walls of the elements), a surface hierarchical equation like Eq. (2) is evaluated. Occurrence of more than two like-valued elements (serial numbers of elements) in any column in the mapped matrix indicates the presence of a vertical wall at that location in the axial direction. Presence of vertical walls at any axial location creates volume segments in the axial direction. Furthermore, starting and ending of any element’s open end also has to be tracked down in order to partition the axial volume in such discontinuities. Therefore, from existence of vertical walls, and starting or ending of the open end of elements, the volume of the muffler gets divided into several length segments, and with each such length segment, one or more area segments will be associated. For the muffler shown in Fig. 2, these segments have been illustrated in Fig. 5. As explained earlier, the cross-sections are assumed circular for all the elements involved in the construction of any muffler. Consequently, from the surface hierarchical equation, at any point along the length of the muffler, the areas enclosed between any pair of surfaces can be calculated; i.e., areas AS1, AS2 etc. in Fig. 5 can be calculated using the surface hierarchical equations along with the radial dimensions of the corresponding elements. With the known physical properties of the medium, known lengths of the volume segments and known areas of the volume segments, separate transfer matrix equations [5] can be written relating the upstream and downstream variables for each such volume segments. For the case in hand, there will be eleven such volume segments (see Fig. 5) and hence, as many transfer matrices. However, these separate equations have to be combined together to take into account the boundary conditions at every discontinuities. This has been achieved through a volume stitching process discussed in the next section. Besides, in order to be compatible with the present algorithm, a slight modification has been made into these transfer matrix equations as will be discussed in a subsequent section.
Fig. 5. Volume segmentation is done both length and area wise.
walls of the elements), a surface hierarchical equation like Eq. (2) is evaluated. Occurrence of more than two like-valued elements (serial numbers of elements) in any column in the mapped matrix indicates the presence of a vertical wall at that location in the axial direction. Presence of vertical walls at any axial location creates volume segments in the axial direction. Furthermore, starting and ending of any element’s open end also has to be tracked down in order to partition the axial volume in such discontinuities. Therefore, from existence of vertical walls, and starting or ending of the open end of elements, the volume of the muffler gets divided into several length segments, and with each such length segment, one or more area segments will be associated. For the muffler shown in Fig. 2, these segments have been illustrated in Fig. 5. As explained earlier, the cross-sections are assumed circular for all the elements involved in the construction of any muffler. Consequently, from the surface hierarchical equation, at any point along the length of the muffler, the areas enclosed between any pair of surfaces can be calculated; i.e., areas AS1, AS2 etc. in Fig. 5 can be calculated using the surface hierarchical equations along with the radial dimensions of the corresponding elements. With the known physical properties of the medium, known lengths of the volume segments and known areas of the volume segments, separate transfer matrix equations [5] can be written relating the upstream and downstream variables for each such volume segments. For the case in hand, there will be eleven such volume segments (see Fig. 5) and hence, as many transfer matrices. However, these separate equations have to be combined together to take into account the boundary conditions at every discontinuities. This has been achieved through a volume stitching process discussed in the next section. Besides, in order to be compatible with the present algorithm, a slight modification has been made into these transfer matrix equations as will be discussed in a subsequent section.Fig. 5. Volume segmentation is done both length and area wise.
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