The geometrical situation for a squared cell PF given by a pair of inlet and outlet channels loaded with soot and ash is sketched in the following figure. The inlet channel is plugged at the end of the filter and the outlet channel is plugged at the inlet.
BOOST/FIRE can handle many different inlet channel geometries, e.g. squared, hexagonal, octagonal, n-gonal, rectangular, etc. shaped channels. For the squared cell PF in the following figure, the diameters of the inlet channel and outlet channel are given by d1 and d2, respectively.
Figure 12. Pair of PF Channels
As shown in the sketch, soot is split into two distinct layers – a depth filtration and a cake filtration layer – and therefore the model distinguishes between soot located within the porous filter wall and soot forming a cake above it. With the introduction of distinct depth and cake layer balance equations the regeneration behavior of coated filters can additionally be modeled in a more precise way. An increased soot combustion rate can be simulated in the depth filtration layer by applying catalytically supported regeneration mechanisms. As soon as the amount of soot in the depth layer is smaller than the maximum load, two different approaches can be chosen.
Either soot from the cake layer slides down and therefore also converts catalytically, or the cake layer remains and cavities develop until the entire depth layer is converted. The second approach typically shows smaller overall conversion rates since only the amount of soot in the depth layers is exposed to the faster catalytic reaction scheme.
Ash, which may be present in aged filters, is treated in the model as ash plug at the end of the inlet channel, as ash cake layer over the entire length of the filter, or as combination of both.
The presence of an ash layer increases the overall pressure drop as there is additional flow resistance and a reduced free frontal surface in the inlet channel. In models where soot needs to be described by a depth and cake filtration layer, the ash layer can be understood as barrier between the two soot layers. Thus, soot cannot be deposited in the depth filtration layer as soon as an ash layer is present. The existence of an ash plug leads to a reduced effective filtration length and therefore to an increased pressure drop. The effective filtration length is also reduced by the length of the inlet and outlet plugs. Compared to the entire length, these two lengths are typically negligible. In order to provide a most generic model, they are included in BOOST/FIRE.
Assuming that radial gradients (within the channels and the entire monolith) are of minor importance, a transient 1D model in axial direction z is sufficient to describe the entire thermo and fluid dynamics.
A comparison of the time scales given by the entire filter system, in particular by the flow velocities and by the growth of the soot layer, reveals that the gas phase processes can be assumed as quasi-steady. Thus, the present model distinguishes between two sub-models for:
• filter flow
• soot deposition/ regeneration
The two sub-models, discussed in the following, are also treated separately in their numerical solution procedures. The individual solutions are updated on a given time step basis and coupled with the help of a lumped filter model (see Section Modeling Glueing Zones in SIC PFs page [57]).
3.2.2.1. Cell Structures of Particulate Filters
The Honeycomb-type cell structure of particulate filters consist of hundreds (thousands) of individual channels. The most common PF types have squared inlet and outlet channels, as shown in the following figure. There, the numbers of inlet and outlet channels are identical and the geometry is defined by the channel diameters d1 and d2, respectively.
In the case of filters composed of individual segments, like the SiC filter, the growth of ash/soot behaves differently depending on the location of the considered inlet channel. Inlet channels directly adjacent to the cement stripes have either three or, if they are located in the corners, two active filtration walls. Consequently, the PF model should be able to consider the impact of the different channel geometries on loading and regeneration.
Figure 13. PF Structure made of Single Segments
The PF model of BOOST/FIRE can handle many different channel geometries. While the channel shape of the outlet channels guiding the gas out of the PF is of minor importance, the channel shape of the inlet channels is significant, since it determines the formation and structure of the soot and ash layers. Thus, for the correct calculation of the soot and ash heights, a detailed specification of the inlet channel geometry is necessary. For the calculation of the outlet channel flow velocity, the heat transfer and the outlet channel pressure drop, it is enough to specify the outlet channel cross-section and the outlet channel perimeter. The general PF approach of BOOST/FIRE can handle any inlet channel geometry which can be reproduced by multiple reflection of a general quadrilateral (see the following figure), here called General Symmetry Element (GSE). The GSE is the geometrical base of all PF channel geometries in BOOST/FIRE.
The following figure shows the GSE and the unity cell for a PF with octagonal inlet channels. The unity cell determines the smallest repetitive element for reflection to represent the PF geometry
42
In the present example the unity cell is defined by a square and contains a single outlet channel and a single inlet channel (four times one fourth of an inlet channel). The general symmetry element of the octagonal inlet channel is drawn on the right hand side in the following figure. It is defined by the center corner angle + , the right corner angle , the left corner angle , and the side lengths on the wall, l1 and l2. The octagonal inlet channel requires eight GSEs to obtain the specified geometry. Thus, the number of general symmetry elements per inlet channel nC is eight.
Figure 14. Unity Cell and General Symmetry Element in a PF with Octagonal Inlet Channels
Another example for hexagonal inlet and outlet channels is shown in the following figure. The unity cell is an equilateral hexagon containing a single outlet and two inlet channels. The inlet channel with the side lengths, a and b, is produced by reflecting the GSE six times. PFs with hexagonal channel shape are further discussed by Becker et al. [5 page [95]].
Figure 15. Unity Cell and General Symmetry Element in a Hexahex PF
By variation of the specification parameters of the general symmetry element, one can obtain many different inlet channel geometries, e.g. squared, hexagonal, octagonal, n-gonal, rectangular, etc. The following figure shows the soot and ash layers in an arbitrary general symmetry element where sc and ac represent soot cake and ash heights respectively. sd represents the soot height in the depth filtration layer. The free perimeter of the empty inlet channel is defined by the equation
(94) and the cross-section of the inlet channel is defined by the cross-section of the single general symmetry element AGSE and the number of GSE per inlet channel nC, as described by
(95) To consider the case where not the entire side length is available for filtration, the reduced side lengths and are applied. They can be expressed as fraction of the actual side lengths, as described by
(96)
where Feff,1 and Feff,2 are filtration efficiency factors. These factors are model parameters and allow to take into account the reduced soot and ash deposition along channel walls which are not located adjacent to an outlet channel. For example, since in the inlet channel of the Hexahex PF, depicted in the previous figure, is located adjacent to another inlet channel, it can be expected that only a fraction of its side length is used for filtration.
Figure 16. Soot and Ash Layers in the General Symmetry Element
The next two figures show the general symmetry elements together with the soot and ash layers for selected types of inlet channels.
The following figure demonstrates that the three different channel types in segmented PFs leading to four, three, and two active filtration walls can be applied by adequate specifications of the GSEs.
Figure 17. General Symmetry Elements of PF Inlet Channels with 4 (left), 3 (middle), and 2 (right) Filtration Walls
The following figure shows the GSEs for two different hexagonal and for one octagonal inlet channel.
Figure 18. General Symmetry Elements of Hexagonal PF inlet Channels a-b-a-b-a-b (left), a-a-b-b-a-a (middle), and Octagonal (right)
The following table gives an overview of the different channel geometries and their specification parameters, which can be specified by the GSE approach in the PF model of BOOST/FIRE.
Table 4-1: GSE Parameters for Different Inlet Channel Geometry Types Nr. Inlet channel
geometry type
n c l1 l2
1 Square (a a a a) 4 90o 90o 90o
2 Rectangular (a b a b) 4 90o 90o 90o
3 Hexagonal (Hexahex) (a b a b a b)
6 60o 90o 90o
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4 Hexagonal Hex3
(a a b a a b)
4 a 90o 60o 90o
5 Octagonal, alternating (a b a b a b a b)
8 45o 90o 90o
6 Octagonal, double side (a a b b a a b b)
4 a b 90o 67.5o 67.5o
7 N-gonal, equilateral (a a . . . a)
N 90o 90o
8 N-gonal, alternating a b (a b a . . . . b)
N 90o 90o
9 Square, 2 sided (a a) 1 a a 90o 90o 90o
10 Square, 3 sided (a a a) 2 a 90o 90o 90o
11 Rectangular, 2 sided (a b) 1 a b 90o 90o 90o
12 Rectangular,
3 sided (a b a)
2 a 90o 90o 90o
Based on the inlet channel geometry, given by the general symmetry element, the PF solver of BOOST/FIRE determines the free channel cross-section. Furthermore, for a given ash mass distribution overlain with a soot mass distribution (varying during the loading and regeneration process), the heights of the soot and ash layers are calculated. The layer heights strongly influence the transition velocity of the exhaust gas through the filter wall and consequently the pressure drop of the PF as well as the progress of the regeneration.
3.2.2.2. Main Geometry Parameters of Particulate Filters
The cell density of a PF is defined by the CPSI number indicating the total number of Channels Per Square Inch. One can distinguish between the cell density of the inlet channels CPSIinl and the cell density of the outlet channels CPSIout, where the sum has to satisfy the equation
(97) The cell density per square meter CPSM is defined by
(98)
In general a unity cell contains Iinl inlet channels and Iout outlet channels. Consequently the CPSM numbers of the inlet and outlet channels are determined by the equations
(99)
and
(100)
By replacing CPSM with CPSI, Eq.99 page [44] and Eq.100 page [44] can be also applied for the cell densities per square inch, CPSIinl and CPSIout.
The cross-section of the unity cell can be calculated by
(101)
The open frontal area of all inlet channels is determined by
(102)
and the open frontal area of the outlet channels is given by
(103)
The total open frontal area is obtained from the sum of OFAInl and OFAOut, as described by (104) The heat transfer between solid and gas as well as the wall flow are mainly influenced by
the overall geometrical surface area (GSA) of the PF. For the determination of the GSA the perimeters of both, the inlet and outlet channels are considered. The GSA of all inlet channels is described by
(105)
and the GSA of the outlet channels is given by
(106)
The total geometrical surface area is obtained by the sum of the individual GSAs for inlet and outlet, GSAInl and GSAOut, as described by
(107)
3.2.2.3. Squared Cell Channels
Symmetric or asymmetric square-shaped channels represent the most common geometry in particulate filters. This section discusses the geometry parameters of the selected channel shape. Other inlet channel geometries, e.g. hexagonal or octagonal have to be treated analogously.
As sketched in the following figure, two different channel diameters are required to describe asymmetric channel structures. In addition, although it is not commonly used for particulate filters, a washcoat thickness is considered in the set of equations given below.
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Figure 19. Squared Cell PF with Asymmetric Cell Structure
As shown, the total thickness of the monolith's wall is given by
(108) where w is the thickness of the substrate wall and wc is the thickness of the washcoat. The repeat distance s for structures with different channel diameters is given by
(109) and determines the side length of the unity cell. d1 and d2 represent the diameters of the inlet and outlet channel, respectively. The cross section of the unity cell simply equals
(110) and the perimeters of the inlet and outlet channels are described by
(111)
The numbers of inlet and outlet channels per unity cell, Iinl and Iout, are two, respectively. By combining Eq.99 page [44], Eq.101 page [45] and Eq.110 page [46], one obtains the correlation between the cell density and the cell distance s by the equation
(112)
With the help of the channel diameter ratio
(113)
the hydraulic diameter of both channels can be evaluated using
(114)
With the calculated hydraulic diameters BOOST/FIRE further evaluates the open frontal area (OFA) and the geometric surface area (GSA) with respect to the inlet channel, with respect to the outlet channel and with respect to both channels. Based on Eq.102 page [45], Eq.103 page [45] and Eq.104 page [45] the open frontal areas are described by
(115)
and the geometrical surface areas, based on Eq.105 page [45], Eq.106 page [45] and Eq.107 page [45]
can be calculated by the equations
(116)
The OFA considering both channel diameters defines the solid fraction of the filter that is used in the energy balance equations. The GSA is used to specify the heat transfer between the gas and solid phase.
3.2.2.4. Soot and Ash Layer Geometries of Squared Channels
This section describes the layer geometries leading to correlations for the soot and ash heights of the squared channels. The correlations for other inlet channel geometries, e.g. hexagonal or octagonal, are derived in an analogous manner.
The solution of the filter wall flow model (see Section Filter Flow Model page [48]) and soot/ash deposition and regeneration model (see Section Deposition and Regeneration of Soot and
Ash page [52]) requires a correlation between the soot/ash loading (mass per filter volume) and
the height of the soot depth layer, ash and soot cake. These values depend, beside the above mentioned geometrical issues, on the number of active filtration walls per inlet channel. For PFs made out of cordierite the value of active filtration walls per inlet channel is normally four. If SiC PFs are considered, this value depends on the position related to the glueing stripes (e.g.
cement) which connect the single segments. As shown in Fig. 13 page [41], inlet channels which are directly adjacent to the cement stripes have either three or, if they are located in the corners, two active filtration walls. Inlet channels which are not in close vicinity to the glueing stripes have four filtration walls.
Simple geometrical considerations are applied to calculate the heights of the soot depth, ash and soot cake layer considering these geometrical constraints.
The height of the soot in the depth filtration layer of a channel with 4 filtration walls is given by (117)
where msd is the soot loading in the depth layer, Afront represents the frontal surface of the filter.
n1 is the number of inlet channels (four, three, or two), sd is the packing density of the soot in the depth layer, nfw is the number of active filtration walls and d1 represents the diameter of the inlet channel. The soot packing density in the depth layer is derived from the maximum depth filtration loading and the corresponding depth layer thickness. Thus, this density should to be viewed as a scaling factor rather than as a real physical density value.
Assuming that the cake layers of soot and ash form a trapezoidal shape (see Fig. 20 page [50]), the soot cake height for an inlet channel is given by
(118)
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where msc is the soot cake loading, and sc represents the packing density of the cake. mac,layer is the local ash loading deposited in the ash layer and ac represents the packing density of the ash cake. The geometrical factors Fnfw,1 and Fnfw,2 depend on the number of active filtration walls in the inlet channel. Fnfw,1 is 1 for four filtration walls, 1.5 for three filtration walls and 2 for two filtration walls. The corresponding values for Fnfw,2 are 1, 2 and 4 respectively.
The height of the ash layer ac required in this equation can be further derived from
(119)
The amount of ash in the layer is calculated by the total amount of ash, mac, and a layer-plug-distribution factor lpd. This factor is defined by
(120)
The amount of ash in the plug is consequently given by
(121) With the known ash load in the plug, the length of the plug lash-plug is calculated. It is
(122)
where VPF is the volume of the entire filter and OFAinl represents the open frontal area evaluated with respect to the inlet channel diameter (see Eq.115 page [46]). The calculated ash plug length together with any specified length of inlet/outlet plugs (see Fig. 12 page [40]) is also used in order to derive the effective filtration length in the PF.