg is the gas constant (J/(mol·K)), and T1 gives the temperature in the inlet channel (K).

The corresponding equation for the outlet channel is

(4)

Note that the source/sink terms in Equation 2 and Equation 4 are proportional to the volumetric flow through the membrane per unit area, or the superficial velocity:

(5)

The pressure fields and thus also this velocity field vary only in the x direction, and each inlet channel is separated from the other inlet channels by an impermeable wall. The inlet and outlet pressures connect the array of inlet channels to each other. If the porous membrane’s permeability, encoded in the variable κm, varies in the filter, a corresponding distribution of the superficial velocity in the inlet and outlet channels results. In addition, the temperature field is connected for the entire system, and heat flows between the different unit cells. For this reason it is appropriate to expand the 1D model to a 3D model that accounts for the impermeability of the walls separating the inlet and outlet channels in adjacent unit cells.

You can express this impermeability with a diagonal friction tensor, k, where only the xx component is nonzero and equal to kl. The model equations then read

(6)

and

(7)

These equations are valid if the permeability and thickness of the porous membrane are constant throughout the process, which is not the case when soot particles deposit on the membrane surface. However, you can derive a model analogous to the one just described by assuming that two porous membranes separate the outlet and inlet channels, one formed by the soot layer and the second one being the porous membrane just described. This approach results in the following equations:

(8)

and

(9)

Here a and b are defined by the equations

(10)

(11)

where κs denotes the soot layer’s permeability (m2), and δs equals its thickness (m). The superficial flow velocity through the membrane is now given by

(12)

At the filter entrance, the boundary condition for the inlet channels specifies a given pressure. All other boundaries are insulating, that is,

(13)

where n denotes the normal vector to the boundary.

The boundary conditions for the outlet channels specify the pressure at the filter exit and impose insulation at all other boundaries:

(14)

Mass Balances

The soot particles are present only in the inlet channel because they are deposited as a film on the porous membrane. A balance for soot particles introduces the deposition of soot particles as a sink to the flux of particles in the open channel. This gives the transport equation

(15)

where cs denotes the soot-particle concentration (mol/m3), and Ds is the diffusivity of soot particles (m2/s).

The model defines the concentration of oxygen using two mass balances, one for the inlet and another for the outlet channel. The mass balance in the inlet channel contains the term where oxygen reacts in the combustion reaction with soot. The mass balance in the inlet channel is represented by the equation

(16)

where co2 denotes the oxygen concentration (mol/m3), Do2 gives the diffusion coefficient (m2/s), and Rs equals the reaction rate for the surface reaction (mol/(m2·s)).

The corresponding balance for oxygen in the outlet channel yields

(17)

Note the addition of the flow term of oxygen over the membrane as a source term in the outlet channel balance.

The balance of soot results in an expression for the growth or consumption of the soot layer:

(18)

where δs denotes the soot layer’s thickness (m), ρs represents its effective density (kg/m3), and Ms equals the molecular weight of carbon (kg/mol).

The boundary condition for co2,1 sets the concentration at the inlet, while insulation is used for all other boundaries. For co2,2, use a convective flux condition at the outlet and set all other boundaries to insulation. The initial conditions for both co2,1 and co2,2 are 0.

Energy Balance

The energy balance in the reactor consists of three equations: one for the inlet channels, one for the porous membranes, and one for the outlet channels.

The balance for the inlet channels reads

(19)

(20)

where T1 is the temperature in the inlet channels (K), Tm is the temperature in the membrane (K), Cp1 gives the heat capacity of the gas in the inlet channel (J/(kg·K)), h equals the heat transfer coefficient (W/(m2·K)), and q1 is the flux vector.

In the membrane, the energy equation is given by

(21)

(22)

Here the subscript “m” refers to the membrane so that Tm is its temperature, T2 equals the temperature in the outlet channels (K), Cp2 refers to the heat capacity of the gas in the outlet channels (J/(kg·K)), and h is the heat transfer coefficient between the membrane and the channels.

The last equation determines the temperature in the outlet channel:

(23)

(24)

The boundary conditions for the inlet channel are a given temperature at the inlet and insulation at all other boundaries. At the outlet channel, use a convective flux condition at the outlet and set all other boundaries to insulation. The initial condition specifies a temperature equal to the inlet temperature of the gas.

For the membrane, the heat flux through the boundaries is given by

(25)

where houter denotes the heat transfer coefficient, and Tamb equals the ambient temperature. The initial temperature is set equal to the inlet temperature.

Homogenized Filter Geometry

The balance equations covered above are implemented on a 3D filter geometry. The cross section of the filter is an oval, 5.86 inches in width, 4.66 inches in height, and the length of the filter is 8 inches. Mirror symmetry reduces the geometry to one quarter of the filter.

Figure 4: Symmetry reduces the modeling geometry to one quarter of the full filter.

Mesh

Boundary layer meshing is used to create a denser mesh in the radial direction of the filter, near the outer surface. When extruding the cross sectional mesh you can further control the distribution to get a denser mesh near the filter inlet and outlet.

Figure 5: Boundary layer meshing and mesh extrusion make it possible to efficiently discretize the geometry.

Results and Discussion

Figure 6 shows the gas velocity along the inlet and outlet filter channels. At the start of the simulation (t = 0), a 25-μm layer of soot is evenly distributed on top of the porous membranes of the inlet channels. The pressure difference over the filter is 50 Pa.

The top panel in Figure 7 shows the pressure difference (p1 – p2) across an inlet/outlet channel pair located at the center of the filter. The bottom panel illustrates the buildup of soot during 15 minutes of simulated operation with no carbon oxidation present. As expected, soot builds up primarily at the channel’s entrance and end, because this is where the pressure gradient and, consequently, the trans-membrane velocity have their highest values.

Figure 6: Gas velocity (m/s) in the inlet channels (v1, top) and outlet channels (v2, bottom).

Figure 7: Pressure difference (top) and soot-layer thickness (bottom) along a channel located at the filter centerline. No soot oxidation is present.

The top image of Figure 8 shows the filter temperature distribution, Tm, when the inlet gas temperature is 550 K, and the ambient temperature is set to 300 K. Furthermore, the effect of soot oxidation is disregarded. The temperature gradients due to cooling of the filter’s outer surface are evident. The bottom image shows Tm, when soot oxidation is taken into account. Filter temperatures exceeding that of the inlet gas are observed in the front end of the filter.

Figure 9 illustrates the time evolution of the soot layer’s thickness (0–900 s). The layer grows due to the deposition of particles from the gas phase, and it is removed by oxidation. Results are shown for a channel located at the filter periphery (top) and at the center of the filter (bottom).

Figure 8: Filter temperature, Tm, when no oxidation takes place (top), and when soot oxidation is taken into account (bottom).

Figure 9: Soot-layer thickness along a channel located at the filter periphery (top) and at the filter centerline (bottom).

Clearly, soot removal is less efficient in the peripheral parts of the filter as lower temperature in these areas decreases the oxidation rate significantly.

Finally, Figure 10 shows the pressure drop across a centrally placed channel pair as function of time (0–900 s). In accordance with the diminishing soot layer illustrated in Figure 9, the pressure drop is reduced in the front end of the filter.