By performing a sensitivity analysis, landfill managers can assess financial impacts of the proposed base and pipe slopes.
Ali Khatami, Ph.D., P.E., SCS Engineers
Landfill design concepts have significantly matured over the past two decades to the point that optimization of the landfill base grades has become a required exercise during design of landfills. The optimization concept is tied to minimizing the construction cost and maximizing the airspace developed for the landfill. Two parameters that play very important roles in optimizing landfill designs are the leachate collection system pipe (LCS pipe) slope and base slope. The pattern selected for a disposal cell (including the LCS pipe and base slopes) may depend on geology, ground water elevations, geometry of the property and many more parameters that fall outside the scope of this article. However, the general pattern used by designers over the past four decades has been the herringbone pattern. The herringbone pattern may be fixed (fixed LCS pipe slope and fixed base slope) over the width and length of a panel, where a panel is defined as the area bounded by a berm and a LCS pipe, a ridge in the middle of the disposal cell and a LCS pipe, or any other arrangement that isolates an area from which leachate flows to a LCS pipe. This type of pattern is referred to as a single-segment pattern. Optimization of the LCS pipe and base slopes for a single-segment pattern was discussed in an article published in the August 2014 issue of 国产麻豆. Equations were developed that enable the engineer to change the pipe and/or base slope and calculate an approximate value for the volume between two different patterns (different LCS pipe and/or base slopes). This type of quick calculation provides the ability to perform sensitivity analysis for the LCS pipe and base slope before selecting the final values.
This article deals with a double-segment pattern within a single panel. The double-segment pattern consists of two different herringbone patterns, one following the other in sequence, as shown in Figure 1. The LCS pipe slope in the first segment may be different than the LCS pipe slope in the second segment, and similarly, the base slope in the first segment may be different than the base slope in the second segment. Similar to the model presented in the previous article, the model developed for this article is also a rectangular-shaped area. The formulation developed in this article also allows the base slopes in one panel to be different than the base slopes in another, as shown in Figure 2.
Formulation
The volume between an initial pattern (Pattern 1) and a trial pattern (Pattern 2) may be calculated using differential calculus. Elemental volumes presented in Figures 3a and 3b are used to define the mathematical relationship between the x, y, and z coordinates of the panel geometry within each segment. For a double-segmented pattern, each segment will be analyzed by its own elemental volume because the distance in the y direction and slopes (pipe and base) in one segment vary independent of the other segment.
The point of origin is assumed to be at the lowest point of the panel. The x axis extends along the width of the panel and the y axis extends along the length of the panel or the leachate collection pipe. Generally, the closest point of the elemental volume in each segment on Pattern 1 to the origin of coordinates is located at coordinates x, y and z, and the closest point of the elemental volume located in each segment on Pattern 2 to the origin of coordinates is located at coordinates x, y and z鈥�. The pipe slope and the base slope in Segment 1 of Pattern 1 are 伪 and 尾, respectively, and the pipe slope and the base slope in Segment 1 of Pattern 2 are 伪’ and 尾’, respectively. The pipe slope and base slope in Segment 2 of Pattern 1 are 未 and 纬, respectively, and the pipe slope and base slope in Segment 2 of Pattern 2 are 未鈥� and 纬鈥�, respectively.
Pattern 1 within Segment 1 may be mathematically defined as: