Patent Publication Number: US-2018029030-A1

Title: Honeycomb ceramic substrates, honeycomb extrusion dies, and methods of making honeycomb ceramic substrates

Description:
This is a divisional application of U.S. application Ser. No. 14/033,883 filed on Sep. 23, 2013, the content of which is relied upon and incorporated herein by reference in its entirety. 
    
    
     FIELD 
     The following description relates generally to substrates, extrusion dies for making substrates and methods of making substrates and, more particularly, to honeycomb ceramic substrates, honeycomb extrusion dies and methods of making honeycomb ceramic substrates. 
     BACKGROUND 
     In the automotive industry, honeycomb ceramic substrates are often employed to support a catalyst to reduce harmful emissions from a combustion engine. Typically, such ceramic substrates include a lattice of walls defining flow channels including a rectangular (e.g., square) or other cross sectional shape. 
     SUMMARY 
     In the examples described herein, a honeycomb ceramic substrate may be created by extrusion from a batch of ceramic or ceramic-forming material using a die with die pins arranged in a shape designed to optimize the flow channel structure of the resultant substrate with respect to a variety of attributes, such as, for example, open frontal area, geometric surface area, and strength. The optimized flow channel structure may have channel walls with concave inner surfaces and concave corner portions, thereby providing a flow channel structure that is, for example, elliptical in nature. This optimized flow channel structure may result in improved product performance in substantially all areas of measurement, including coating efficiency by minimizing corner coating build-up. 
     In a first example aspect, a honeycomb ceramic substrate includes a lattice of intersecting walls defining a honeycomb network of flow channels extending between an inlet end and an outlet end of the honeycomb substrate. Each flow channel is defined by a plurality of channel walls of the intersecting walls with at least two of the plurality of channel walls including concave inner surfaces facing a center of the corresponding flow channel from central portions of the concave inner surfaces to concave corner portions facing the center of the corresponding flow channel. The concave corner portions are where each of the plurality of channel walls intersects with another one of the plurality of channel walls. 
     In one example of the first aspect, a peripheral cross-sectional shape of at least one of the flow channels is substantially defined by the equation: 
     
       
         
           
             
               
                 
                   
                      
                     
                       x 
                       a 
                     
                      
                   
                   n 
                 
                 + 
                 
                   
                      
                     
                       y 
                       b 
                     
                      
                   
                   m 
                 
               
               = 
               1 
             
             , 
           
         
       
     
     wherein a and b are rectangular-fitted half-lengths along an x direction and a y direction, respectively, of the inner surfaces of channel walls defining each flow channel on either side of a y axis and an x axis, respectively, 
     wherein x and y represent coordinates (x, y) of the inner surfaces of the channel walls defining each flow channel in the x direction and the y direction, respectively, 
     wherein −a≦x≦a, 
     wherein −b≦y≦b, and 
     wherein n and m are exponents defining a degree of curvature of the channel walls. In one example, at least one of n and m are in a range of from about 2.5 to about 10. In another example, a and b are independently in a range of from about 330 microns to about 1.829 mm. In still another example, n and m are varied across the plurality of flow channels. 
     In another example of the first aspect, the channel walls are continuously curving around the center of the corresponding flow channel. 
     The first aspect may be provided alone or in combination with any one or more of the examples of the first aspect discussed above. 
     In a second example aspect, a method of making a honeycomb ceramic substrate is provided. The method includes extruding a ceramic or ceramic-forming batch material through a honeycomb extrusion die to form green honeycomb substrate including a lattice of intersecting walls defining a honeycomb network of flow channels extending between an inlet end and an outlet end of the green honeycomb substrate. Each flow channel is defined by a plurality of channel walls of the intersecting walls with at least two of the plurality of channel walls including concave inner surfaces facing a center of the corresponding flow channel from central portions of the concave inner surfaces to concave corner portions facing the center of the corresponding flow channel. The concave corner portions are where each of the plurality of channel walls intersects with another one of the plurality of channel walls. The method further includes drying the green honeycomb substrate, and firing the green honeycomb substrate into the honeycomb ceramic substrate. 
     In one example of the second aspect, a peripheral cross-sectional shape of at least one of the flow channels is substantially defined by the equation: 
     
       
         
           
             
               
                 
                   
                      
                     
                       x 
                       a 
                     
                      
                   
                   n 
                 
                 + 
                 
                   
                      
                     
                       y 
                       b 
                     
                      
                   
                   m 
                 
               
               = 
               1 
             
             , 
           
         
       
     
     wherein a and b are rectangular-fitted half-lengths along an x direction and a y direction, respectively, of the inner surfaces of channel walls defining each flow channel on either side of a y axis and an x axis, respectively, 
     wherein x and y represent coordinates (x, y) of the inner surfaces of the channel walls defining each flow channel in the x direction and the y direction, respectively, 
     wherein −a≦x≦a, 
     wherein −b≦y≦b, and 
     wherein n and m are exponents defining a degree of curvature of the channel walls. In one example, at least one of n and m is in a range of from about 2.5 to about 10. In another example, a and b are independently in a range of from about 330 microns to about 1.829 mm. In another example, n and m are varied across the plurality of the flow channels. 
     In another example of the second aspect, the channel walls are continuously curved around the center of the corresponding flow channel. 
     The second aspect may be provided alone or in combination with any one or more of the examples of the second aspect discussed above. 
     In a third example aspect, a honeycomb extrusion die configured to extrude a honeycomb ceramic substrate from a batch of ceramic or ceramic-forming material is provided. The honeycomb extrusion die includes a plurality of die pins arranged in a matrix and spaced from one another to define a lattice of intersecting slots defined between the die pins at an outer face of the die pins. An outer periphery at the outer face of at least one of the die pins includes a plurality of sides joined by corresponding corner portions with at least two convex sides facing away from a center of the corresponding die pin from central portions of the convex sides to the corresponding corner portions of the convex sides. At least one corner portion is convex facing away from the center of the corresponding die pin. 
     In one example of the third aspect, at least one wall slot is defined between facing sides of two adjacent die pins of the plurality of die pins. Each of the facing sides are convex facing each other from central portions thereof to corresponding corner portions of the facing sides. The wall slot is concave toward central portions of the two adjacent die pins. 
     In another example of the third aspect, a shape of the outer periphery of the outer face of the at least one of the die pins is substantially defined by the equation: 
     
       
         
           
             
               
                 
                   
                      
                     
                       x 
                       a 
                     
                      
                   
                   n 
                 
                 + 
                 
                   
                      
                     
                       y 
                       b 
                     
                      
                   
                   m 
                 
               
               = 
               1 
             
             , 
           
         
       
     
     wherein a and b are rectangular-fitted half-lengths along an x direction and a y direction, respectively, of the sides of the die pins on either side of a y axis and an x axis, respectively, 
     wherein x and y represent coordinates (x, y) of the sides of the die pins in the x direction and the y direction, respectively, 
     wherein −a≦x≦a, 
     wherein −b≦y≦b, and 
     wherein n and m are exponents defining a degree of curvature of the sides of the die pins. In one example, at least one of n and m is in a range of from about 2.5 to about 10. In another example, a and b are independently in a range of from about 330 microns to about 1.829 mm. In another example, n and m are varied across the plurality of the die pins. 
     In a further example of the third aspect, the sides of each of the die pins are symmetric to each other. 
     In still a further example of the third aspect, the sides of each of the die pins are continuously curved around the center of the corresponding die pin. 
     The third aspect may be provided alone or in combination with any one or more of the examples of the third aspect discussed above. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other features, aspects and advantages of the present disclosure are better understood when the following detailed description of the disclosure is read with reference to the accompanying drawings, in which: 
         FIG. 1  is a perspective view illustrating an example of a honeycomb ceramic substrate in accordance with example aspects of the disclosure; 
         FIG. 2  is a schematic sectional view illustrating an example of the honeycomb ceramic substrate in accordance with example aspects of the disclosure along line  2 - 2  of  FIG. 1 ; 
         FIG. 3  is an enlarged view illustrating an example of the honeycomb ceramic substrate in accordance with example aspects of the disclosure taken at view  3  of  FIG. 2 ; 
         FIG. 4  is an enlarged view of  FIG. 3 ; 
         FIG. 5  is an enlarged view of  FIG. 4 ; 
         FIGS. 6-12  are enlarged views similar to  FIG. 3  but illustrating various alternative example honeycomb ceramic substrate configurations; 
         FIG. 13  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a change in inertia of the flow channel in accordance with example aspects of the disclosure; 
         FIG. 14  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a resistance of the flow channel to chipping as evidenced by an effective additional web attachment length in accordance with example aspects of the disclosure; 
         FIG. 15  is a enlarged view illustrating an example of the honeycomb ceramic substrate in accordance with example aspects with respect to effective additional channel wall thickness; 
         FIG. 16  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a percentage of reduction in an open frontal area of the flow channel in accordance with example aspects of the disclosure; 
         FIG. 17  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on an effective corner radius for washcoat efficiency of the flow channel in accordance with example aspects of the disclosure; 
         FIG. 18  is a flow diagram illustrating an example of a method of making a honeycomb ceramic substrate in accordance with example aspects of the disclosure; 
         FIG. 19  is a schematic view illustrating an example of an extrusion apparatus in accordance with example aspects of the disclosure; 
         FIG. 20  is an enlarged partial schematic sectional view illustrating an example of a die member in accordance with example aspects of the disclosure taken at view  20  of  FIG. 19 ; 
         FIG. 21  is a partial sectional view of the die member along line  21 - 21  of  FIG. 20 ; 
         FIG. 22  is an enlarged view of portions of  FIG. 21 ; and 
         FIGS. 23-25  are enlarged views illustrating examples of the honeycomb ceramic substrate in accordance with example aspects of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The disclosure will now be described more fully hereinafter with reference to the accompanying drawings in which example embodiments are shown. Whenever possible, the same reference numerals are used throughout the drawings to refer to the same or like parts. However, the disclosure may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. These example embodiments are provided so that this disclosure will be both thorough and complete. 
       FIG. 1  is a perspective view illustrating an example of a honeycomb ceramic substrate  102 . The honeycomb ceramic substrate  102  is not necessarily drawn to scale and illustrated only one example schematic representation of a honeycomb ceramic substrate  102 . The honeycomb ceramic substrate  102  includes an inlet end  104  and an outlet end  106  positioned opposite from the inlet end  104 . A lattice of intersecting walls defining a honeycomb network of flow channels  108  extend between the inlet end  104  and outlet end  106 . In one example, substantially all of the flow channels  108  are not plugged and therefore provide for an unobstructed pass flow from the inlet end  104  to the outlet end  106  of the honeycomb ceramic substrate  102 . 
       FIG. 2  is a schematic sectional view illustrating an example of the honeycomb ceramic substrate  102  in accordance with example aspects of the disclosure along line  2 - 2  of  FIG. 1 . As shown in  FIG. 2 , the flow channels  108  can be formed by a plurality of channel walls  110  of the intersecting walls extending longitudinally between the inlet end  104  and outlet end  106  of the honeycomb ceramic substrate  102 . The flow channels  108  and the channel walls  110  can each extend in a substantially parallel orientation longitudinally between the inlet end  104  and the outlet end  106 . As further illustrated, the honeycomb ceramic substrate  102  may include an outer skin defining an outer surface  112  that can extend longitudinally between the inlet end  104  and outlet end  106 . As shown, the outer surface  112  can comprise a circular cylindrical shape having a circular cross-sectional profile. In further examples, the outer surface  112  may have an elliptical, polygonal or other shape. For example, although not shown, the outer surface  112  may have a polygonal shape such as triangular, rectangular (e.g., square) or other polygonal shape. Moreover, as shown, the honeycomb ceramic substrate  102  can comprise a single monolithic substrate although the substrate may comprise a segmented substrate wherein many substrates are mounted parallel to one another to provide the desired overall cross sectional configuration. Whether a single monolithic or segmented substrate, various geometries may be incorporated in accordance with aspects of the disclosure. For example, the substrates may comprise a rectangular (e.g., square) cross-sectional outer periphery or other polygonal shape having three or more sides. In further examples, the substrates may have an outer cross-sectional periphery that is circular, oval, or other curved shape. 
     The honeycomb ceramic substrate  102  can have a variety of cell densities, such that a larger or smaller number of flow channels  108  can be provided per unit area. For instance, the channel density can be in the range of from about 7.75 channels/cm 2  (50 channels/in 2 ) to about 232.5 channels/cm 2  (1500 channels/in 2 ) of the honeycomb ceramic substrate  102  cross-section. As such, the examples shown in  FIGS. 1 and 2  are not intended to be limiting, as various ranges of cell densities may be provided in accordance with aspects of the disclosure. 
     In further examples, the channel wall constructions forming the flow channels  108  can have different configurations.  FIG. 3  is an enlarged view illustrating an example of the honeycomb ceramic substrate  102  in accordance with example aspects of the disclosure taken at view  3  of  FIG. 2 . For illustration purposes,  FIG. 3  shows a grouping of nine flow channels  108 .  FIG. 4  is an enlarged view of  FIG. 3  illustrating an example of a flow channel  108  of the honeycomb ceramic substrate  102  in accordance with example aspects of the disclosure.  FIG. 5  is an enlarged view of  FIG. 4 , illustrating demonstrating further features of the example flow channel  108 .  FIGS. 6-12  are enlarged views of alternative honeycomb ceramic substrates  102   a - g , respectively, that are similar to the honeycomb ceramic substrate  102  of  FIGS. 3-5  but illustrating various alternative flow channel  108   a - g  configurations in accordance with aspects of the disclosure. In each of the example honeycomb ceramic substrates, for example as illustrated in  FIGS. 1-12 , the flow channel structure may have channel walls with concave inner surfaces and concave corner portions. Indeed, the channel walls including concave inner surfaces and concave corner portions are noticeably shown in  FIGS. 1-6 and 10 , and would also be more noticeably illustrated in enlarged views of the channel walls and corner portions of the flow channel structure illustrated in  FIGS. 7-9, 11 and 12 . This optimized flow channel structure may result in improved product performance in substantially all areas of measurement, including coating efficiency by minimizing corner coating build-up. 
     In the examples shown in  FIGS. 3-12 , each flow channel  108 ,  108   a - g  is defined by a plurality of intersecting channel walls  110  with at least two of the plurality of channel walls  110  including concave inner surfaces  116  facing a center  125  of the corresponding flow channel  108  from central portions  120  of the concave inner surfaces  116  to concave corner portions  118  facing the center  125  of the corresponding flow channel  108 . The concave corner portions  118  are where each of the plurality of channel walls  110  intersects with another one of the plurality of channel walls  110 . The arrangement of flow channels  108  illustrated in  FIGS. 3-6 and 10  are generally elliptical with concave channel walls  110  and the concave corner portions  118  continuously curving around the center  125  of the corresponding flow channel  108 . Further examples, e.g., as illustrated in  FIGS. 7-9, 11 and 12 , may have flow channels  108  with a square-like or rectangle-like configuration with concave channel walls  110  and concave corner portions  118  continuously curving around the center  125  of the corresponding flow channel  108 . 
     As mentioned above, the flow channels  108  of the honeycomb ceramic substrate  102  may be a superellipse, also known as a Lamé curve, generally elliptical or even square-like or rectangular-like with concave channel walls and concave corner portions in order to obtain an optimized flow channel shape for an open frontal area (OFA) and a geometric surface area (GSA) of the flow channels  108 . By controlling the length of the channel walls  110  and a degree of curvature of the channel walls  110  and corner portions  118 , various desired flow channel shapes may be obtained. For example, a peripheral cross-sectional shape of at least one of the flow channels  108  may be substantially defined by Equation (I). 
     
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           x 
                           a 
                         
                          
                       
                       n 
                     
                     + 
                     
                       
                          
                         
                           y 
                           b 
                         
                          
                       
                       m 
                     
                   
                   = 
                   1 
                 
               
               
                 
                   ( 
                   I 
                   ) 
                 
               
             
           
         
       
     
     As is illustrated in  FIGS. 4 and 5 , a and b are rectangular-fitted half-lengths along an x direction and a y direction, respectively, of the inner surfaces  116  of channel walls  110  defining each flow channel  108  on either side of a y axis and an x axis, respectively. In other words, a and b represent the half-lengths along an x and y direction, respectively, if the channel walls  110  were truly straight to provide a square or rectangular flow channel  108 . The dimensions a and b serve to define the density of flow channels  108  and the channel wall  110  thickness between the flow channels  108  in the honeycomb ceramic substrate  102 . The references x and y represent coordinates (x, y) of the inner surfaces  116  of the channel walls  110  defining each flow channel  108  in the x direction and the y direction, respectively. Further, −a≦x≦a and −b≦y≦b. Moreover, n and m are exponents defining a degree of curvature of the channel walls  110 . 
     In an example, at least one of n and m may be in a range of from about 2.5 to about 10. In another example, a and b may be independently in a range of from about 330 microns (0.013 inches) to about 1.829 mm (0.072 inches). In a further example, a thickness of the plurality of channel walls  110  between adjacent ones of the flow channels  108  may be in a range of from about 25.4 microns (0.001 inches) to about 482.6 microns (0.019 inches). 
     In yet another example, the channel walls  110  in each flow channel  108  may have substantially identical lengths. In further examples, at least two of the channel walls  110  in each flow channel  108  may have a length that is the same. Additionally, the channel walls  110  of each flow channel  108  may be symmetric to each other. 
     Equation (I) results in a flow channel  108  having gently curved inner surfaces  116  with more pronounced curved corners  118 . Even with corners  118  that have a more pronounced curvature than the gently curved inner surfaces  116 , each of the channel walls  110  of the flow channels  108  are continuously curved toward a center  125  of the corresponding flow channel  108  throughout the length of the channel wall between corresponding concave corner portions. In further examples, the entire inner surface of the flow channel is continuously concave about the entire periphery of the flow channel, wherein the inner surface is defined by gently concave channel walls  110  and more pronounced concave corner portions  118  that seemlessly transition with one another about the inner periphery of the inner surface of the flow channels. 
     The exponents n and m of Equation (I) are tuned, depending on the flow channel density and a thickness of the channel walls  110  between adjacent flow channels, to meet the above-referenced attributes in order to minimize coating buildup in the corner portions  118  and maximize open frontal area. The examples illustrated in  FIGS. 6-12  demonstrate the power that an adjustment of the exponents n and m may implement on a curvature degree of the flow channels  108  and a concavity of the inner surfaces  116 . More particular,  FIGS. 6-12  illustrate that, according Equation (I), the concavity of the inner surfaces  116  may be maintained at the same time a curvature degree of the flow channels  108   a - g  is decreased. 
     For example,  FIGS. 6-9  are representative of honeycomb ceramic substrates  102   a - g  having a flow channel density of 69.75 channels/cm 2  (450 channels/in 2 ) and a thickness of the plurality of channel walls  110  between adjacent ones of the flow channels  108  of 635 microns (0.025 inches). However,  FIGS. 6-9  each represent honeycomb ceramic substrates  102  in which exponents n and m have been tuned.  FIG. 6  represents a honeycomb ceramic substrate  102   a  with n and m equaling 3.6.  FIG. 7  represents a honeycomb ceramic substrate  102   b  with n and m equaling 4.0.  FIG. 8  represents a honeycomb ceramic substrate  102   c  with n and m equaling 10.  FIG. 9  represents a honeycomb ceramic substrate  102   d  with n and m equaling 50. As can be seen by a comparison of  FIGS. 6-9 , both a degree to which a flow channel  108  is curved and a concavity of the inner surfaces  116  can be changed by varying the exponents n and m. 
       FIGS. 10 and 11  are representative of honeycomb ceramic substrates  102   e - f  having a flow channel density of 93 channels/cm 2  (600 channels/in 2 ) and a thickness of the plurality of channel walls  110  between adjacent ones of the flow channels  108  of 571.5 microns (0.0225 inches). However,  FIG. 10  represents a honeycomb ceramic substrate  102   e  with n and m equaling 3.6, while  FIG. 11  represents a honeycomb ceramic substrate  102   f  with n and m equaling 42. 
       FIG. 12  represents a honeycomb ceramic substrate  102  having a flow channel density of 62.78 channels/cm 2  (405 channels/in 2 ) and a thickness of the plurality of channel walls  110  between adjacent ones of the flow channels  108  of 1.0668 mm (0.042 inches). However,  FIG. 12  represents a honeycomb ceramic substrate  102  with n equaling 4 and m equaling 8, thereby producing flow channels  108  with a rectangular-like shape, rather than a square-like shape. 
     The concavity of the inner surfaces  116  may compensate for some of the open area of the flow channel  108  that is lost due to the curved flow channel shape. Flow channel density and the thickness of the channel walls  110  between adjacent flow channels  108  may additionally be selected to optimize flow channel shape for a desired GSA. Further, the concavity of the inner surfaces  116  may provide the flow channels  108  with increased strength and resistance to buckling and rotational failures by increasing a moment of inertia, or decreasing a slenderness ratio, of each of the channel walls  110 . This increased strength may afford reduced thicknesses of the channel walls  110  between adjacent flow channels  108 , thereby further optimizing open frontal area and reduction of weight of the honeycomb ceramic substrate  102 . 
     In an example, Table 1 compares square flow channels in a honeycomb ceramic substrate having a channel density of 93 channels/cm 2  (600 channels/in 2 ) with elliptical flow channels in a honeycomb ceramic substrate having n and m equal to 3.5 with a channel density of 93 channels/cm 2  (600 channels/in 2 ). 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 Flow Channel Density 
                 93 
                 channels/cm 2   
                 93 
                 channels/cm 2   
               
            
           
           
               
               
               
            
               
                   
                 (600 channels/in 2 ) 
                 (600 channels/in 2 ) 
               
               
                   
                 Square 
                 n and m = 3.5 
               
            
           
           
               
               
               
               
               
            
               
                 Channel Wall Thickness 
                 76.2 
                 microns 
                 76.2 
                 microns 
               
            
           
           
               
               
               
            
               
                 Between Flow Channels 
                 (0.003 inches) 
                 (0.003 inches) 
               
            
           
           
               
               
               
               
               
            
               
                 Equivalent Fillet Radius 
                 76.2 
                 microns 
                 254 
                 microns 
               
            
           
           
               
               
               
            
               
                 of Channel Walls 
                 (0.003 inches) 
                 (0.01 inches) 
               
               
                 MIF (uncoated) 
                 0.69 
                 1.24 
               
               
                   
               
            
           
         
       
     
     While an equivalent fillet radius of the channel walls is greater in the elliptical flow channel than in the square flow channel, a mechanical integrity factor (MIF) of the elliptical flow channel is greater than an MIF of the square flow channel. The MIF, represented by Equation (II) below, is a dimensionless structural property that is directly proportional to a load carrying capability parallel to the channel walls and along a diagonal of the flow channel, where “t” is the channel wall thickness between flow channels, “l” is the distance between channel wall centers, and “R” is the effective corner radius, as is illustrated in  FIGS. 4 and 5 . The MIF is derived by equating a maximum bending stress at a midpoint of the channel walls or at an intersection of the channel walls to channel wall strength. 
       MIF*100 =t 1*( t /( l−t− 2 *R ))*100   (II)
 
     Further, the equivalent fillet radius of the channel walls is defined by the group of equations listed below in Table 2, where “a”, “b”, and “c” represent the three sides of a triangle, “A” is the area of the triangle, and “r” is the equivalent fillet radius of the channel walls. 
     Table 2 
         a=sqrt (( x 1 −x 2) 2 +( y 1 −y 2) 2 ) 
         b=sqrt (( x 2 −x 3) 2 +( y 2 −y 3) 2 ) 
         a=sqrt (( x 3 −x 1) 2 +( y 3 −y 1) 2 ) 
         s =( a+b+c )/2 
         A=sqrt ( s *( s−a )*( s−b )*( s−c )) 
         r=a*b*c /(4 *A ) 
     Table 3 compares other strength advantages of elliptical flow channels over square flow channels based on other product variants. For example, a slenderness ratio is used to compare the strength of the channel walls. Smaller slenderness ratios are more resistant to failures at equivalent loadings. Additionally to be noted is the fact that the channel wall thickness between the elliptical flow channels is less than the channel wall thickness between the square flow channels. When the square and elliptical flow channels having equivalent flow channel densities are compared, the elliptical flow channel having a thinner channel wall has a lower slenderness ratio than the comparable square flow channel having a thicker channel wall. In other words, the elliptical flow channel designs have the lower slenderness ratios, which indicates a higher resistance to buckling loading. Buckling failure is often seen in the extrusion of substrates having thin channel walls when adjacent channel wall velocities are not uniform. Buckling failure is one of the principle reasons for rejected extruded ware. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                 Flow Channel 
                 93     
                 93     
                 69.75   
                 62     
                 62     
                 139.5    
                 139.5    
               
               
                 Density - 
                 (600)     
                 (600)     
                 (450)     
                 (400)     
                 (400)     
                 (900)     
                 (900)     
               
               
                 channels/cm 2   
               
               
                 (channels/in 2 ) 
               
               
                 Channel Wall 
                 88.9   
                 57.15   
                 57.15   
                 88.9   
                 106.68   
                 68.58   
                 57.15   
               
               
                 Thickness 
                 (0.0035) 
                  (0.00225) 
                  (0.00225) 
                 (0.0035) 
                 (0.0042) 
                 (0.0027) 
                  (0.00225) 
               
               
                 Between Flow 
               
               
                 Channels - 
               
               
                 microns 
               
               
                 (inches) 
               
               
                 Flow Channel 
                 Square 
                 Elliptical 
                 Elliptical 
                 Square 
                 Square 
                 Square 
                 Elliptical 
               
               
                 Shape 
               
               
                 Distance 
                 1.036  
                 1.036  
                 1.197  
                 1.270  
                 1.270  
                 0.847  
                 0.847  
               
               
                 Between 
                 (0.0408) 
                 (0.0408) 
                  (0.04714) 
                 (0.05)  
                 (0.05)  
                  (0.03333) 
                  (0.03333) 
               
               
                 Channel Wall 
               
               
                 Centers - mm 
               
               
                 (inches) 
               
               
                 Radius of 
                 269.2    
                 342.9    
                 403.9    
                 315.0    
                 327.7    
                 215.9    
                 241.3    
               
               
                 Gyration X - 
                 (0.0106) 
                 (0.0135) 
                 (0.0159) 
                 (0.0124) 
                 (0.0129) 
                 (0.0085) 
                 (0.0095) 
               
               
                 microns 
               
               
                 (inches) 
               
               
                 Slenderness 
                 1.92  
                 1.51  
                 1.48  
                 2.02  
                 1.94  
                 1.96  
                 1.75  
               
               
                 Ratio 
               
               
                   
               
            
           
         
       
     
     The Slenderness Ratio is the quotient of the distance between the channel wall centers, the radius of gyration around the x-axis, and 2. The Slenderness Ratio is applied to determine the critical buckling stress 6 between the channel wall centers, as is shown in Equation (III) below. 
       σ=π 2   E /(Slenderness Ratio) 2    (III)
 
     where E is the Young&#39;s modulus of the channel wall material. 
       FIG. 13  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a change in inertia of the flow channel in accordance with example aspects of the disclosure. As shown  FIG. 13 , the vertical axis is the percent change in inertia while the horizontal axis demonstrates the exponents m and n (that are the same). Change in inertia is a measurement used to determine resistance of a flow channel to buckling of the channel walls. For example, when a flow channel has a higher percentage of change in inertia, it is more likely to be a stronger flow channel and have channel walls that are less subject to buckling. It is noted an elliptical flow channel having a lower value of m and n has a greater percentage of change in inertia. 
       FIG. 14  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a resistance of the flow channel to chipping as evidenced by an effective additional channel wall thickness attachment length in accordance with example aspects of the disclosure. As shown in  FIG. 14 , the vertical axis is the effective additional channel wall thickness attachment length (in inches) and the horizontal axis demonstrates the exponents m and n (that are the same). Example attachment lengths are schematically illustrated in  FIG. 15 . For example, when the channel wall thickness has an effective additional attachment length L 2  of 127 microns (0.005 inches) or greater, a clear resistance to chipping is shown when compared to smaller attachment lengths L 1 . It is noted that an elliptical flow channel having a lower value of m and n has a greater effective additional attachment length. 
       FIG. 16  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on a percentage of reduction in an OFA of the flow channel in accordance with example aspects of the disclosure. As shown in  FIG. 16 , the vertical axis is the percent difference in open area while the horizontal axis demonstrates the exponents m and n (that are the same). While lower n and m values promote greater strength in flow channels than higher n and m values,  FIG. 16  demonstrates that higher n and m values provide a greater flow channel OFA, which relates to the catalyst performance of a flow channel. 
       FIG. 17  is a graphical view illustrating an example of an impact of exponents defining a degree of curvature of a flow channel of the honeycomb ceramic substrate on an effective corner radius for washcoat efficiency of the flow channel in accordance with example aspects of the disclosure. As shown in  FIG. 17 , the vertical axis is the effective corner radius (inches) while the horizontal axis demonstrates the exponents m and n (that are the same). For example, measurements of coated substrates show a minimum of a 254 micron (0.010 inch) washcoat radius after coating. It is noted that an elliptical flow channel having a lower value of m and n has a greater uniformity of washcoat coating. 
     In addition, Table 4 highlights several performance attributes of an elliptical flow channel having a channel density of 69.75 channels/cm 2  (450 channels/in 2 ), a channel wall thickness between flow channels of 63.5 microns (0.0025 inches), and m and n values equal to 3.6 in comparison with alternative hexagonal and square flow channels. Of note, the elliptical flow channel substantially matches the hexagonal flow channel&#39;s exhaust performance characteristics while providing close to 40% increased strength over the hexagonal flow channel. 
     
       
         
           
               
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                   
                   
                 Elliptical, n 
               
               
                 Flow Channel Shape 
                 Hexagonal 
                 Square 
                 and m = 3.6 
               
               
                   
               
             
            
               
                 Flow Channel 
                  93 
                  93 
                    69.75 
               
               
                 Density - 
                 (600) 
                 (600) 
                 (450) 
               
               
                 channels/cm 2   
               
               
                 (channels/in 2 ) 
               
               
                 Channel Wall 
                   76.2 
                   76.2 
                   63.5 
               
               
                 Thickness Between 
                     (0.003) 
                     (0.003) 
                     (0.0025) 
               
               
                 Flow Channels - 
               
               
                 microns (inches) 
               
               
                 Equivalent Fillet 
                   76.2 
                   76.2 
                   304.8 
               
               
                 Radius of Channel 
                     (0.003) 
                     (0.003) 
                     (0.012) 
               
               
                 Walls - microns 
               
               
                 (inches) 
               
               
                 OFA (coated) 
                    0.84 
                    0.78 
                    0.84 
               
               
                 GSA (coated) 
                   31.6 
                   35.3 
                   31.3 
               
               
                 MIF (uncoated) 
                    0.46 
                    0.69 
                    0.64 
               
               
                 Resistance to Flow 
                 491 
                 709 
                 452 
               
               
                 (RTF) (coated) 
               
               
                   
               
            
           
         
       
     
     The OFA, the GSA, and RTF are calculated as is provided in Equations (IV), (V), and (VI) below, respectively. The OFA is used to compare substrates for back pressure. The GSA is used to compare substrates for conversion efficiency. For example, a higher GSA translates into a higher conversion efficiency or capability for the substrate. The RTF is a measure of the resistance to flow through the channels. 
       OFA=(1 −t/L ) 2 −(4−π)( R/L ) 2    (IV)
         where “t” is the channel wall thickness between flow channels, “L” is the distance between channel wall centers, and “R” is the effective corner radius, as is illustrated in  FIGS. 4 and 5 .       

       GSA=(4( L−t )/ L   2 )−(((8−2π) R )/ L   2 )   (V)
         where “t” is the channel wall thickness between flow channels, “L” is the distance between channel wall centers, and “R” is the effective corner radius, as is illustrated in  FIGS. 4 and 5 .       

       RTF=2 f /((OFA* Dh   2 ) w )   (VI)
         where “f” is the fanning friction factor, “Dh” is the hydraulic diameter of the flow channel, and “w” is the width       

     Table 5 highlights further variants of the elliptical flow channel concept. As Table 5 shows, the elliptical flow channel is designed to allow thinner channel wall thicknesses while maintaining equivalent product strength and improving OFA and GSA on a 900 channel/inch 2  substrate. 
     
       
         
           
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                 Flow Channel Shape 
                 Square 
                 Elliptical 
               
               
                   
               
             
            
               
                 Flow Channel Density - 
                   139.5 
                   139.5 
               
               
                 channels/cm 2   
                 (900) 
                 (900) 
               
               
                 (channels/in 2 ) 
               
               
                 Channel Wall Thickness 
                    68.58 
                    57.15 
               
               
                 Between Flow Channels - 
                     (0.0027) 
                      (0.00225) 
               
               
                 microns (inches) 
               
               
                 Equivalent Fillet Radius 
                   50.8 
                   124.46 
               
               
                 of Channel Walls - 
                     (0.002) 
                     (0.0049) 
               
               
                 microns (inches) 
               
               
                 OFA (coated) 
                   84.2 
                  86 
               
               
                 GSA (coated) 
                   43.3 
                   43.8 
               
               
                 MIF (uncoated) 
                    0.74 
                    0.72 
               
               
                 RTF (coated) 
                 867 
                 857 
               
               
                   
               
            
           
         
       
     
     Equation (I) may also be translated into x and y coordinates to yield Equation (VII) and Equation (VIII), respectively. 
     
       
         
           
             
               
                 
                   
                     
                       x 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ± 
                         a 
                       
                       · 
                       
                         
                            
                           
                             cos 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                         
                         
                           ( 
                           
                             2 
                             n 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       a 
                     
                     = 
                     
                       
                         a 
                          
                         
                             
                         
                          
                         for 
                          
                         
                             
                         
                          
                         0 
                       
                       ≤ 
                       t 
                       ≤ 
                       
                         
                           π 
                           2 
                         
                          
                         
                             
                         
                          
                         and 
                       
                     
                   
                    
                   
                      
                   
                    
                   
                     
                       
                         
                           3 
                            
                           π 
                         
                         2 
                       
                       ≤ 
                       t 
                       ≤ 
                       
                         2 
                          
                         π 
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     a 
                     = 
                     
                       
                         
                           - 
                           a 
                         
                          
                         
                             
                         
                          
                         for 
                          
                         
                             
                         
                          
                         
                           π 
                           2 
                         
                       
                       &lt; 
                       t 
                       &lt; 
                       
                         3 
                          
                         
                           π 
                           / 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   VII 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ± 
                         b 
                       
                       · 
                       
                         
                            
                           
                             sin 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                         
                         
                           ( 
                           
                             2 
                             m 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       b 
                     
                     = 
                     
                       
                         
                           b 
                            
                           
                               
                           
                            
                           for 
                            
                           
                               
                           
                            
                           0 
                         
                         ≤ 
                         t 
                         ≤ 
                         
                           π 
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           b 
                         
                       
                       = 
                       
                         
                           
                             - 
                             b 
                           
                            
                           
                               
                           
                            
                           for 
                            
                           
                               
                           
                            
                           π 
                         
                         &lt; 
                         t 
                         &lt; 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   VIII 
                   ) 
                 
               
             
           
         
       
     
       FIG. 18  is a flow diagram illustrating an example of a method  300  of making a honeycomb ceramic substrate in accordance with example aspects of the disclosure.  FIG. 19  is a schematic view illustrating an example of an extrusion apparatus  400  in accordance with example aspects of the disclosure.  FIG. 20  is an enlarged partial schematic sectional view illustrating an example of a honeycomb extrusion die  408  in accordance with example aspects of the disclosure taken at view  20  of  FIG. 19 . 
     Referring to  FIGS. 18-20 , the method  300  includes extruding  302  a ceramic or ceramic-forming batch material  402  through a honeycomb extrusion die  408  to form green honeycomb substrate of potentially unlimited length. The extruding  302  may be performed by introducing the ceramic or ceramic-forming batch material  402  into an input portion  404  of an extruding device  406 . Once the desired length is achieved, a cutter (not shown) can be used to sever the extruded ceramic or ceramic-forming substrate to provide the substrate with the desired length. 
     As shown, in one example, the extruding device  406  can include a twin-screw extruder including twin screws  410   a ,  410   b  configured to be rotated by respective motors  412   a ,  412   b  to mix and compress the batch  402  of ceramic or ceramic-forming batch material as it travels along a path  414  toward the honeycomb extrusion die  408 . The extruding device  406  includes an extrusion axis wherein the ceramic-forming substrate can be extruded from the honeycomb extrusion die  408  along an extrusion direction substantially parallel to the extrusion axis. 
     As shown in  FIG. 20 , the die member  408  includes feed holes  416  configured to feed batch material  402  in direction  418 , along the path  414 , toward a plurality of die pins  420 . The die pins  420  are arranged in a matrix and spaced apart from one another to define a lattice of intersecting slots  422  defined between the die pins  420  at an outer face of the die pins  420 . As shown in  FIGS. 21-22 , an outer periphery  501  at an outer face  503  of at least one of the die pins  420  includes a plurality of sides  505   a - d  joined by corresponding corner portions  507   a - d  with at least two convex sides facing away from a center  509  of the corresponding die pin  420  from central portions of the convex sides to the corresponding end portions of the convex sides. At least one corner portion  507   a - d  is convex facing away from the center  509  of the corresponding die pin  420 . The slots  422  are designed to form the channel walls  110  of the honeycomb ceramic substrate  102  as the ceramic-forming batch material  402  is drawn into the honeycomb ceramic substrate  102 . 
     At least one wall slot  422  may be defined between facing sides (e.g.,  505   a / 505   c ,  505   b / 505   d ,  505   c / 505   a ,  505   d / 505   b ) of two adjacent die pins of the plurality of die pins  420 . As shown, each of the facing sides may be convex facing each other from central portions thereof to corresponding end portions of the facing sides. Consequently, the wall slot  422  defined therebetween, may be concave toward central portions of the two adjacent die pins. 
     For example, with reference to the die pin shown in  FIG. 22 , a shape of the outer periphery  501  of the outer face  503  of one of the die pins  420  can be substantially defined by Equation (I). 
     
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           x 
                           a 
                         
                          
                       
                       n 
                     
                     + 
                     
                       
                          
                         
                           y 
                           b 
                         
                          
                       
                       m 
                     
                   
                   = 
                   1 
                 
               
               
                 
                   ( 
                   I 
                   ) 
                 
               
             
           
         
       
     
     As is somewhat inversely illustrated in  FIGS. 4 and 5 , a and b are rectangular-fitted half-lengths along an x direction and ay direction, respectively, of the sides  505   a - d  of the die pins  420  on either side of a y axis and an x axis, respectively. x and y represent coordinates (x, y) of the sides of the die pins  420  in the x direction and the y direction, respectively. Further, −a≦x≦a and −b≦y≦b. Moreover, n and m are exponents defining a degree of curvature of the sides of the die pins  420 . 
     In an example, at least one of n and m may be in a range of from about 2.5 to about 10. In another example, a and b may be independently in a range of from about 330 microns (0.013 inches) to about 1.829 mm (0.072 inches). In a further example, the die pins  420  may be arranged in the matrix and spaced from one another to have a die pin density in a range of from about 7.75 die pins/cm 2  (50 die pins/in 2 ) to about 232.5 die pins/cm 2  (1500 die pins/in 2 ). In an additional example, a thickness of the intersecting slots  422  between adjacent ones of the die pins  420  may be in a range of from about 25.4 microns (0.001 inches) to about 482.6 microns (0.019 inches). 
     In yet another example, the sides of each of the die pins  420  may have a length that is the same. Further, at least two of the sides of each of the die pins  420  may have a length that is the same. In still another example, the sides of each of the die pins  420  may be symmetric to each other. In addition, the sides and corner portions of each of the die pins  420  may be continuously curved around the center of the corresponding die pin  420 . 
     In addition, the shape of the die pin  420  either may be the same or varied along an entire length of the die pin  420 . For example, the shape of the die pin  420  on the outer face  503  may extend a depth of 127 microns (0.005 inches) from the outer face  503  along the length of the die pin  420 , with the remaining length of the die pin  420  being formed in a different shape. In another example, the shape of die pin  420  on the outer face  503  may extend a depth of 30% to 50% of the length of the die pin  420  from the outer face  503  along the length of the die pin  420  while the remaining length of the die pin  420  is formed in a different shape. For instance, in one example, the die pins may be formed by Electrical Discharge Machining (EDM) wire machining the entire length of the die pin. After forming the initial die pin shape, a subsequent machining step may be carried out by plunge EDM machining an electrode having the desired shape of the die pin at the outer face of the die pin. In such examples, the plunge EDM may extend a depth of 127 microns (0.005 inches) and/or to a depth of from about 30% to about 50% of the length of the die pin. As such, the shape of the die pin at the outer face may comprise the shape of the electrode while the remaining shape is defined by the initial wire EDM machining procedure. 
     Turning back to  FIG. 18 , the method  300  can further include the step of drying  304  the green honeycomb substrate. Additionally, the method  300  can include the step of firing  306  the green honeycomb substrate into the honeycomb ceramic substrate  102 . 
       FIGS. 23-25  are enlarged views illustrating examples of the honeycomb ceramic substrate  102  in accordance with example aspects of the disclosure. Referring to  FIGS. 23-25 , the honeycomb ceramic substrate  102  is illustrated having flow channels  108  and channel walls  110  of varied shape and size. For example, exponents m and n of Equation (I) can be varied across the die pins  420  of a honeycomb extrusion die  408  to create a honeycomb ceramic substrate  102  with varying thicknesses of the channel walls  110  and varying areas of the flow channels  108 . 
     Referring to  FIG. 23 , as an example, values of m and n may be decreased progressively from an internal portion of the honeycomb ceramic substrate  102  to a periphery of the honeycomb ceramic substrate  102 . Such a design may serve to strengthen a peripheral portion of the honeycomb ceramic substrate  102  while maintaining a thickness of the channel walls  110  across the honeycomb ceramic substrate  102 . The example illustrated in  FIG. 24  shows that the values of m and n may also be increased progressively from an internal portion of the honeycomb ceramic substrate  102  to a periphery of the honeycomb ceramic substrate  102 . This may provide a more uniform gas flow in a catalytic chamber by manipulating the OFA of the flow channels  108 . In addition, as is illustrated in  FIG. 25 , exponents m and n can be increased or decreased abruptly in a specific section of the honeycomb ceramic substrate  102 . This may serve to increase strength in the specific section of the honeycomb ceramic substrate  102  in which such an abrupt change is applied. 
     It will be apparent to those skilled in the art that various modifications and variations can be made to the present disclosure without departing from the spirit and scope of the disclosure. Thus, it is intended that the present disclosure cover the modifications and variations of this disclosure provided they come within the scope of the appended claims and their equivalents.