Filter Flow Model

As sketched in Fig. 12 ^{page [40]}, the entire filter can be split into several flow regions. These are regions for the inlet/outlet flow, regions for the channel flow neighboring plugs and the region for the effective filtration flow. The first two regions are described by algebraic corrections summarized in Section Overall Pressure Drop and Pressure Drop Contributions ^{page [51]}. The latter is described by 1D steady-state balance equations of continuity and momentum for a representative pair of inlet and outlet channels.

3.2.3.1. 1D Continuity and Momentum Balance 3.2.3.1.1. General Conservation Equations

The steady-state continuity equations of the gas phase in the inlet and outlet channel is

(123)

(124)

where _g,n is the density of the gas phase and v_g,n is the gas velocity in the inlet (n=1) and outlet (n=2) channel, respectively. z is the spatial coordinate in axial direction.

A_F,n represents the free channel cross section that is available for the gas flow. This cross section is constant in the outlet channel but changes in the inlet channel depending on the local height of the ash and soot cakes. It is essential to have A_F,1 inside the spatial derivative. Soot loadings up to 20 g/l lead to significant reductions of the free inlet channel cross-section (see Millet et al. [49 ^{page [96]}]) and therefore the complete neglect of the cake height may lead to over-simplified models.

P_S,n is the wet perimeter of the free channel cross section of the nth channel. This value is constant in the outlet channel but also changes in the inlet channel as a function of the cake height. v_w,n represents the wall velocity lateral to the axial direction. The difference between the wall velocity of inlet v_w,1 and of the outlet channel v_w,2 at the same axial position can be derived by continuity considerations as discussed later in this section.

The steady-state momentum balance equations of the gas phase in the pair of channels is given by

(125)

(126)

where _g,n represents the pressure in the inlet and outlet channel, respectively. The frictional pressure loss along the channels is approached as a linear function of the local channel velocity scaled by a friction coefficient F_n and the dynamic viscosity . The last term of Eq.125 ^{page [49]}

describes a momentum sink due to mass sinks in the inlet channel. The term can be understood as additional contribution to the channel friction and therefore is often dropped in literature.

Nevertheless, omitting this term in inviscid flows (F₁ 1) leads to an increase of the stagnation pressure over the filter length, and therefore breaks the second law of thermodynamics.

3.2.3.1.2. Squared Channel Structure

Under the assumption of a squared cell structure, the free channel cross section and the wet perimeter of the two channels are derived from geometrical considerations according to

(127)

(128) (129) (130) d_n is the diameter of the individual channels. _sc and _ac represent the heights of the soot and ash cake and F_nfw,A, F_nfw,B and F_nfw,C are again geometrical factors, which distinguish between inlet channels with four, three and two active filtration walls. The corresponding values for a channel with four, three or two active filtration walls are 4, 3 and 2 for F_nfw,A , 8, 4, and 2 for F_nfw,B and 4, 2 and 1 for F_nfw,C, respectively.

For non-squared channel geometries (see Fig. 17 ^{page [43]} and Fig. 18 ^{page [43]}), the required geometrical parameters, A_F,1, A_F,2, P_S,1, and P_S,2, are determined in an analogous manner.

3.2.3.2. Wall Flow and Wall Pressure Drop

50

flow velocity. For squared channels, the cake (ash or soot) builds a trapezoidal shape, as sketched in the following figure. Deviations from this cake geometry, given especially in the corners of non-squared channels, are lumped into the trapezoidal shape. This approximation seems reasonable since the cake grows over the entire internal surface area of the inlet channel (compare experimental investigations from Ogyu et al., [52 ^{page [97]}]). This issue is also addressed in more detail by Konstandopoulos et al. [39 ^{page [96]}], who concluded that the flow follows the path with the lowest resistance which is not necessarily the smallest geometrical distance between the inlet and outlet channel.

Figure 20. Soot and Ash Cake

Assuming that changes of the gas density are negligible over the cake height, a continuity consideration leads to the flow velocity expressed by

(131)

where x describes the coordinate from top of the cake down to the wall. The wall velocity entering the outlet channel is one solution of the previous equation where additionally changing gas densities in the inlet and outlet channel are taken into account

(132)

The total 'wall pressure drop' is the sum of individual pressure drops given by the wall, a soot depth filtration layer, an ash cake layer and a soot cake layer. The application of Darcy's law over the different layers with their individual flow velocities leads to

(133)

where k_w, k_sd, k_ac and k_sc represent the permeability of the wall, the soot depth layer, the ash cake and soot cake layer, respectively. Assuming that the depth filtration height is small compared to the entire wall thickness, the latter is held constant in the applied pressure drop correlation.

Soot Permeability

When setting the soot permeability to "Formula" in the GUI, k_sc is calculated according to [41 ^page

[96]]:

(134)

where f( ) is the Kuwabara function, is the porosity, d_primary is the primary particle diameter, SCF is the Stokes-Cunningham Factor, Kn the Knudsen number and the gas mean free path.

is evaluated using the following equation:

(135)

where _g is the kinematic viscosity of the exhaust gas, and MW is the molecular weight.

3.2.3.3. Overall Pressure Drop and Pressure Drop Contributions

The set of flow model equations (Eq.123 ^{page [48]}to Eq.133 ^{page [50]}) can be solved together with the following four boundary conditions

(136) (137) (138) (139) where the inlet velocity and filter back pressure are assumed to be known. The geometrical situation leads to the boundary conditions of the velocities in the inlet and outlet channel. The inlet velocity at the end of the inlet channel, and the outlet velocity at the beginning of the outlet channel is zero. The solution of Eq.123 ^{page [48]} to Eq.133 ^{page [50]} leads to the spatial distribution of the velocities, pressures in the inlet and outlet channels (v_g,1(z), v_g,2(z), p _g,1(z), p_g,2(z)), and of the wall velocity v_w(z).

The pressure drop over the entire effective filtration length is one solution of the calculated pressure profiles. It is given by

(140) where p _g,1 (z = 0) is the inlet channel pressure at the inlet and p _g,2 (z = l_eff)is the outlet channel pressure at the end of the effective filter length. This pressure drop can be further split into its individual contributions.

Therefore, the different pressure drops over the wall, soot depth layer, ash, and soot cake (see Eq.133 ^{page [50]}) are simply evaluated as mean values over the effective filter length. The viscous pressure drop of the inlet and outlet channel is given by

(141)

(142)

52

where the pressure differences from one position z in the filter to inlet and outlet, respectively, are evaluated and averaged over the entire effective filter length.

Outside the effective filtration length there are two regions where pure channel flow can be assumed. This is the entrance region in the inlet channel adjacent to the inlet channel plug and the rear region in the outlet channel neighboring the outlet and ash plug. Assuming laminar flow, the pressure drops for both regions are given by

(143)

(144)

where is a general channel shape factor. For squared channels is typically 0.89, for hexagonal channels 0.95, and for octagonal channels 0.98. is the length of the inlet and outlet plug and represents the length of the ash plug.

At the inlet and outlet of the filter, changing cross sections lead to contraction and expansion effects of the flow. In order to describe additional pressure losses caused by the flow acceleration and slowdown, two correlations are considered in the present model. These are

(145)

(146)

where the given dynamic pressure is scaled by a general friction loss coefficient .

The total pressure drop over the entire filter is simply the sum of the individual pressure losses given by Eq.140 ^{page [51]} to Eq.145 ^{page [52]}. The pressure drops given by the plug regions are typically small compared to the proper filter pressure drop.

The inlet and outlet pressure losses, on the contrary, have a decisive impact on higher mass flows because of their quadratic velocity dependency.