Patent Publication Number: US-11654592-B2

Title: Rectangular outlet honeycomb structures, particulate filters, extrusion dies, and method of manufacture thereof

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a Continuation of U.S. application Ser. No. 16/095,596 filed on Oct. 22, 2018, which claims the benefit of International Application No. PCT/US2017/029159, filed on Apr. 24, 2017, which claims the benefit of priority of U.S. Provisional Application Ser. No. 62/326,384 filed on Apr. 22, 2016 and U.S. Provisional Application Ser. No. 62/452,765 filed on Jan. 31, 2017, the contents of which are relied upon and incorporated herein by reference in their entireties. 
    
    
     FIELD 
     The present specification relates to particulate filters, such as filters comprising a plugged honeycomb structure, and honeycomb structures comprised of porous ceramic walls, such as used in filtering particles from a fluid stream such as from an engine exhaust stream. 
     BACKGROUND 
     Filters, such as plugged porous ceramic honeycomb filters, for example diesel particulate filters (DPFs), have been employed in exhaust after-treatment systems. 
     SUMMARY 
     The present disclosure relates to bodies comprising honeycomb structures, or honeycomb structure bodies or honeycomb bodies, which can comprise plugs such that the honeycomb bodies are particulate filters, such as filters comprising a plugged honeycomb structure, or a plugged honeycomb structure body, comprising a honeycomb structure comprised of porous ceramic walls, such as those used in filtering particles from a fluid stream such as an engine exhaust stream. The honeycomb structure of the honeycomb structure body comprises a matrix of cells comprised of porous walls disposed in relationship to each other, in which a plurality of subsets of cells can be identified, wherein the walls of the cells define channels and which can be, or are, plugged according to a selected plugging pattern, wherein a select subset, or subsets, of cells are referred to herein as repeating structural units (otherwise referred to herein as repeat blocks, repeat units, or unit blocks). Cell walls may serve as a shared boundary wall for two or more cells. Cell walls and plugs define cell channels, i.e. inlet and outlet channels, and a repeating structural units can be characterized as comprising a repeating cell pattern, or a repeating structural unit (i.e. focusing on structural walls) and/or a repeating channel unit (i.e. focusing on channels). Channels are adapted to allow fluid flow (for example, exhaust gas flow that may be comprised of gas and particulates), given appropriate conditions, such as fluid biometric flow, presence of plugs, particulate loading, and cell wall structure. 
     Thus, the honeycomb structure comprises a built up structure that comprises identifiable multiple repeating structural units (or multiple repeating cell pattern), or a plurality of repeating units (or plurality of repeating cell patterns), whether the plurality of repeating units is made up of a monolithic array of walls, or whether some portion of the honeycomb structure is an assembly of smaller components or segments bonded together to form a larger matrix of cells or groups of cells. In one set of embodiments, the intersecting walls are comprised of porous ceramic material, such as cordierite, cordierite magnesium aluminum titanate, mullite, aluminum titanate, silicon carbide, alumina, and combinations thereof. 
     In one aspect, the present disclosure relates to particulate filters comprised of a honeycomb structure comprising a large ratio of inlet to outlet cells and associated channels while also comprising some outlet cells, or channels, having a large hydraulic diameter relative to the inlet cells, or channels, for good pressure drop performance (both clean and soot loaded). The particulate filters, honeycomb bodies, and honeycomb structures also have a relatively large fraction of the inlet cell (or channel) surface bordering an outlet channel in order to provide favorable clean pressure drop performance, as well as soot-loaded pressure drop performance. In some embodiments, the particulate filters, honeycomb bodies, and honeycomb structures disclosed herein provide for both good ash storage and low pressure drop. 
     In another aspect, a honeycomb structure is provided comprising a matrix of intersecting porous cell walls extending in an axial direction between an inlet and outlet end of the honeycomb structure, the matrix defining a plurality of inlet cells and outlet cells, and corresponding inlet channels and outlet channels defined by respective inlet cells and respective outlet cells, wherein at least a portion of the outlet channels are larger in cross-sectional area than any of the inlet channels, and at least some of the outlet channels comprise a rectangular shape. 
     In another aspect, an extrusion die is provided. The extrusion die comprises an outlet face of a die body comprising a matrix of intersecting slots comprising a partial slot type, the matrix defining a die repeat unit, wherein the partial slot type extends less than entirely across the outlet face, the die repeat unit comprising: four or more die pins made up of a first die pin type and a second die pin type, the first die pin type larger in cross-sectional area than the second die pin type and comprises a rectangular shape in cross-section with two first sides of length Lo and two second sides of width Wo, wherein Lo&gt;Wo, and a slot of the partial slot type terminating with a T-intersection on at least one of the first sides, and the second die pin type comprising a side length Li that is less than half the length of the first side of length Lo. 
     In yet another aspect, a method of manufacturing a honeycomb body comprising a honeycomb structure is provided. The method comprises providing an extrusion die comprising: an outlet face of a die body comprising a matrix of intersecting slots comprising a partial slot type, the matrix defining a die repeat unit, wherein the partial slot type extends less than entirely across the outlet face, the die repeat unit comprising: four or more die pins made up of a first die pin type and a second die pin type, the first die pin type larger in cross-sectional area than the second die pin type and comprises a rectangular shape in cross-section with two first sides of length Lo and two second sides of width Wo, wherein Lo&gt;Wo, and a slot of the partial slot type terminating with a T-intersection on at least one of the first sides, and the second die pin type comprising a side length Li that is less than half the length of the first side of length Lo; extruding a batch mixture through the matrix of intersecting slots to form a green body; and heating or firing the green body to form a body comprising a honeycomb structure which comprises a matrix of intersecting porous cell walls extending axially between an inlet end and an outlet end of the honeycomb structure, the matrix defining a plurality of inlet cells and outlet cells, and corresponding inlet channels and outlet channels defined by respective inlet cells and respective outlet cells, wherein at least a portion of the outlet channels are larger in cross-sectional area than any of the inlet channels, and at least some of the outlet channels comprise a rectangular shape in cross-section. 
     Additional features and advantages will be set forth in the detailed description which follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the embodiments described herein, including the detailed description which follows, the claims, as well as the appended drawings. 
     It is to be understood that both the foregoing description and the following detailed description describe various embodiments and are intended to provide an overview or framework for understanding the nature and character of the claimed subject matter. The accompanying drawings are included to provide a further understanding of the various embodiments, and are incorporated into and constitute a part of this specification. The drawings illustrate the various embodiments described herein, and together with the description serve to explain the principles and operations of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1 A  schematically depicts and illustrates a representative portion of a known honeycomb structure for a particulate filter having a plugging pattern with a square cell array structure, or matrix, or web of walls with equal size inlet cells and outlet cells (plugs of the outlet cells shown hatched). 
         FIG.  1 B  illustrates a repeating structural unit of the honeycomb structure of  FIG.  1 A . 
         FIG.  2 A  schematically depicts and illustrates a representative portion of a honeycomb structure for a particulate filter having plugs disposed in a known plugging pattern with a square cell array structure, or matrix, or web of walls, in an asymmetric design of larger inlet cells and smaller outlet cells (plugs of the outlet cells shown hatched). 
         FIG.  2 B  illustrates a repeating structural unit of the honeycomb structure of the asymmetric design of  FIG.  2 A . 
         FIG.  3 A  illustrates a representative portion of a comparative plugged honeycomb structure of a particulate wall-flow filter having plugs disposed in a plugging pattern which results in a higher number of inlet cells than outlet cells (shown hatched). 
         FIG.  3 B  illustrates a repeating structural unit of the honeycomb structure of  FIG.  3 A . 
         FIG.  4 A  shows a representative portion of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  4 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  4 A  according to one or more embodiments of the disclosure. 
         FIG.  4 C  is a partial cross-sectioned side view of a filter comprising the plugged honeycomb structure of  FIG.  4 A  taken along section line  4 C- 4 C and illustrating the plugging pattern and relatively-larger outlet channels according to one or more embodiments of the disclosure. 
         FIG.  4 D  illustrates an inlet end view of a plugged honeycomb structure of a filter comprising a repeating structural unit of  FIG.  4 B  with plugs shown as hatched according to one or more embodiments of the disclosure. 
         FIG.  4 E  illustrates an outlet end view of a plugged honeycomb structure of a filter comprising a repeating structural unit of  FIG.  4 B  with plugs shown as hatched according to one or more embodiments of the disclosure. 
         FIG.  5 A  shows a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  5 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  5 A  according to one or more embodiments of the disclosure. 
         FIG.  5 C  shows a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  5 D  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  5 C  according to one or more embodiments of the disclosure. 
         FIG.  6 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  6 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  6 A  according to one or more embodiments of the disclosure. 
         FIG.  6 C  illustrates rounded corners of a honeycomb structure according to one or more embodiments of the disclosure. 
         FIG.  6 D  illustrates beveled corners of a honeycomb structure according to one or more embodiments of the disclosure. 
         FIG.  7 A  illustrates a representative portion of an example embodiment a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  7 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  7 A  according to one or more embodiments of the disclosure. 
         FIG.  8 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  8 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  8 A  according to one or more embodiments of the disclosure. 
         FIG.  9 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  9 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  9 A  according to one or more embodiments of the disclosure. 
         FIG.  10 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  10 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  10 A  according to one or more embodiments of the disclosure. 
         FIG.  11 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  11 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  11 A  according to one or more embodiments of the disclosure. 
         FIG.  12 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  12 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  12 A  according to one or more embodiments of the disclosure. 
         FIG.  13 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  13 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  13 A  according to one or more embodiments of the disclosure. 
         FIG.  14 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  14 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  14 A  according to one or more embodiments of the disclosure. 
         FIG.  15 A  illustrates a representative portion of an example embodiment a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  15 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  15 A  according to one or more embodiments of the disclosure. 
         FIG.  16 A  illustrates a representative portion of an example of a plugged honeycomb structure of a particulate filter according to one or more embodiments of the disclosure. 
         FIG.  16 B  illustrates a repeating structural unit of the plugged honeycomb structure of  FIG.  16 A  according to one or more embodiments of the disclosure. 
         FIG.  16 C  illustrates a plugged honeycomb structure of a particulate filter wherein less than all the honeycomb structure comprises a structural repeating unit according to one or more embodiments of the disclosure. 
         FIG.  16 D  illustrates an alternative embodiment plugged honeycomb structure of a particulate filter wherein less than all the honeycomb structure includes a structural repeating unit according to one or more embodiments of the disclosure. 
         FIG.  17 A  shows a photograph of the inlet and outlet faces of a comparative plugged honeycomb structure ( 17 A, design A). 
         FIG.  17 B  shows a photograph of the inlet and outlet faces of a comparative plugged honeycomb structure ( 17 B, design B). 
         FIG.  17 C  shows a photograph of the inlet and outlet faces of plugged honeycomb structure according to embodiments disclosed herein ( 17 C, design C), wherein the inlet face of example  17 C is shown after soot loading. 
         FIG.  18 A  schematically depicts clean pressure drop as a function of (cold) flow rate for the structures  FIGS.  17 A,  17 B, and  17 C . 
         FIG.  18 B  schematically depicts soot-loaded pressure drop as a function of soot load at 743 scfm cold flow for the structures of  FIGS.  17 A,  17 B, and  17 C . 
         FIG.  18 C  schematically depicts ash-loaded pressure drop as a function of soot load at 743 scfm cold flow for the structures of  FIGS.  17 A,  17 B, and  17 C , at an ash load of 20 g/liter ash. 
         FIG.  19 A  shows a photograph of the plugged inlet end (upper picture) and outlet end (lower picture) of comparative design  22 A having an asymmetric honeycomb structure similar to that shown in  FIGS.  2 A and  2 B  with a geometry of 300 cells per square inch, cell wall thickness of 7 mils (“300/7 geometry”), and an inlet-to-outlet cross-sectional area ratio of 1.7:1. 
         FIG.  19 B  shows a photograph of the plugged inlet end (upper picture) and outlet end (lower picture) of comparative design  22 B having a symmetric honeycomb structure similar to that shown in  FIGS.  1 A and  1 B  with a geometry of 200 cells per square inch, cell wall thickness of 8 mils (“200/8 geometry”), and an inlet-to-outlet cross-sectional area ratio of 1:1. 
         FIG.  19 C  shows a photograph of the plugged inlet end (upper picture) and outlet end (lower picture) of design  22 C having a honeycomb structure  500  similar to that shown in  FIGS.  5 C and  5 D  with a geometry of 400 cells per square inch, cell wall thickness of 8 mils (“400/8 geometry”), and an inlet to outlet cross-sectional area ratio of 1.8:1. 
         FIG.  19 D  shows a photograph of the plugged inlet end (upper picture) and outlet end (lower picture) of design  22 D having a honeycomb structure  500  similar to that shown in  FIGS.  5 C and  5 D  with a geometry of 300 cells per square inch, cell wall thickness of 8 mils (“300/8 geometry”), and an inlet to outlet cross-sectional area ratio of 1.9:1. 
         FIG.  20    graphically shows soot-loaded pressure drop as a function of soot load at 625 m 3 /h and 200° C. for the plugged honeycomb bodies of  FIGS.  22 A,  22 B,  22 C,  22 D . 
         FIG.  21 A  illustrates a front end view of an example extrusion die configured to manufacture an embodiment of a particulate filter comprising the honeycomb structure of  FIG.  11 A  according to one or more embodiments. 
         FIG.  21 B  illustrates a partial, cross-sectioned side view of an example of an extrusion die of  FIG.  21 A  taken along section line  19 B- 19 B according to one or more embodiments. 
         FIG.  21 C  illustrates an end view of an example of a die repeating unit of the extrusion die of  FIG.  21 A  according to one or more embodiments. 
         FIG.  22 A  schematically illustrates an example of a plugged honeycomb structure comprising one or more cell walls that are thicker than nearby cell walls of the same honeycomb structure according to one or more embodiments. 
         FIG.  22 B  schematically illustrates an example of a repeating structural unit comprising one or more cell walls that are thicker than nearby cell walls of the same repeating structural unit according to one or more embodiments. 
         FIG.  22 C  schematically illustrates another example of a plugged honeycomb structure comprising multiple cell walls that are thicker than nearby cell walls of the same honeycomb structure and that extend in two orthogonal directions according to one or more embodiments. 
         FIG.  23    illustrates a flowchart of a method of manufacturing a honeycomb structure according to one or more embodiments. 
         FIG.  24    shows modeled pressure drop performance of four different filter (plugged honeycomb body) designs for different soot loads, each plugged honeycomb structure having a diameter of 10.5 inches and length of 7.5 inches, cell density of 350 cells per square inch, honeycomb matrix wall thickness of 9.5 mils, wall porosity of 45% and wall median pore diameter of 14 micrometers. 
         FIG.  25    shows the modeled pressure drop performance of four different filter designs for different soot loads, each plugged honeycomb structure having diameter of 10.5 inches and length of 7.5 inches, wall porosity of 55% and wall median pore diameter of 12 micrometers. 
     
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to embodiments of particulate filters, honeycomb structure bodies, such as porous ceramic honeycomb articles, porous ceramic wall-flow diesel particulate filters and honeycomb structures thereof, embodiments of which are illustrated in the accompanying drawings. Whenever possible, the same reference numerals will be used throughout the drawings to refer to the same or like parts. 
     As seen in the known structure of  FIGS.  1 A and  1 B , every other channel is plugged in a checkerboard pattern at one face, and the opposite channels are plugged at the other face. In such structures, 50% of the channels are inlet channels and in each inlet channel, all four walls of the inlet cell defining the inlet channel are shared with neighboring outlet cells that define neighboring outlet channels. All inlet channels have the same cross-sectional area as the outlet cells. Therefore, 100% of the inlet walls in the representative portion of the honeycomb structure can be considered to be “filtration walls,” that are configured to directly filter particulates Of course when considering an entire honeycomb structure of a filter, not every wall need be a filtration wall, such as the border provided by an outer peripheral skin, which may be nonporous or less porous than the bulk of the walls of the honeycomb structure. 
       FIGS.  2 A and  2 B  illustrate another known plugged honeycomb structure that provides an increased inlet volume (for example, as compared to the structure of  FIG.  1 A  having equal-sized inlet and outlet channels) wherein the size of the inlet channels is large relative to the outlet channels in order to provide increased ash storage in the larger inlet channels of the inlet cells. Such structures are referred to as having asymmetric cells, or asymmetric cell sizes. Larger hydraulic diameter ratios of such asymmetric cell designs can provide reduced frequency of filter cleaning intervals, however very large hydraulic diameter ratios can result in larger pressure drop that results as the size of the outlet channels (e.g. the hydraulic diameter of the outlet channels) decreases. That is, if the inlet-to-outlet cell (or inlet-to-outlet channel) size ratio is increased to a large degree, the smaller size of the outlet channels can result in quite a large penalty in pressure drop. In addition, die design and manufacture can become more difficult and costly as the size of the outlet channels is further reduced. For example, small pins would be used to produce the small outlet channels and such small pins may not be well anchored to the die and may be more relatively easy to break off during extrusion or handling. Furthermore, very small channels can become difficult to plug due to their small size. In addition, the position of the wall separating one pair of inlet &amp; outlet channels from another in one wall becomes more offset from the wall separating a nearby pair of inlet &amp; outlet channels in the row below. With such an offset, the load carrying capacity of the structure in that direction may become lowered relative to a structure having straighter walls. Furthermore, extrusion dies for such asymmetric cell designs tend to be relatively costly and complicated to manufacture such as by electrical discharge machining (EDM) and more complex electrodes. 
       FIGS.  3 A and  3 B  shows a representative portion of another known plugged honeycomb structure that can provide an increased ash storage capacity by virtue of plugs disposed in a plugging pattern which has a higher number of inlet channels than outlet channels. The comparative structure has an inlet:outlet channel count ratio of 3:1 (number of inlet channels to number of outlet channels). Inlet channels are depicted by no shading, wherein the outlet channels are depicted as being shaded (hatched). In this honeycomb structure, there are 2 types of inlet cells, which are labeled A and B. In A-type inlet cells, there are 2 walls shared with neighboring outlet cells (one wall shared with each neighboring outlet channel). In B-type inlet cells, there is one shared wall with a neighboring outlet cell (of a neighboring outlet channel), the other three walls are shared with other neighboring inlet cells. In this structure, there are one A-type inlet cell and 2 B-type inlet cells in each repeating structural unit (otherwise referred to as a unit block, a repeating unit, or a repeating channel unit). Thus, this honeycomb structure employs, on average, 33% of inlet walls for direct filtration. Although this structure provides an increased inlet volume (or area) by 50% and therefore results in a large increase in ash storage capacity, this structure would suffer from a pressure drop penalty because all the outlet flow is restricted to a small number of relatively small outlet channels. 
     For both of the designs illustrated in  FIGS.  2 A- 2 B and  3 A- 3 B , a pressure drop penalty develops due to either the small hydraulic diameter of the outlet cells, or the reduced number of outlet channels available to carry the flow. In both of those cases, the pressure drop penalty due to the outlet channel can be relatively severe. 
     In contrast to known filters and honeycomb structures represented by  FIGS.  1 A- 1 B,  2 A- 2 B or  3 A- 3 B , the filters and plugged honeycomb structure bodies disclosed herein comprise honeycomb structures which help to mitigate these issues by comprising one or more outlet cells (or channels) of increased hydraulic diameter relative to the inlet cells (or channels). For example, in various embodiments disclosed herein, a combination of square and rectangular cells achieves an advantageous geometry of the honeycomb structure, wherein the inlet cells and channels are squares and at least some of the outlet cells and channels are rectangles (i.e., non-squares). “Rectangle” or “non-square rectangle” as used herein means a quadrilateral with four right angles and two sides of longer length than the other two sides, and “rectangular” means having a non-square rectangular shape or outline, or the shape or shape of a quadrilateral with four right angles and two sides of longer length than the other two sides. 
     In the example embodiments disclosed below, the ratio of inlet-to-outlet volume (or ratio of cross-sectional areas) is given, as well as the average fraction of the inlet cell surface that is shared with an outlet cell. In one or more embodiments, both the ratio of inlet-to-outlet volume and the average fraction of the inlet cell surface that is shared with an outlet cell are relatively high values. Higher ratios of inlet-to-outlet volume may provide increased ash storage capacity. Higher fractions (e.g., ratios) of shared inlet:outlet walls serves to mitigate pressure drop increases for both clean and soot loaded pressure drops. 
     In one or more embodiments, the honeycomb structure comprises a plurality of repeating structural units (e.g. unit blocks) which when disposed adjacent to each other to form at least a portion of the honeycomb structure. The repeating structural units can be characterized as having one or more of A-type inlet cells having 2 walls shared with neighboring outlet cells, B-type inlet cells having one shared wall with a neighboring outlet cell, or X-type inlet cells (or channels) having no shared walls with a neighboring outlet cell, or combinations thereof. As used herein, “neighboring cell” refers to a cell which is directly adjacent (abutting) another cell. Various embodiments may comprise A-type inlet cells, B-type inlet cells, or X-type inlet cells, or combinations thereof. In the Figures, the inlet channels corresponding to inlet cells are depicted by having no shading, and outlets channels corresponding to outlet cells are depicted as including hatched shading. In one set of embodiments, plugs are disposed in at least some of the channels at or near an inlet end or inlet face, and plugs are disposed in at least some of the channels at or near the outlet end or outlet face, opposite the inlet end. 
     A first embodiment of a honeycomb structure  400  comprising a portion of a honeycomb structural body of a particulate filter according to embodiments of the disclosure is shown in  FIGS.  4 A- 4 E . The honeycomb structure  400  has an inlet-to-outlet volume ratio of 1.5:1 and having an Inlet/Outlet ratio approximately equal to 3/1, and each inlet cell  401  of the repeating structural unit  400 U (shown in  FIG.  4 B  and outlined in a dotted line in  FIG.  4 A ) shares two of its walls (i.e. all inlet cells  401  have two shared outlet walls) with neighboring outlet cells, so that the average filtration area is approximately 50%. The inlet cells  401  labeled A 1  and A 0  have either two opposing walls shared with outlet cells (labeled A 0 ) or two orthogonal walls shared with adjacent outlet cells (labeled A 1 ). In one set of embodiments, and in the embodiment of  FIGS.  4 A- 4 E , the repeating structural unit  400 U (unit block) comprises a rectangular outlet cell  402  (and associated rectangular outlet channel), and square inlet cells  401  (and associated square inlet channels). In the case of the  FIGS.  4 A- 4 E  embodiment, the outlet length Lo measured in cross-section from end wall-to-end wall along a long dimension of the rectangular outlet cell  402  can be a wall thickness more than twice (for example, more than 2.01 times, or more than 2.1 times, or more than 2.5 times, or more than 3 times, etc.) the inlet length Li measured end wall-to-end wall thereof; in some embodiments, the wall thickness may be twice or more than twice (for example greater than or equal to two times) the inlet length Li measured end wall-to-end wall. The outlet width Wo may be equal to the inlet length Li. For example, Lo may range between about 0.075 inch (1.91 mm) and about 0.125 inch (3.18 mm) and Li and Wo may range between about 0.035 inch (0.89 mm) and about 0.055 inch (3.18 mm). Other values of Lo and Li may be used. 
     As shown in  FIG.  4 C , the honeycomb structure  400  comprises inlet cells  401  and outlet cells  402  formed by a matrix of intersecting porous cell walls  403  (a few labeled). The inlet cells  401  may be unplugged at the inlet end  404  and may be plugged with outlet-end plugs  408  at or near the outlet end  406 . The outlet cells  402  may be plugged at or near the inlet end  404  with inlet-end plugs  407  and may be unplugged at the outlet end  406 . The plugging may be to a depth of about 5 to 20 mm, although this can vary. Any suitable plugging cement may be used for the particular ceramic material making up the cell walls  403 . 
     As can be seen in  FIGS.  4 D and  4 E , the repeating structural unit  400 U is repeated throughout the honeycomb structure  400  of the honeycomb body of the particulate filter  405 . Plugs are shown as hatched in all inlet and outlet views shown herein. In particular, the outlet cells  402  of rectangular cross-section may be interspersed uniformly within the honeycomb structure  400 , in some embodiments. Within the honeycomb structure  400 , there are two kinds of cell walls  403  present, namely, first walls that extend entirely across the honeycomb structure  400  of the particulate filter  405  (e.g., from one portion of the skin  409  to another portion of the skin  409 ) and partial walls  403 P that do not extend all the way across the honeycomb structure  400 , but that only extend part way across the honeycomb structure  400  and which terminate within the honeycomb structure  400 . Cell walls  403 X are of the first type of walls in this embodiment and they extend entirely across a width of the particulate filter  405  in the x direction from one side to the other, and partial walls  403 P that extend in a second direction (e.g., in the y direction) orthogonal to the first direction x. The partial walls  403 P within the honeycomb structure  400  terminate at a side of an outlet cell  402  at a T-intersection with a side wall of the outlet cell  402 , i.e., at mid span. Thus, in this embodiment of  FIGS.  4 A- 4 E , the cell walls  403 X of the first type extend in the first direction x and the partial walls  403 P extend in the second direction y. The partial walls  403 P extend across two adjacent repeating structural units  400 U in this embodiment. 
     Another embodiment of honeycomb structure  500  is shown in  FIGS.  5 A and  5 B  which comprises inlet cells  501  and outlet cells  502  formed by a matrix of intersecting porous cell walls  503  (a few labeled). The honeycomb structure  500  has comprises an inlet-to-outlet volume ratio of 2:1 and having an Inlet/Outlet ratio approximately equal to 4/1, and half the inlet cells  501  (labeled A 0 ) share two walls with a neighboring outlet cell  502  and the other half of the inlet cells  501  (labeled B 2 ) share one wall (plus two shared corners) with neighboring outlet cells so the average filtration area is approximately 37.5%. In one set of embodiments, the repeating structural unit  500 U (unit block) has a rectangular outlet cell  502  (and associated rectangular outlet channel), and square inlet cells  501  (and associated square inlet channels). In this embodiment, each adjacent repeating structural unit  500 U′ positioned above and below the repeating structural unit  500 U is shown as being staggered by one position along the row so that the outlet cell  502  of repeating structural unit  500 U is not vertically aligned with an outlet cell  502 ′ of an adjacent repeating structural unit  500 U′, i.e., they do not align in the same column, as shown. In this embodiment, the vertical walls  503 X and horizontal walls  503 Y each extends all the way across the honeycomb structure  500  of the honeycomb body. The partial walls  503 P in this embodiment extend only one inlet width and terminate on sides of the outlet cells  502  at T-intersections. 
       FIGS.  5 C and  5 D  disclose an embodiment where the structure of  FIGS.  5 A- 5 B  has been turned on its side. In this embodiment, all of the cell walls  503  of the honeycomb structure  500 ″ at a given x location are interconnected to form a first straight line of walls, and only a fraction of the walls at a given y location are interconnected to form a second straight line of walls. In this embodiment, all of the walls  503 Y′ extending in they direction may extend fully and entirely across the honeycomb structure  500 ″ of the honeycomb body. In the x direction, some of the walls  503 X′ may extend fully across the honeycomb structure  500 ″, while the partial walls  503 P′ do not extend fully across the honeycomb structure  500 ″. All the other structure of this embodiment is the same as for  FIGS.  5 A- 5 B . 
     Another embodiment of a honeycomb structure  600  is shown in  FIGS.  6 A and  6 B  which comprises inlet cells  601  and outlet cells  602  formed by a matrix of intersecting porous cell walls  603  (a few labeled). The honeycomb structure  500  has an inlet-to-outlet volume ratio of 2:1, and half the inlet cells  601  (labeled A 0 ) share two walls with neighboring outlet cells  602  and the other half of the inlet cells  601  (labeled B 2 ) share one wall (preferably, plus two shared corners) with a neighboring outlet cell  602  so the average filtration area is 37.5% and the Inlet/Outlet ratio approximately equal to 16/4. In one set of embodiments the repeating structural unit  600 U (e.g., unit block) has a plurality of rectangular outlet cell  602  (and associated rectangular outlet channel), and square inlet cells  601  (and associated square inlet channels). In this embodiment, the repeating structural unit  600 U is shown staggered (e.g., by one column) from the directly adjacent repeating structural unit  600 U′. In this embodiment, all the walls are partial walls  603 P. Furthermore, in this embodiment, the outlet cells  602  with the repeating structural units  600 U are arranged in different orientations with the long dimension of the outlet cells  602  being oriented vertically along the x direction in some instances, and oriented with the long dimension of the outlet cell  602  oriented along the y direction in other instances. 
     The honeycomb structure  700  shown in  FIGS.  7 A and  7 B  which comprises inlet cells  701  and outlet cells  702 R,  702 S formed by a matrix of intersecting porous cell walls  703  (a few labeled). The honeycomb structure  700  has an inlet-to-outlet volume ratio of 2:1 and an Inlet/Outlet ratio approximately equal to 8/3, and 75% of the inlet cells  701  (labeled A 0  or A 1 ) share two walls with neighboring outlet cells and 25% of the inlet cells  701  (labeled B 1 ) share one wall (preferably, plus one shared corner) with a neighboring outlet cell  702 R so the average filtration area is 43.75%. In one set of embodiments the repeating structural unit  700 U (e.g., unit block) has one rectangular outlet cell  702 R, two square outlet cells  702 S, and the inlet cells  701  are square. In the depicted embodiment, the repeating structural unit  700 U is stacked upon, but not staggered from, an adjacent repeating structural unit  700 U′. This embodiment comprises a combination of the first type of walls  703 X and  703 Y that extend all the way across the honeycomb structure  700  of the honeycomb body, and partial cells  703 P that are two cells long. Moreover, in this embodiment, some of the outlet cells  702 S comprise a square cross-sectional shape and some inlet cells  701 R comprise a rectangular cross-sectional shape. 
     The honeycomb structure  800  shown in  FIGS.  8 A and  8 B  which comprises inlet cells  801  and outlet cells  802  formed by a matrix of intersecting porous cell walls  803  (a few labeled). The honeycomb structure  800  has an inlet-to-outlet volume ratio of 2.125:1 and an Inlet/Outlet ratio approximately equal to 17/4, and approximately 47% of the inlet cells  801  (labeled A 0  and A 2 ) share two walls with a neighboring outlet cell, approximately 47% share one wall with neighboring outlet cells  802 , and approximately 6% of the inlet cells  801  (labeled X 4 ) have no shared walls (preferably, and four shared corners) with neighboring outlet cells  802 , so the average filtration area is approximately 35.3%. In one set of embodiments the repeating structural unit  700 U (unit block) has a rectangular outlet cell  802  (and associated rectangular outlet channel), and square inlet cells  801  (and associated square inlet channels). In the depicted embodiment, the repeating structural unit  800 U is stacked upon, but not staggered from, an adjacent repeating structural unit  800 U′. This embodiment comprises a combination of the first type of walls  803 X and  803 Y that extend all the way across the honeycomb structure  800  of the honeycomb body in both the x and y directions, and partial walls  803 P that are four cells long and extend part way in both the x and y directions; each partial wall  803 P terminating at a side wall of an outlet cell  802  with a T-intersection. Furthermore, in this embodiment, the outlet cells  802  with the repeating structural units  800 U are arranged in different orientations with the long dimension of the outlet cells  802  being oriented vertically along the x direction in some instances, and oriented with the long dimension of the outlet cell  802  oriented along the y direction in other instances. 
     The honeycomb structure  900  shown in  FIGS.  9 A and  9 B  which comprises inlet cells  901  and outlet cells  902  formed by a matrix of intersecting porous cell walls  903  (a few labeled). The honeycomb structure  900  has an inlet-to-outlet volume ratio of 2.5:1 and an Inlet/Outlet ratio approximately equal to 5/1, and 20% of the inlet cells  901  (labeled A) share two walls with neighboring outlet cells  902 , and 80% of the inlet cells  901  (labeled B) share one wall (plus one shared corner) with a neighboring outlet cell  902 , so the average filtration area is 30%. In one set of embodiments, the repeating structural unit  800 U (unit block) has a rectangular outlet cell  902  (and associated rectangular outlet channel), and square inlet cells  901  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  903 X that extend all the way across the honeycomb structure  900  of the honeycomb body in the x direction, and partial walls  903 P of varying length that extend part way in both the y directions; each partial wall  903 P terminating at a side wall of an outlet cell  902  with a T-intersection. 
     The honeycomb structure  1000  shown in  FIG.  10    which comprises inlet cells  1001  and outlet cells  1002  formed by a matrix of intersecting porous cell walls  1003  (a few labeled). The honeycomb structure  1000  has an inlet-to-outlet volume ratio of 3:1 and an Inlet/Outlet ratio approximately equal to 6/1, and 33% of the inlet cells  1001  (labeled A) share two walls with a neighboring outlet cell  1002 , 33% of the inlet cells  1001  (labeled B) share one wall with a neighboring outlet cell  1002 , and 33% of the inlet cells  1001  (labeled X) have no common wall (preferably, plus two shared corners) with a neighboring outlet cell  1002 , so the average filtration area is 25%. In one set of embodiments, the repeating structural unit  1000 U (e.g., unit block) has a rectangular outlet cell  1002  (and associated rectangular outlet channel), and square inlet cells  1001  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1003 X and  1003 Y that extend all the way across the honeycomb structure  1000  of the honeycomb body in both the x and y directions, and partial walls  1003 P that are one cell wide and extend part way in the y directions; each partial wall  1003 P terminating at a side wall of an outlet cell  1002  with a T-intersection. 
     The honeycomb structure  1100  shown in  FIG.  11    which comprises inlet cells  1101  and outlet cells  1102  formed by a matrix of intersecting porous cell walls  1103  (a few labeled). The honeycomb structure  1100  has an inlet-to-outlet volume ratio of 3:1 and an Inlet/Outlet ratio approximately equal to 6/1, and all of the inlet cells  1101  (labeled B) share one wall with a neighboring outlet cell  1102  (preferably, wherein two cells (labeled B 0 ) in the repeating structural unit  1100 U (unit block) have a shared wall and no shared corners, and wherein four cells (labeled B 1 ) in the repeating structural unit  1100 U have a shared wall and one shared corner), so the average filtration area is 25%. In one set of embodiments the repeating structural unit  1100 U (unit block) has a rectangular outlet cell  1102  (and associated rectangular outlet channel), and square inlet cells  1101  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1103 X and  1103 Y that extend all the way across the honeycomb structure  1100  of the honeycomb body in both the x and y directions, and partial walls  1103 P that are four cells long and extend part way in both the x and y directions; each partial wall  1103 P terminating at a side wall of an outlet cell  1102  with a T-intersection. 
     The honeycomb structure  1200  shown in  FIG.  12    which comprises inlet cells  1201  and outlet cells  1202  formed by a matrix of intersecting porous cell walls  1203  (a few labeled). The honeycomb structure  1200  has an inlet-to-outlet volume ratio of 3:1 and an Inlet/Outlet ratio approximately equal to 6/1, and all of the inlet cells  1201  share one wall with a neighboring outlet cell  1202 , (preferably, wherein some inlet cells  1201  (labeled B 1 ) share one wall and one corner, and other inlet cells  1201  (labeled B 0 ) share one wall and no corners) so the average filtration area is 25%. In one set of embodiments, the repeating structural unit  1200 U (unit block) has a rectangular, non-square outlet cell  1202  (and associated rectangular outlet channel), and square inlet cells  1021  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1203 X and  1203 Y that extend all the way across the honeycomb structure  1200  of the honeycomb body in both the x and y directions (every other set of walls), and partial walls  1203 P that extend part way in both the x and y directions; each partial wall  1203 P terminating at a side wall of an outlet cell  1202  with a T-intersection. Furthermore, in this embodiment, the outlet cells  1202  and repeating structural units  1200 U,  1200 U′ are arranged in different orientations throughout the honeycomb structure  1200  with the long dimension of the outlet cells  1202  being oriented vertically along the x direction in some instances, and oriented with the long dimension of the outlet cell  1202  oriented along they direction in other instances. 
     The honeycomb structure  1300  shown in  FIG.  13    which comprises inlet cells  1301  and outlet cells  1302  formed by a matrix of intersecting porous cell walls  1303  (a few labeled). The honeycomb structure  1300  has an inlet-to-outlet volume ratio of 3.5:1 and an Inlet/Outlet ratio approximately equal to 7/1, and 86% of the inlet cells  1301  (labeled B 0  and B 1 ) share one wall with a neighboring outlet cell  1302 , and 14% of the inlet cells  1301  (labeled X 2 ) have no common wall (and two shared corners) with a neighboring outlet cell  1302 , so the average filtration area is 21.4%. In one set of embodiments, the repeating structural unit  1300 U (unit block) has a rectangular outlet cell  1302  (and associated rectangular outlet channel), and square inlet cells  1301  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1303 X and  1303 Y that extend all the way across the honeycomb structure  1300  of the honeycomb body in both the x and y directions, and partial walls  1303 P that extend part way in they direction; each partial wall  1303 P being two cells wide and terminating at a side wall of an outlet cell  1302  with a T-intersection. 
     The honeycomb structure  1400  shown in  FIG.  14    which comprises inlet cells  1401  and outlet cells  1402  formed by a matrix of intersecting porous cell walls  1403  (a few labeled). The honeycomb structure  1400  has an inlet-to-outlet volume ratio of 4:1 and an Inlet/Outlet ratio approximately equal to 8/1, and 75% of the inlet cells  1401  (labeled B 1  or B 0 ) share one wall (and either one shared corner, or no shared corner) with a neighboring outlet cell  1402 , and 25% of the inlet cells  1401  (labeled X 1 ) have no common wall (and one shared corner) with a neighboring outlet cell  1402 , so the average filtration area is 18.75%. In one set of embodiments, the repeating structural unit  1400 U (unit block) has a rectangular outlet cell  1402  (and associated rectangular outlet channel), and square inlet cells  1401  (and associated square inlet channels). This embodiment comprises a combination of the first type of cell walls  1403 X and  1403 Y that extend all the way across the honeycomb structure  1400  of the honeycomb body in both the x and y directions, and partial walls  1403 P that extend part way in the y direction; each partial wall  1403 P being four cells wide and terminating at a side wall of an outlet cell  1402  with a T-intersection. 
     The honeycomb structure  1500  shown in  FIG.  15    which comprises inlet cells  1501  and outlet cells  1502  formed by a matrix of intersecting porous cell walls  1503  (a few labeled). The honeycomb structure  1500  has an inlet-to-outlet volume ratio of 4.5:1 and an Inlet/Outlet ratio approximately equal to 9/1, and two thirds of the inlet cells  1501  (labeled B 0 ) share one wall (and no corner) with a neighboring outlet cell, and one third of the inlet cells have no common wall (one with two shared corners labeled X 2 , the remainder with one shared corner labeled X 1 ) with a neighboring outlet cell, so the average filtration area is 16.7%. In one set of embodiments, the repeating structural unit  1400 U (unit block) has a rectangular outlet cell  1502  (and associated rectangular outlet channel), and square inlet cells  1501  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1503 X that extend all the way across the honeycomb structure  1400  of the honeycomb body in the x direction, and partial walls  1503 P that extend part way in the y direction; each partial wall  1503 P terminating at a side wall of an outlet cell  1502  with a T-intersection. 
     The honeycomb structure  1600  shown in  FIG.  16    which comprises inlet cells  1601  and outlet cells  1602  formed by a matrix of intersecting porous cell walls  1603  (a few labeled). The honeycomb structure  1600  has an inlet-to-outlet volume ratio of 5:1 and an Inlet/Outlet ratio approximately equal to 10/1, and 60% of the inlet cells  1601  (labeled B 0 ) share one wall (and no shared corner) with a neighboring outlet cell  1602 , and 40% of the inlet cells  1601  (labeled X 1 ) have no common wall (and one shared corner) with a neighboring outlet cell  1602 , so the average filtration area is 15%. In one set of embodiments, the repeating structural unit  1600 U (unit block) has a rectangular outlet cell  1602  (and associated rectangular outlet channel), and square inlet cells  1601  (and associated square inlet channels). This embodiment comprises a combination of the first type of walls  1603 X and  1603 Y that extend all the way across the honeycomb structure  1600  of the honeycomb body in both the x and y directions, and partial walls  1603 P that extend part way in the y direction; each partial wall  1603 P being five cells wide and terminating at a side wall of an outlet cell  1602  with a T-intersection. 
     Thus, in various embodiments, a filter, or particulate filter, is disclosed herein, comprising a honeycomb body comprising a honeycomb structure (e.g., honeycomb structures  400 - 1600 ) comprising a matrix of intersecting porous cell walls (e.g., porous cell walls  403 - 1600 P) extending axially between an inlet end (e.g., inlet end  404 ) and an outlet end (e.g.,  406 ) of the honeycomb structure, the matrix of intersecting porous cell walls (e.g., cell walls  403 - 1603 ) defining a plurality of inlet cells (e.g., inlet cells  401 - 1601 ) and outlet cells (e.g., outlet cells  402 - 1602 ), and corresponding inlet channels and outlet channels defined by respective inlet and outlet cells, wherein at least a portion of the outlet channels are larger in cross-sectional area than any of the inlet channels, and wherein at least some of the outlet channels (e.g., corresponding to outlet cells  402 - 1602 ) are rectangular. In some embodiments, each of the outlet channels (e.g., outlet cells  402 - 1602 ) is larger in cross-sectional area than any of the inlet channels (e.g., inlet channels  401 - 1601 ). In some embodiments, some of the outlet channels (and outlet cells  702 S of  FIGS.  7 A- 7 B ) have cross-sectional area equal to the cross-sectional area of the inlet channels (and inlet cells  701 ). 
     In some embodiments, a filter, particulate filter, or honeycomb body is disclosed herein, comprising a honeycomb structure (e.g., honeycomb structure  400 - 1600 ) comprising an matrix of interconnected porous walls (e.g., interconnected cell walls  403 - 1603 ) comprising an array of cells comprised of inlet cells (e.g., inlet cells  401 - 1601 ) and outlet cells (e.g., outlet cells  402 - 1602 ) defining an array of inlet channels and outlet channels, respectively (“array of channels”), each inlet cell or inlet channel having an inlet hydraulic diameter, and each outlet cell or outlet channel having an outlet hydraulic diameter, wherein at least a portion of the outlet cells or outlet channels have an outlet hydraulic diameter larger than the inlet hydraulic diameter of any of the inlet cells or inlet channels, and wherein at least some of the outlet channels (e.g., outlet cells  402 - 1602 ) have a rectangular cross-section. The rectangular shape is defined by a perimeter of the intersecting porous walls  403  of that particular outlet cell  402 - 1602 . In some embodiments, each of the outlet cells, or outlet channels, has an outlet hydraulic diameter larger than the inlet hydraulic diameter of any of the inlet channels, or inlet cells. In some embodiments, some of the outlet cells, or outlet channels, have an outlet hydraulic diameter equal to the inlet hydraulic diameter of an inlet channel, or inlet cell, such as shown in  FIGS.  7 A- 7 B . 
     In a first set of embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the cell walls (e.g., cell walls  403 ) extend in an axial direction (z-direction as shown in  FIG.  4 C ), the walls being disposed in an x-y grid arrangement as viewed in cross-section in a plane perpendicular to the axial direction (See  FIGS.  4 A and  4 B ), the matrix comprising a first group of walls (e.g., first group of cell walls  403 X) aligned parallel to the x-direction, and a second group of parallel walls (e.g., second group of walls  403 Y) aligned parallel to the y-direction, wherein the x-direction is orthogonal to the y-direction. 
     In some of the first set of embodiments, all of the walls at a given x location in the matrix are interconnected to form a first straight line of walls that may extend entirely across the honeycomb filter body, and only a fraction of the walls at a given y location are interconnected to form a second straight line of walls orthogonal in direction to the first straight line of walls, i.e., they are partial walls. In some of the first set of embodiments, the walls corresponding to at least 3 consecutive cells interconnect end-to-end to form a first straight line of a plurality of walls, and only a fraction of the walls at a given y location interconnect to form a second straight line of a plurality of walls. In some of the first set of embodiments, all of the walls at a given x location are interconnected to form a first straight line of walls, and only a fraction of the walls at a given y location are interconnected to form a second straight line of walls, i.e., they are partial walls. In some of the first set of embodiments, a plurality of the walls at a given x location in the matrix are interconnected to form a first straight line. In some of the first set of embodiments, at a plurality of y locations in the x-y grid and for a given x location in the x-y grid, a plurality of the walls at the y locations in the matrix are interconnected to form straight lines. In some of the first set of embodiments, not all the walls that extend in the x-direction at a selected y-location in the matrix are disposed end-to-end in a straight line, i.e., they are partial walls. In some of the first set of embodiments, some of the walls that extend in the x-direction at a selected y-location in the matrix are disposed end-to-end in a straight line. 
     In a second set of embodiments of the filters, honeycomb bodies, and honeycomb substrates disclosed, the matrix of porous cell walls extends in an axial direction, the matrix comprising a first group of parallel walls, and a second group of parallel walls which are oriented orthogonally with respect to the walls of the first group as viewed in a plane perpendicular to the axial direction. In some of these embodiments, the walls of the first group are interconnected continuously across the entire width of the matrix of walls of the honeycomb body; in some of these embodiments, the walls of the second group are not continuously interconnected across the entire width of the matrix of walls of the honeycomb body, i.e., they are partial walls. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the inlet channels and the outlet channels axially extend parallel to one another. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the inlet cells and the outlet cells extend in axial parallel arrangement to one another. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the inlet channels (as well as the corresponding inlet cells) have a polygonal cross-sectional shape in a plane perpendicular to the axial direction. In some of these embodiments, at least one vertex of the polygonal cross-sectional shape comprises a rounded portion; in other embodiments, at least one vertex of the polygonal cross-sectional shape comprises a beveled portion as is shown in  FIGS.  6 C and  6 D . In some embodiments, the outlet channels (or outlet cells) have a polygonal cross-sectional shape in a plane perpendicular to the axial direction; in some of these embodiments at least one vertex of the polygonal cross-sectional shape comprises a rounded portion (e.g., a radius), and in other embodiments at least one vertex of the polygonal cross-sectional shape comprises a beveled portion (a chamfer) as is shown in  FIGS.  6 C and  6 D . 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures, the at least some of the inlet channels, or inlet cells of the inlet channels, have a square cross-sectional shape in a plane perpendicular to the axial direction z. In some embodiments of the filters, honeycomb bodies, and honeycomb structures, at least some of the outlet channels, or the outlet cells of the outlet channels, have a rectangular cross-sectional shape in a plane perpendicular to the axial direction. 
     In some embodiments of the filters and honeycomb structures disclosed herein, at least some of the inlet channels, or the inlet cells of the inlet channels, have a square cross-sectional shape in a plane perpendicular to the axial direction, and at least some of outlet channels, or the outlet cells of the outlet channels, have a rectangular cross-sectional shape in a plane perpendicular to the axial direction. For example, for some repeating structural units (e.g., repeating structural units  400 U,  500 U, and  900 U- 1600 U), one of the cells of the repeating structural unit may be an outlet cell having a rectangular cross-sectional shape (e.g., outlet cells  402 ,  502 , and  902 - 1602 ) and 3 or more cells may be inlet cells having a square cross-sectional shape (including 3, 4, 5, 6, 7, 8, 9, and 10 inlet cells). 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, there are 3 or more times as many inlet channels (or inlet cells) as outlet channels (or outlet cells), i.e., the I/O ratio is greater than or equal to 3/1, including greater than or equal to 4/1, greater than or equal to 5/1, greater than or equal to 6/1, greater than or equal to 7/1, greater than or equal to 8/1, or even greater than or equal to 9/1. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, an inlet open frontal area of the honeycomb structure (OFAin) comprises the sum of the areas of the inlet channels in a plane perpendicular to the axial direction, and an outlet open frontal area of the honeycomb structure (OFAout) comprises the sum of the areas of the outlet channels in a plane perpendicular to the axial direction, and wherein OFAin&gt;OFAout. In some of these embodiments, a ratio of OFAin:OFAout is between 1.5 and 5.0. In some embodiments, a product of the ratio OFAin:OFAout and the average fraction of the perimeter of inlet cells or channels sharing a wall with outlet cells or channels is at least 0.67 but less than 1.0. 
     In some or all of the embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, all of the porous cell walls (e.g., porous cell  403 - 1603 ) in the matrix have the same thickness. The wall thickness may range from about 0.10 mm to about 0.41 mm, for example. Other wall thicknesses are possible. 
     In some or all of the embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, all of the porous cell walls (e.g., cell walls  403 - 1603 ) in the matrix have the same average thickness. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, one or more inlet channels are disposed between any two outlet channels in the matrix; in some embodiments, each outlet channel is spaced away from any other outlet channel by one or more inlet channels. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the cell walls comprise a plurality of first walls and a plurality of second walls, wherein the first walls have a first average thickness and the second walls have a second average thickness, and the second average thickness is greater than the first average thickness (See  FIGS.  20 A- 20 C ). The second average thickness may be 20% greater than the first average thickness, or even more than 20%, for example. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the matrix comprises a plurality of repeating unit blocks, each unit block comprising a group of cells defining at least one outlet channel and a plurality of surrounding inlet channels adjacent and abutting at least one the outlet channel of the repeating unit block. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the honeycomb structure comprises a plurality of repeating structural units (e.g., repeating structural units  400 U- 1600 U), each unit comprising a group of interconnected walls defining a respective outlet cell, or outlet channel, and a plurality of surrounding or abutting inlet cells, or surrounding or abutting inlet channels, adjacent the respective outlet channel or outlet cell. In some embodiments, the honeycomb structure comprises a portion comprising a first plurality of first repeating structural units, and another portion which is either free of repeating structural units or which comprises a plurality of second repeating structural units wherein the first and second repeating structural units differ. In some embodiments, the honeycomb structure can comprise a portion consisting of square cells, which in some embodiments such portion consists of square cells which are of the same size or substantially similar size to each other. For example, 
     In some of these embodiments, each repeating structural unit comprises walls extending in an axial direction (z-direction), the walls comprised of a first group of side walls aligned parallel to a first direction and a second group of side walls aligned parallel to a second direction, wherein the first direction is orthogonal to the second direction, and the first and second directions are each orthogonal to the axial direction (e.g., direction z). In some embodiments, each repeating structural unit (e.g., repeating structural units  400 U- 1600 U), comprises: an outlet-defining set of walls comprised of a first set of the walls of the first group interconnected with a second set of the walls of the second group to collectively define a corresponding outlet channel, or outlet cell, in the repeating structural unit, the outlet cell or channel having a plurality of corners, and at least one wall, other than the outlet-defining set of walls which interconnects with a T-intersection with one of the outlet-defining set of walls at an intermediate location (e.g., half way) between corners of the corresponding outlet channel or cell. In some of these embodiments, each repeating structural unit (e.g., repeating structural units  400 U- 1600 U), comprises: an outlet-defining set of walls comprised of a first set of the walls of the first group interconnected with a second set of the walls of the second group to collectively define a corresponding outlet channel, or outlet cell, in the repeating structural unit, the outlet cell or outlet channel having a plurality of corners, and at least one wall, other than the outlet-defining set of walls which interconnects with one of the outlet-defining set of walls at a location spaced away from the corners of the corresponding outlet channel, or outlet cell. 
     In some embodiments, the repeating structural unit (e.g., repeating structural units  400 U- 1600 U), comprises a plurality of inlet channels or inlet cells. In some embodiments, at least one of the walls in the repeating structural unit is shared by one of the inlet cells and one of the outlet cells in the repeating structural unit. In some embodiments, a plurality of the walls in the repeating structural unit are shared by at least one inlet cell and at least one outlet cell. In some of these embodiments, at least one of the inlet channels shares one of the outlet-defining set of walls with the corresponding outlet channel or outlet cell. In some embodiments, each of a plurality of the inlet cells or channels shares one of the outlet-defining set of walls with the corresponding outlet cell or channel. 
     In some embodiments, the repeating structural unit (e.g., repeating structural units  600 U- 800 U), comprises a plurality of outlet channels or cells; in some of these embodiments (e.g., repeating structural units  700 U), the repeating structural unit comprises at least two outlet cells, or outlet channels, having differing cross-sectional areas, or hydraulic diameters. In some embodiments, a sidewall of the first group interconnects with a sidewall of the second group. In some embodiments, sidewalls of the first group interconnect with a sidewall of the second group. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the walls are disposed in an x-y grid arrangement as viewed in a plane perpendicular to the axial direction (z-direction), the array of walls comprising a first group of walls (e.g.,  403 X- 1603 X) aligned parallel to the x-direction, and a second group of parallel walls (e.g.,  403 Y- 1603 Y) aligned parallel to the y-direction, wherein the x-direction is orthogonal to the y-direction. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the matrix comprises a plurality of repeating channel units, each channel unit comprising a respective outlet channel and a plurality of surrounding or abutting inlet channels adjacent the respective outlet channel. In some embodiments, the plurality of surrounding or abutting inlet channels is defined by a plurality of shared sidewall inlet cells. In some embodiments, the plurality of surrounding or abutting inlet channels is defined by a plurality of shared corner portion inlet cells. In some embodiments, the plurality of surrounding or abutting inlet channels is defined by a plurality of shared sidewall inlet cells, or a plurality of shared corner portion inlet cells, or both. In some embodiments, a respective one of the cell walls is disposed between each shared sidewall inlet cell and a respective adjacent outlet cell. In some embodiments, each of the inlet cells surrounds a respective outlet cell in its respective repeating unit, wherein the respective shared walls of the repeating unit are disposed between the outlet cell and each of the inlet cells. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, each of the outlet cells, or outlet channels, is completely surrounded by inlet cells, or inlet channels. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, no outlet channel in the matrix is adjacent to another outlet channel. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, no outlet cell in the array of cells is adjacent to another outlet cell. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, each of the inlet channels in at least one repeating structural unit are of equal cross-sectional shape and size. In some embodiments of the filters and honeycomb structures disclosed herein, each of the inlet cells in at least one repeating structural unit are of equal cross-sectional shape and/or size. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, all of the inlet channels are of equal cross-sectional shape and/or size (excluding partial channels intersecting with the skin). In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, each of the inlet cells are of equal cross-sectional shape and/or size (excluding partial channels intersecting with the skin). 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, all of the inlet channels in at least one repeating structural unit are of equal cross-sectional shape and/or size. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, the walls extend axially between an inlet end and an outlet end of the honeycomb structure. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, all of the walls in the matrix have the same thickness at an axial location disposed between the inlet end and the outlet end. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, at least a majority of the inlet channels are plugged at or near the outlet end. 
     In some embodiments of the filters, honeycomb bodies, and honeycomb structures disclosed herein, at least a majority of the outlet channels are plugged at or near the inlet end. 
     In some embodiments, the honeycomb structure comprises one or more portions comprising repeating structural units, and one or more portions not comprising repeating structural units, and in particular embodiments one or more portions free of repeating structural units comprising a non-square rectangular cell. For example, as shown in partial view in  FIGS.  16 C- 16 D , some portions that may not constitute the presence of repeating structural units  1600 U could comprise inlet channels and outlet channels, or blocked channels, which may be present in one or more locations in the honeycomb structure such as at or near the outer periphery, or at the centerline, or at other select locations throughout the honeycomb structure. For example, the one or more portions not comprising repeating structural units  1600 U may be incomplete units  1600 I having part of the shape of a repeating structural units  1600 U that are located at the outer periphery adjacent to the skin  1609  as shown in  FIG.  16 C . In  FIG.  16 D , the one or more portions not comprising repeating structural units  1600 U may be a group of blocked cells  1600 N (e.g., outlet and/and inlet cells) that are located within the honeycomb structure  1600 B, but do not have the same plugging pattern as the repeating structural units  1600 U. Such a group of blocked cells  1600 N may be located at or near the outer periphery, or at the centerline, or at other select locations of the honeycomb body. In some embodiments, the honeycomb structure comprises more than one distinct such group. In some embodiments, the honeycomb structure comprises one or more portions comprising first repeating structural units, and one or more portions comprising second repeating structural units, wherein the first and second repeating structural units differ from one another. 
     In one set of embodiments disclosed herein, particulate wall flow filters (e.g., diesel and/or gas particulate wall flow filters) comprise a honeycomb body with a honeycomb structure comprising: OFAin is greater than OFAout, and the ratio OFAin:OFAout that is between 1.5 and 5.0 (including 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0); ratio of the area of inlet cells, or inlet channels, to area of the outlet cells, or outlet channels. In some embodiments, the ratio OFAin:OFAout is greater than 2, or even greater than 3, all inlet cells and their inlet channels may be of equal cross-sectional shape and size. In some embodiments, at least some of the outlet cells and their channels have a larger hydraulic diameter than any of the inlet cells or channels. In some embodiments, the product of the ratio OFAin:OFAout and the average fraction of the perimeter of inlet cells sharing a wall with outlet cells is at least 0.67 but less than 1.0. In some embodiments, the honeycomb structure is comprised of walls disposed such that inlet cells define square inlet channels and outlet cells define rectangular outlet channels. In some embodiments, for a particular repeating structural unit of the honeycomb structure, a cross-sectional area of at least one of the outlet cells is twice that of at least one of the inlet cells. In some embodiments, the cross-sectional area of any of the outlet cells is twice that of any of the inlet cells. In some embodiments, the web thickness, or thickness of the walls of the honeycomb structure that define the channels are of constant or substantially constant wall thickness; in other embodiments, the honeycomb structure comprises non-equal web thicknesses (or wall thicknesses), such as to provide enhanced isostatic strength. 
     Example embodiments of the honeycomb structures  400 - 1600  of the present disclosure can in some instances comprise a relatively high level of open and interconnected total porosity. For example, honeycomb structures  400 - 1600  comprising a total porosity, % P, of at least 35%, at least 40%, at least 45%, at least 50%, or even at least 60%, as determined by mercury porosimetry, can be provided. 
     In addition to the relatively high total porosities, the honeycomb structures  400 - 1600  of the present disclosure can also comprise median pore diameter, d 50 , of the pores of at least 8 μm, of at least 10 μm, or even of at least 12 μm. Further, the median pore diameter, d 50 , of the pores may not exceed 30 μm, may not exceed 25 μm, and in some embodiments may not exceed 20 μm. In still another embodiment, the median pore diameter, d 50 , of the pores can be in the range of from 8 μm to 30 μm, from 10 μm to 25 μm, from 12 μm to 20 μm, or even from 12 μm to 18 μm. 
     In addition to the relatively high total porosities and specified median pore sizes, the honeycomb structures  400 - 1600  of the present disclosure can also comprise a relatively narrow pore size distribution as evidenced by a minimized percentage of relatively fine and/or relatively coarse pores. To this end, pore size distributions can be expressed by a pore fraction. For example, the quantity d 50  represents the median pore size based upon pore volume, and is the pore diameter at which 50% of the open porosity of the ceramic structure has been intruded by mercury. The quantity d 90  is the pore diameter at which 90% of the pore volume is comprised of pores whose diameters are smaller than the value of d 90 ; thus, d 90  is also equal to the pore diameter at which 10% by volume of the open porosity of the ceramic structure has been intruded by mercury. Still further, the quantity d 10  is the pore diameter at which 10% of the pore volume is comprised of pores whose diameters are smaller than the value of d 10 ; thus, d 10  is equal to the pore diameter at which 90% by volume of the open porosity of the ceramic structure has been intruded by mercury. The values of d 10  and d 90  are expressed in units of micrometers. 
     The relatively narrow pore size distribution of the exemplary embodiments of the honeycomb structures  400 - 1600  can, in one embodiment, be evidenced by the width of the distribution of pore sizes finer than the median pore size, d 50 , further quantified as pore fraction. As used herein, the width of the distribution of pore sizes finer than the median pore size, d 50 , are represented by a “d factor ” or “d f ” value which expresses the quantity (d 50 −d 10 )/d 50 . To this end, the ceramic honeycomb structures of the present disclosure can comprise a d f  value of less than 0.50, less than 0.40, less than 0.35, or even less than 0.30. In some embodiments, the d f  value of the disclosed honeycomb structure is less than 0.25, or even less than 0.20. To this end, a relatively low value of d f  indicates a low fraction of fine pores, and low values of d f  can be beneficial for improving low soot-loaded pressure drop when the honeycomb structures  400 - 1600  are utilized in filtration applications, such as for diesel or uas particulate filters. 
     Relatively narrow pore size distribution of the disclosed honeycomb structures  400 - 1600  can optionally or additionally, in some embodiments, be evidenced by the width of the distribution of pore sizes that are finer or coarser than the median pore size, d 50 , further quantified as a pore fraction. As used herein, the width of the distribution of pore sizes that are finer or coarser than the median pore size, d 50 , are represented by a “d Breadth ” or “d B ” value which expresses the quantity (d 90 −d 10 )/d 50 . To this end, the honeycomb structures  400 - 1600  of the present disclosure in some embodiments, may comprise a d B  value that is less than 1.50, less than 1.25, less than 1.10, or even less than 1.00. In some example embodiments, the value of d B  is less than 0.8, less than 0.7, or even less than 0.6. A relatively low value of d B  can provide a relatively higher filtration efficiency as well as and higher strength in honeycomb structures used to filter particulates from exhaust flows for diesel and/or gas combustion engines. 
     To this end, a combination of the aforementioned porosity values, median pore diameter values, and either d f  or d B  can aid in providing low clean and soot-loaded pressure drop while maintaining useful filtration efficiency when the ceramic honeycomb bodies of the present disclosure are used in diesel or gas exhaust filtration applications. 
     Further, one or more embodiments of the honeycomb structures  400 - 1600  described herein may exhibit a low coefficient of thermal expansion resulting in excellent thermal shock resistance (TSR). As will be appreciated, TSR is inversely proportional to the coefficient of thermal expansion (CTE). That is, a honeycomb structure with low thermal expansion will typically have higher thermal shock resistance and can survive the wide temperature fluctuations that are encountered in, for example, diesel or gas exhaust filtration applications. Accordingly, in one or more embodiments, the honeycomb structures  400 - 1600  may be characterized by having a relatively low coefficient of thermal expansion (CTE) in at least one direction and as measured by dilatometry, that is less than or equal to about 25.0×10″ 7 /° C., less than or equal to 20.0×10″ 7 /° C.; less than or equal to 15.0×10″ 7 /° C.; less than or equal to 10.0×10″ 7 /° C.; or even less than or equal to 8.0×10″ 7 /° C., across the temperature range of from 25° C. to 1000° C. 
     Still further, it should be understood that embodiments of the above-described honeycomb structures  400 - 1600  can exhibit any desired combination of the aforementioned properties. For example, in some embodiments, CTE (25-1000° C.) is less than or equal to 12×10″ 7 /° C. (or even is less than or equal to 10×10″ 7 /° C.), the porosity % P is at least 40% or even at least 45%, the median pore diameter is at least 10 μm (or at least 12 μm), and the value of d f  is less than 0.35 (or even less than 0.30). In some embodiments, d B  may be less than 1.0, less than 0.85, or even less than 0.75. The honeycomb bodies of the present disclosure can have any shape or geometry suitable for a particular application, such as round (See  FIGS.  4 D- 4 E ) Other outer peripheral shapes such as oval, racetrack, square, rectangular, triangular, octagonal, hexagonal, or the like can be used. The honeycomb structure  400 - 1600  can further have cellular densities from about 70 cells/in 2  (10.9 cells/cm 2 ) to about 400 cells/in 2  (62 cells/cm 2 ). Other cell densities may be used. The length of the honeycomb bodies may be any suitable length for the application. In some embodiments, the honeycomb bodies made up of the honeycomb structures  1600  may be rectangular or hexagonal in outer cross-sectional shape and may be adhered together, such as with suitable cement mixture, to form a larger honeycomb body (a so-called segmented structure). 
     In some embodiments disclosed herein, the particulate wall flow filters comprising the honeycomb structure described herein provides very high ash storage volume for practically maintenance-free particulate wall flow filter use over the lifetime of a vehicle, and provides reduced pressure drop, compared to other known designs. 
     Preferably, extrusion dies used to produce the various honeycomb structures and honeycomb bodies for the particulate wall flow filters can be produced with a minimum of plunge EDM manufacturing processes, or even no plunge EDM steps (such as provided by, for example, wire EDM techniques), thereby tremendously reducing the extrusion die manufacturing costs. In some embodiments, divots or plenums may be added to some, most, or all of the pins on die designs of this type. 
     In some embodiments disclosed herein, a porous ceramic wall-flow particulate filter comprises a plugged, wall-flow honeycomb filter body comprised primarily of a ceramic and having a plurality of longitudinal cell walls defining a plurality of cell channels extending from an inlet end to an outlet end of the body, wherein at least some of the cell channels are plugged, such as at the inlet end, or at the outlet end, between the inlet and outlet ends, or a combination thereof. The porous ceramic filter may be formed from a suitable ceramic-forming batch mixture that is extruded through an extrusion die to form a green body comprised of a matrix of intersecting walls, such as a honeycomb structure, wherein the green body is dried and fired to form a ceramic structure. In some embodiments, the ceramic may be comprised of a cordierite crystalline phase, or primarily of cordierite and optionally comprising other minor crystalline phases. In some embodiments a ceramic wall-flow particulate filter is formed with plugs disposed in at least some of the channels at one end (e.g., an inlet end), or at least some of the channels at the opposite end (e.g., outlet end), such as shown plugged on each end face in the patterns described herein. The ceramic wall-flow particulate filter can be designed to be coated such as with a washcoat, but could alternatively be utilized as a bare (uncoated) filter. In some embodiments, the wall-flow honeycomb filter body and honeycomb structure (e.g., honeycomb structures  400 - 1600 ) may be comprised primarily of cordierite, cordierite magnesium aluminum titanate, mullite, aluminum titanate, silicon carbide, alumina, or other suitable ceramic materials exhibiting open interconnected porosity. In some embodiments disclosed herein, the porous ceramic wall-flow particulate filter and honeycomb body comprises a honeycomb structure comprising a plurality of intersecting walls defining parallel channels extending in an axial direction between an inlet end and an outlet end. 
     Embodiments of the present disclosure may be used as, for example, a diesel particulate filter or a gas particulate filter to filter exhaust streams emanating from diesel or gas combustion engines. 
     Depicted in  FIG.  17 C  are inlet side and outlet side views, respectively, of a ceramic body for use in a particulate wall-flow filter according to one or more embodiments described herein. The honeycomb structure  500  of the honeycomb body shown is the same as is described in  FIGS.  5 A- 5 B  herein. However, any of the other honeycomb structures (e.g., honeycomb structures  400  and  600 - 1600 ) described herein may be substituted. The ceramic body may generally have a honeycomb structure  500  comprising a matrix of intersecting cell walls  503  defining parallel channels extending in an axial direction (e.g., z direction) between an inlet end and an outlet end and may comprise any suitable ceramic material such as described above. 
     EXAMPLES 
     Various honeycomb bodies (e.g., plugged honeycomb structure bodies) including various honeycomb structures were fabricated, namely 2″ diameter×6″ axial length parts were produced of porous aluminum titanate and tested in order to compare the pressure drop of designs disclosed herein versus comparative designs. Three different honeycomb structures were evaluated:  FIG.  17 A ) an asymmetric honeycomb structure similar to that shown in  FIGS.  2 A and  2 B  with a geometry of 300 cells per square inch, cell wall thickness of 7 mils (aka “300/7 geometry”), and an inlet-to-outlet cross-sectional area ratio of 1.7:1;  FIG.  17 B ) an asymmetric honeycomb structure similar to that shown in  FIGS.  2 A and  2 B  with a geometry of 400 cells per square inch, cell wall thickness of 7 mils (aka “400/7 geometry”), and an inlet-to-outlet cross-sectional area ratio of 2.2:1; and  FIG.  17 C ) a honeycomb structure  500  similar to that shown in  FIGS.  5 A and  5 B  with a geometry of 400 cells per square inch, cell wall thickness of 8 mils (aka “400/8 geometry”), and an inlet to outlet cross-sectional area ratio of 2.2:1. 
       FIGS.  17 A- 17 C  shows photographs of the plugged inlet ends (upper picture) and outlet ends (lower picture) of the representative parts of the three designs (A, B, C), respectively, wherein design A is shown in  FIG.  17 A , design B is shown in  FIG.  17 B , and design C (a presently disclosed design) is shown in  FIG.  17 C . The plugged honeycomb parts of designs A, B, and C were tested for cold pressure drop as a function of flow rate (see  FIG.  18 A ), and were then loaded with printex soot and retested for soot loaded pressure drop at various levels up to about 5 g/l (see  FIG.  18 B ). After soot loaded pressure drop values were recorded, the soot was burned out and the parts were loaded with ash material obtained from a filter removed from a truck, at a level of 20 g/l. Pressure drop testing was again performed as a function of soot loading and the results are shown in  FIG.  19 C . 
     As can be seen from the pressure drop testing illustrated in the Figures, the honeycomb structure of design C as disclosed herein offers a distinct advantage in terms of pressure drop performance for a given ash storage capacity over, for example, designs A and B. The high ash storage capacity of designs A and B are disadvantaged in pressure drop due to the relatively small hydraulic diameter of the outlet channels. The embodiments disclosed herein can thus provide a pressure drop advantage in the clean and soot-loaded states over known designs. 
     Various embodiments of honeycomb bodies (plugged honeycomb structure bodies) were fabricated, for example 10.5″ diameter×7.5″ axial length bodies were made of porous cordierite and tested to evaluate the pressure drop performances. For example, four honeycomb structures were evaluated:  FIG.  22 A ) shows Sample  22 A having an asymmetric honeycomb structure similar to that shown in  FIGS.  2 A and  2 B  with a geometry of 300 cells per square inch, cell wall thickness of 7 mils (“300/7 geometry”), and an inlet-to-outlet cross-sectional area ratio of 1.7:1;  FIG.  22 B ) shows Sample  22 B having a symmetric honeycomb structure similar to that shown in  FIGS.  1 A and  1 B  with a geometry of 200 cells per square inch, cell wall thickness of 8 mils (“200/8 geometry”), and an inlet-to-outlet cross-sectional area ratio of 1:1;  FIG.  22 C ) shows Sample  22 C having a honeycomb structure  500  similar to that shown in  FIGS.  5 C and  5 D  with a geometry of 400 cells per square inch, cell wall thickness of 8 mils (“400/8 geometry”), and an inlet to outlet cross-sectional area ratio of 1.8:1; and  FIG.  22 D ) shows sample  22 D having a honeycomb structure  500  similar to that shown in  FIGS.  5 C and  5 D  with a geometry of 300 cells per square inch, cell wall thickness of 8 mils (“300/8 geometry”), and an inlet to outlet cross-sectional area ratio of 1.9:1. 
       FIGS.  22 A- 22 D  shows photographs of the plugged inlet ends (upper picture) and outlet ends (lower picture) of the representative parts of the four designs (A, B, C, D), respectively, wherein design  22 A is shown in  FIG.  22 A , design  22 B is shown in  FIG.  22 B , design  22 C (a presently disclosed design) is shown in  FIG.  22 C , design  22 D is shown in  FIG.  22 D . The plugged honeycomb parts of designs  22 A,  22 B,  22 C, and  22 D were tested for pressure drop as a function of soot load, results shown graphically in  FIG.  23   , on a test rig up-fitted with a diesel fired burner used for generating soot and a blower supplying air to control temperature and flow into the tested honeycomb parts. The parts were loaded with soot generated from the above mentioned test rig for determining soot loaded pressure drop at various levels up to about 5 g/l. 
     As seen in  FIG.  23   , the honeycomb structures of designs  22 C and  22 D as disclosed herein provided improved soot loaded pressure drop performance over, for example, design  22 B. The higher inlet to outlet cross-sectional area ratio of designs  22 C and  22 D would also result in a higher ash storage capacity over, for example, designs A and B and an end of life ash load of designs  22 C and  22 D would provide an advantage of soot loaded pressure drop over, for example, designs  22 A and  22 B. Note that high ash storage capacity of designs like  22 A and  22 B can result in higher pressure drop due to relatively small hydraulic diameter of the outlet channels. The embodiments disclosed herein can thus provide an improvement in pressure drop in soot and ash-loaded states over known designs. 
       FIG.  23    also illustrates a difference between design  22 C and design  22 D in that the slope for the soot load vs. pressure drop data for design  22 D is greater than that of designs  22 C and  22 A. A higher slope for soot load vs. pressure drop can provide an improved ability to estimate filter soot load estimation based on pressure drop, resulting in improved pressure drop based particulate filter diagnostics. Among the four designs tested, design  22 B offers the highest slope for soot load vs. pressure drop but lacks in ash storage capacity compared to designs  22 A,  22 C, and  22 D. Design  22 D offers a greater slope for soot load vs. pressure drop as compared to design C as a result of the lower cell density (300 cells per square inch vs. 400 cells per square inch) although both designs have honeycomb structure  500  similar to that shown in  FIGS.  5 C and  5 D . FIGS 
       FIG.  24    shows modeled pressure drop performance of four different filter (plugged honeycomb body) designs having diameter of 10.5 inches and length of 7.5 inches, cell density of 350 cells per square inch, honeycomb matrix wall thickness of 9.5 mils, wall porosity (average porosity) of 45% and wall median pore diameter of 14 micrometers (microns). Incorporated into the pressure drop modeling was a thin ash layer that was inferred to exist in the filter which acts to stop soot particles from entering into the filter wall with the result that soot is present only within the channels and there is no contribution to pressure drop from deep bed filtration. The pressure drop was modeled at soot loadings of 0 g/L and 6 g/L, and therebetween, for an exhaust flow rate of 1250 m 3 /hr and a gas temperature of 200 C. For a design  24 A for a filter body with a known honeycomb structure as shown in  FIGS.  1 A and  1 B , the pressure drop increased from 2.36 kPa at 0 g/L soot load to 7.15 kPa at 6 g/L corresponding to pressure drop vs. soot load slope of 0.798 kPa/(g/L of soot). For a design  24 B for a filter body with a honeycomb structure with an asymmetric design as shown in  FIGS.  2 A and  2 B  the pressure drop increased from 3.06 kPa at 0 g/L soot load to 6.67 kPa at 6 g/L corresponding to pressure drop vs. soot load slope of 0.601 kPa/(g/L of soot). For a design  24 C a filter body as disclosed herein with a honeycomb structure with an asymmetric design having outlet to inlet cross-sectional area ratio of 2:1 as shown in  FIGS.  5 A and  5 B  the pressure drop increased from 1.41 kPa at 0 g/L soot load to 5.44 kPa at 6 g/L corresponding to pressure drop vs. soot load slope of 0.671 kPa/(g/L of soot). For a design  24 D for a filter body with a honeycomb structure with an asymmetric design as shown in  FIGS.  10 A and  10 B  and having outlet to inlet cross-sectional area ratio of 3:1 the pressure drop increased from 1.69 kPa at 0 g/L soot load to 6.41 kPa at 6 g/L corresponding to pressure drop vs. soot load slope of 0.786 kPa/(g/L of soot). The pressure drop performance can be quantified by a defined parameter Ω which is the ratio of the clean pressure drop (corresponding to the case of soot loading of 0 g/L) of an asymmetric design filter as disclosed herein to the clean pressure drop of a symmetric filter having similar CPSI, wall thickness, diameter, length and wall microstructure, and parameter a as the ratio of the pressure drop vs. soot load slope of an asymmetric design filter as disclosed herein to the clean pressure drop of a symmetric filter having similar CPSI, wall thickness, diameter, length and wall microstructure. For the asymmetric design disclosed herein having outlet to inlet cross-sectional area ratio of 2:1, parameters Ω and σ were calculated to be 0.597 and 0.84 respectively. For the asymmetric design disclosed herein having outlet to inlet cross-sectional area ratio of 2:1, parameters Ω and σ were calculated to be 0.716 and 0.98 respectively. In some embodiments, the asymmetric design filters disclosed herein have Ω less than 0.85 and σ less than 1. In other embodiments, the asymmetric design filters disclosed herein have Ω less than 0.75 and σ less than 1. In still other embodiments, the asymmetric design filters disclosed herein have Ω less than 0.65 and σ less than 1. In yet other embodiments, the asymmetric design filters disclosed herein have Ω less than 0.65 and σ less than 0.9. In other embodiments, the asymmetric design filters disclosed herein have Ω less than 0.6 and σ less than 0.85.  FIG.  25    shows the modeled pressure drop performance of four different filter designs having diameter of 10.5 inches and length of 7.5 inches, (average) wall porosity of 55% and wall median pore diameter of 12 micrometers (microns). The pressure drop was modeled at soot loadings of 0 g/L and 6 g/L and therebetween for exhaust flow rate of 1250 m 3 /hr and gas temperature of 200 C. For a design  25 A for a filter body with known honeycomb structure as shown in  FIGS.  1 A and  1 B  and having cell density of 200 cells per square inch and honeycomb matrix wall thickness of 8 mils, the pressure drop increased from 1.197 kPa at 0 g/L soot load to 8.03 kPa at 6 g/L, corresponding to pressure drop vs. soot load slope of 1.139 kPa/(g/L of soot load).The bulk density for this example was about 296 g/L. For a design  25 B for a filter body with an asymmetric honeycomb structure as shown in  FIGS.  2 A and  2 B  and having cell density of 300 cells per square inch and wall thickness of 7 mils, the pressure drop increased from 2.1 kPa at 0 g/L soot load to 5.71 kPa at 6 g/L, corresponding to pressure drop vs. soot load slope of 0.6 kPa/(g/L of soot load).The bulk density for this example was about 297 g/L. For a design  25 C for a filter body with an asymmetric honeycomb structure as shown in  FIGS.  2 A and  2 B  and having cell density of 300 cells per square inch and wall thickness of 9 mils, the pressure drop increased from 2.43 kPa at 0 g/L soot load to 6.37 kPa at 6 g/L, corresponding to pressure drop vs. soot load slope of 0.66 kPa/(g/L of soot load).The bulk density for this example was about 361 g/L. For a design  25 D as disclosed herein for a filter body with an asymmetric honeycomb structure as shown in  FIGS.  5 A and  5 B  (having inlet to outlet cross-section area of 2:1) and having cell density of 350 cells per square inch and wall thickness of 9.5 mils, the pressure drop increased from 1.46 kPa at 0 g/L soot load to 5.58 kPa at 6 g/L, corresponding to pressure drop vs. soot load slope of 0.686 kPa/(g/L of soot load). The bulk density for this example was about 412 g/L. For a design  25 E as disclosed herein where the filter body has an asymmetric honeycomb structure as shown in  FIGS.  10 A and  10 B  (having outlet to inlet cross-section area of 3:1) and having cell density of 350 cells per square inch and wall thickness of 9.5 mils, the pressure drop increased from 1.76 kPa at 0 g/L soot load to 6.57 kPa at 6 g/L, corresponding to pressure drop vs. soot load slope of 0.801 kPa/(g/L of soot load).The bulk density for this example was about 412 g/L. In some embodiments, filters have higher bulk density and low clean and soot loaded pressure drop as higher bulk density can result in smaller temperature excursions during the regeneration of the filters. In some embodiments, the bulk density of filters disclosed herein can be 10% or more higher than that of a symmetric honeycomb filter (such as that shown in  FIGS.  1 A and  1 B ) while having similar clean filter pressure drop (corresponding to soot loading of 0 g/L). In other embodiments, the bulk density of filters disclosed herein can be 20% or more higher than that of a symmetric honeycomb filter having similar clean filter pressure drop. In still other embodiments, the bulk density of filters disclosed herein is can be 30% or more higher than that of a symmetric honeycomb filter having similar clean filter pressure drop. In yet other embodiments, the bulk density of filters disclosed herein can be 20% or more higher than that of a symmetric honeycomb filter having a similar clean filter pressure drop. 
     In addition to the pressure drop advantage, various embodiments disclosed herein lend themselves to having a much lower cost of extrusion die manufacturing, for example as compared to known asymmetric designs, such as comparing similar cell density. For example, in various embodiments disclosed herein, much of the honeycomb structure  400 - 1600  is comprised of ‘straight line walls’ so that large portions of the extrusion dies, which are used to form honeycomb structures disclosed herein by extrusion, can be cut with less expensive wire EDM or cutting wheel (e.g., abrasive wheel slitting) rather than relying primarily or exclusively on plunge EDM, such as with graphite electrodes. Therefore the use of techniques such as plunge EDM, which can be relatively expensive and time consuming, can be minimized. 
     For example, an embodiment of an extrusion die  1920  is disclosed in  FIGS.  21 A- 21 C . This example may be used to manufacture a green body by an extrusion process by extruding a batch mixture of inorganic and organic components and a liquid vehicle through the extrusion die  1920 . The green body structure can be subsequently dried and fired to produce a honeycomb structure. The extrusion die  1920  that is shown is used to manufacture the honeycomb structure  1100  of  FIG.  11   , however, the general die structure disclosed herein is easily made applicable to the other honeycomb structures  400 - 1000 ,  1200 - 1600  described herein. Referring again to  FIGS.  21 A- 21 C , the honeycomb structures may be formed by extrusion of an extrudable batch mixture, which is described, for example, in any one of U.S. Pat. Nos. 3,885,977, 5,332,703, 6,391,813, 7,017,278, 8,974,724, WO2014/046912, and WO2008/066765, through the extrusion die  1920  to produce a green body. In general, the green body comprises a substantially self-supporting structure formed from the extrudable mixture and is comprised of one or more ceramic-forming materials, or one or more ceramic materials, or both ceramic and ceramic-forming materials. The green body may then be dried and/or heated to dry, sinter, anneal, or otherwise fire the green body to form a structure comprising porous ceramic material. The green body may be dried such as described in U.S. Pat. Nos. 9,038,284, 9,335,093, 7,596,885, or 6,259,078, for example. The green body can be fired, such as described in any one of U.S. Pat. Nos. 9,452,578, 9,446,560, 9,005,517, 8,974,724, 6,541,407, and 6,221,308 to form the honeycomb structure  400 - 1600  to comprise the geometry described herein. The honeycomb extrusion die  1920  comprises a die body  1922  ( FIG.  21 B ), a die inlet face  1924  configured to receive the batch mixture, and a die outlet face  1926  opposite from the die inlet face  1924  configured to expel the plasticized batch in the form of a green body having a green honeycomb structure. The die body comprises a matrix of intersecting slots  1932  including a partial slot type  1932 P and possibly one or more other slot types. The matrix of intersecting slots  1932  defines a die repeat unit  1940 , i.e., a die structural unit that is repeated throughout the extrusion die  1920 . 
     The honeycomb extrusion die  1920  comprises a plurality of feedholes  1930  (a few labeled) extending from the die inlet face  1924  into the die body  1922 , and an intersecting with the matrix of intersecting slots  1932  (a few labeled) extending into the die body  1922  from the die outlet face  1926  and connecting with the plurality of feedholes  1930 . The die may be incorporated into an extruder system such as a ram extruder or screw extruder such as a twin screw extruder which accepts batch mixture material to be extruded. The batch mixture is forced through a plurality of feedholes  1930  and into the matrix of intersecting slots  1932 . The intersecting array of slots  1932  comprises at least slots of a partial slot type  1932 P that extend less than entirely across the die outlet face  1926 . In this embodiment, other slot types such as first slots  1932 X (a few labeled) may be provided that may extend fully across the die outlet face  1926  (e.g., vertically as shown), and second slots  1932 Y (orthogonal to the first slots  1932 X) that may also extend fully across the die outlet face  1926  (e.g., horizontally as shown). The partial slots  1932 P that do not extend fully across the die outlet face  1926 , and in particular form a T-intersection, for example with the first slots  1932 X, as shown. Together, the slots (e.g., slots  1932 X,  1932 Y, and/or  1932 P) correspond to the array of die repeat units  1940  that are repeated across at least some of the die outlet face  1926 . The die repeat units  1940  may be arranged, as shown for example, in a staggered side-by-side abutting relationship in the horizontal direction, and stacked one atop another in the vertical direction. A partial slot  1932 P of the die repeat unit  1940  intersects, with a T-intersection, with one of the other slots, such as first slot  1932 X. 
     The honeycomb extrusion die  1920  may comprise a skin-forming portion  1920  such as comprising a skin-forming mask  1950  (e.g., a mask ring) that interfaces with skin-forming feedholes  1930 S to form an extruded skin on the honeycomb structure or matrix of the extruded green body honeycomb during the extrusion method. The mask ring may have a circular inner periphery shape as shown, but other shapes corresponding to other outer perimeter shapes of the extrudate or green body are possible as provided herein. 
     Die repeat unit  1940  comprises four or more die pins made up of a first die pin type and a second die pin type. The first die pin type is larger in cross-sectional area (in a plane orthogonal to the axial direction or extrusion direction z) than the second die pin type and comprises a rectangular shape in cross-section. The first die pin type comprises two first sides of length Lo and two second sides of width Wo, wherein Lo&gt;Wo, and a partial slot  1932 P terminating with a T-intersection on at least one of the first sides of length Lo. The second die pin type comprises a side length Li parallel to length Lo that is less than half the length of the first side of length Lo. 
     In the depicted embodiment, each of the die repeat units  1940  comprises die pins P 1 -P 7 . Each of the first slots  1932 X and second slots  1932 Y forming a part of the die pins (P 1 -P 7 ) of the structure of the die repeat units  1940  may be formed, for example by wire EDM or by an abrasive cutting wheel process. The partial slots  1932 P may be formed by a plunge EDM process, and even in this case, the EDM electrode has a simple rectangular shape in cross-section. Thus, the overall cost of the extrusion die can be dramatically reduced. In some embodiments, the partial slots  1932 P may be started using an abrasive cutting wheel and then the ends of the partial slots at the T-intersection cleaned up using plunge EDM. 
     As can be seen in  FIG.  21 C , the die repeat units  1940  comprise a structure wherein some of the die pins of the second type (e.g., die pins P 1 -P 3  and P 5 -P 7 ) may comprise square shape in cross-section, and at least one die pin of the first type (e.g., die pin P 4 ) comprises a rectangular shape in cross-section. The at least one die pin (e.g., die pin P 4 ) comprising a rectangular shape in cross-section also comprises a relatively larger cross-sectional area than the pins of the second type (e.g., pins P 1 -P 3  and P 5 -P 7 ). In this depicted embodiment, the die repeat units  1940  have an outer peripheral shape that is rectangular. However, the die repeat units may comprise the outer perimeter shape of the repeating structural unit of the other honeycomb structures  400  and  600 - 1600  described herein. 
     Die repeat units  1940  comprises structure wherein the partial slot  1932 P intersects with another slot adjacent to a long side of the first type (e.g., die pin P 4  intersects with first slot  1932 X) to form a junction that is a T-intersection  1944  rather than a cross intersection  1946  as is present at the intersection of the first and second slots  1932 X and  1932 Y. 
     The extrusion die  1920  may comprise different feedhole patterns (feedholes  1930  shown as dotted circles in  FIG.  21 C ). For example, in a first embodiment the feedholes  1930  may be disposed at every intersection of the slots  1932 X,  1932 Y,  1932 S. Other feedhole designs for feedhole locations may be used. Thus, for each die design adapted to manufacture the honeycomb structures  400 - 1600  described herein, the die repeat units comprise a structure wherein the partial slot  1932 P intersects with one of the first slot  1932 X and/or the second slot  1932 Y at a T-intersection  1944  at the side of a pin of the first type (e.g., pin P 4 ) corresponding to an outlet cell  402 - 1602  in the honeycomb structure  400 - 1600  produced after drying and firing the extruded green body. As such, the partial slot  1932 P terminates at the relatively large pin configured to form an outlet cell, i.e., at the pin that comprises a relatively larger cross-sectional area as compared to at least some of the other pins. In some embodiments, the die repeat unit comprises a rectangular outer perimeter shape (See  FIGS.  5 A- 5 B ,  FIGS.  7 A- 7 B ,  FIGS.  8 A- 8 B ,  FIG.  10 A  through  FIG.  13 B , and  FIGS.  16 A- 16 B . In other embodiments, the outer periphery shape may include more than four sides. 
     In some embodiments disclosed herein, one or more strengthening features can be incorporated into the honeycomb structure. For example, as is shown in  FIGS.  22 A and  22 B , one or more cell walls within the matrix of interconnected porous walls can be provided with an increased cell wall thickness (shown via a thicker line) compared to nearby or surrounding cell walls. Thickness may be 20% greater or more, for example. In some of these embodiments, the thicker cell wall thickness, or web thickness, can be increased on those cell walls  2003 Y that extend continuously across a plurality of cells (e.g., in they direction), or even across the entire honeycomb structure of the honeycomb body. In other embodiments, the wall thickness of walls  2003 X that extend in the x direction can be made thicker, such as shown in  FIG.  22 C . Combinations of thicker walls in the x and y directions can be provided in some embodiments (See  FIG.  22 C ), especially for those walls that extend across multiple repeating structural units  2000 U′, and even across the entire the honeycomb structure  2000 U′ of the honeycomb body. 
       FIG.  23    illustrates a flow chart of method of manufacturing a honeycomb structure (e.g., honeycomb structures  400 - 1600  and  2000 ,  2000 ′). The method  2100  comprises extruding ( 2102 ) a ceramic or ceramic-forming batch mixture through an extrusion die to form a self-standing green ware, drying ( 2104 ) the green ware, and firing ( 2106 ) the green ware to form a porous ceramic article. The extrusion die (e.g., extrusion die  1920 ) can comprise an outlet face (e.g., outlet face  1926 ) of a die body (die body  1922 ) comprising a matrix of pins defining intersecting slots (e.g., slots  1932 ) including a partial slot type (e.g., slot type  1932 P, the matrix defining a die repeat unit (e.g., die repeat unit  1940 ), wherein the partial slot type extends less than entirely across the outlet face. For example, the die repeat unit can include four or more die pins (e.g., die pins P 1 -P 7 ) made up of a first die pin type and a second die pin type. The first die pin type is larger in cross-sectional area than the second die pin type and includes a rectangular shape in cross-section with two first sides of length Lo and two second sides of width Wo, wherein Lo&gt;Wo, and includes a slot of the partial slot type terminating with a T-intersection (e.g., T-intersection  1944 ) on at least one of the first sides. 
     The second die pin type can comprise a side length Li that is less than half the length of the first side of length Lo. The second die pin type (e.g., P 1 -P 3  and P 5 -P 7 ) may be square in cross-section in a plane orthogonal to the extrusion direction. The method  2100  can comprise extruding a batch mixture through such matrix of intersecting slots to form a green body. The green body comprises a honeycomb structure which is largely the geometrical structure of the final fired honeycomb article, although typically the green ware is subject to shrinkage upon firing into the final ceramic article. Thus, in some embodiments, 
     The method  2100  comprises firing the green body to form a ceramic bodies comprising a honeycomb structure (e.g., honeycomb structures  400 - 1600  and  2000 ,  2000 ′). The honeycomb structure comprises a matrix of intersecting porous cell walls (e.g., cell walls  403 - 2003 ,  2003 ′) extending axially between an inlet end and an outlet end of the honeycomb structure, the matrix defining a plurality of inlet cells (e.g., inlet cells  401 - 2001 ) and outlet cells (e.g., outlet cells  402 - 2002 ), and corresponding inlet channels and outlet channels defined by respective inlet cells and respective outlet cells, wherein at least a portion of the outlet channels are larger in cross-sectional area than any of the inlet channels, and at least some of the outlet channels comprise a rectangular shape in cross-section. 
     It will be apparent to those skilled in the art that various modifications and variations can be made to the embodiments described herein without departing from the scope of the claimed subject matter. Thus it is intended that the specification cover the modifications and variations of the various embodiments described herein provided such modification and variations come within the scope of the appended claims and their equivalents.